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Solving optimization problems

1. dconstr2dp1 dconstr2dp2 errorcode 37 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats reqcalcobj reqcalcconstr reqcalcgradobj reqcalcgradconstr lt indl ind2 coefl coef2 defdata gt Meaning of symbols used above is as follows pl pl p3 optimization parameters at which analysis was performed These are input data for a direct analysis but it is requested that they are included in the output file in this way the optimization program can verify that the analysis results refer to the expected set of parameters or verify what errors were perpetrate by transfer via text files Flags that tell whether something has actually been calculated 0 yes 1 no e calcobj flag for the objective function e calcconstr flag for constraint functions e calcgradobj gradient of the objective function e calcgradconstr gradients of constraint functions obj value of the objective functions constr1 constr2 values of the constraint functions dobjdp dobjdp2 derivatives of the objective function with respect to individual parameters components of the objective function gradient dconstrldp1 dconstr2dp1 dconstr2dp2 derivatives of individual constraint functions with respect to individual optimization parameters components of gradients of the constraint functions e g dconstr2dp3 is the derivative
2. INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools e scaling2 Additional scaling factor by which intervals are multiplied in the second direction scalar value argument e printtab if non zero then data is also printed in table form counter value argument e printparam if non zero then a table of parameters in sampled points is printed together with the corresponding table indices and factors defining relative position with respect to point0 and point counter value argument e printlist if non zero then data is also printed in list form counter value argument 6 4 3 3 tabld kindspec point0 point numpt factor printparam printmeas j Obsolete Use taban1d instead Runs a set of direct analyses along a line in the parameter space and prints the requested results to the programme s standard output and output file Aindspec is a string that specifies what kind of table of direct analyses should be made and can be either noncent or cent noncent means that numpt direct analyses with sampling points on a line between point0 and point will be performed while cent means that sampling points will lie on the line whose centre is point0 and one of its two endpoints is point point0 is the base point and point the final point in the parameter space Both points must be specified as vectors of parameters numpt specifies the number of sampling points factor is the factor by which the distances betwee
3. inverse opt cm The program software home directory and some additional files can be downloaded from the download section of the Inverse home page INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Problem and its Solution 6 1 3 Definition of the Problem in the Command file The optimization problem and its solution procedure must be defined in the shell command file which is interpreted by the interpreter The command file typically consists of three parts the preparation part the analysis block and the final action part In the preparation part variables are typically allocated data initialized and functions defined for use at a later time The analysis block defines how direct analysis is performed This block is interpreted every time the direct analysis is performed either run from within some algorithm or as a consequence of user request In the action part the optimization algorithms that lead to problem solution are run Test analyses at different parameter sets or some other tests e g tabulating of the objective function can also be run in this part The preparation part and analysis block can usually be swapped Individual allocations and definitions can be performed right before they are used although the command file usually looks clearer if this is done in one place The user must be careful about putting definitions and allocations in the analysis block because this block is iteratively in
4. Approximation tools numit counter argument is the number of iterations for reducing effects of outliers currently this is not implemented Currently dealing with outliers is not yet implemented The smoothed approximation is stored in matrix element specified by element specification smoothspec 7 UNIFORM FILE INTERFACE BETWEEN OPTIMIZATION AND ANALYSIS PROGRAMS There are currently two distinct fie formats envisaged for use in the uniform file interface The native format is similar to the output format used in Mathematica where data can be combined in arbitrarily nested lists except for the representation of numbers which is the standard form used in programming languages e g 6 02e26 rather than the Mathematica form e g 6 02 10 26 The second format is XML document XML is a versatile format used for storing in text files any kind of data that can be represented by an arbitrary tree structure It is widely used especially for exchange of data over the internet its format is simple which facilitates implementation of parsers but in some cases also kind of verbose and less efficient for exchange of numerical data However due to a small extent of data that is typically exchanged via the uniform interface and due to relatively large computational times for the direct analysis using XML does not represent any narrow throat Possibility of using XML is offered simply because some numerical systems already utilize XML for d
5. either commercial or free software without requesting prior permission or paying royalty fees Warning Because numerical quantities are transferred through text files special care must be taken that all significant digits are written for each numerical value On the contrary accuracy is lost which may cause numerical problems or lead to inaccurate results Practically all programming languages enable writing floating point numbers to text files with arbitrary precision The number of digits must usually be explicitly stated otherwise a default number of digits is assumed which can lead to insufficient accuracy In C for example numbers are usually written to text files by using the printf function where format specification is given The following format specification can be used to preserve the floating point precision double x FILE fp fprintf p 301g x Text that begins with the sign represents the format specification 30 specifies that 30 significant digits should be output when printing a floating point number A large number of digits was specified in order to make sure that no precision is lost Part lg of the specification determines that a floating point is being printed in 36 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats compact format g and that a type long double is expected T In compact format eventual red
6. gt 70P The same as fileanalysis except that XML format is used for transferring data see Section 7 2 3 and description of fileanalysis 33 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions 7 1 1 9 filewriteaninput_xml filename lt cd gt 7 The same as filewriteaninput except that XML file format is used for transferring data see Section 7 2 3 and description of filewriteaninput 7 1 1 10filereadanres_xml filename 7 P The same as filereadanres except that XML file format is used for transferring data see Section 7 2 3 and description of filereadanres 7 1 1 11 filewriteanres_xml filename 7 The same as filewriteanres except that XML file format is used for transferring data see Section 7 2 3 and description of filewriteanres 7 1 1 12filereadaninput_xml filename The same as filereadaninput except that XML file format is used for transferring data see Section 7 2 3 and description of filereadaninput 7 1 1 13parsefilevar filename lt typel varspecl gt lt type2 varspec2 gt lt command gt lt command2 gt Parses the file named filename extracts different data from the file and stores the data into file interpreter variables if this is specified The first argument filename must be the name of the file to be parsed string argument Other arguments are arranged in an a
7. 0 library manual 2007 48
8. 2 parameters 2 conatraints all values and gradients calculated lt Analysis output file created by analysis wrapper gt lt data type analysispoint mode analysis_output ind 1 gt lt ret type counter gt 0 lt ret gt lt reqcalcobj type counter gt 1 lt reqcalcobj gt lt reqcalcconstr type counter gt 1 lt reqcalcconstr gt lt reqcalcgradobj type counter gt 1 lt reqcalcgradobj gt lt reqcalcgradconstr type counter gt 1 lt reqcalcgradconstr gt lt calcobj type counter gt 1 lt calcobj gt lt calcconstr type counter gt 1 lt calcconstr gt lt calcgradobj type counter gt 1 lt calcgradobj gt lt calcgradconstr type counter gt 1 lt calcgradconstr gt lt param type vector dim 2 gt lt vector_el type scalar ind 1 gt 1 6 lt vector_el gt lt vector_el type scalar ind 2 gt 1 lt vector_el gt lt param gt lt obj type scalar gt 0 20088905308774715 lt obj gt lt constr type table eltype scalar dim 2 gt lt table_el type scalar ind 1 gt 0 0 lt table_el gt lt table_el type scalar ind 2 gt 0 0 lt table_el gt 41 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats lt constr gt lt gradobj type vector dim 2 gt lt vector_el type scalar ind 1 gt 0 24138 lt vector_el gt lt vector_el type scalar ind 2 gt 0 0172418 lt vector_el gt lt gradobj gt lt gradconstr type table eltype vector dim 2
9. Igor Gresovnik which ranges analysis results with violated constraints by the sum of constraint residuals This is a non gradient algorithm suitable also for non differentiable and even non continuous functions that have a well defined constrained minimum The basic principle is similar to the Nelder Mead simplex algorithm Arguments are the same as for nlpsimp 6 2 5 nlpsimpbound0 numconstr tolx tolf tolconstr maxit printlevel initial step bignum lt lowbounds upbounds bignum lt kpen kconstr lt numviolations maxresid gt gt gt 70P 24 Performs the basic non linear constraint simplex optimization algorithm of Igor Gresovnik This is a non gradient algorithm suitable also for non differentiable and even non continuous functions that have a well defined constrained minimum The basic framework is similar to the Ne der Mead simplex algorithm e numconstr the number of constraints equality inequality counter argument 12 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms e tolx tolerance on optimal parameters approximate It is a vector value argument a tolerance is specified for each co ordinate If vector dimension is less than the problem dimension then missing components are replaced by the first component For components that are 0 no tolerance is imposed e tolf tolerance on on optimal value of the objective function scalar argument If it is 0 then this tolerance is not imposed e
10. Programs File Formats Examples 1 11 2 22 1 6 1605 1 0 165 2 44 1 2 22 4 44 1 1 5 0 0 2 0 1 1 1 1 3 1 11 2 22 1 6 1605 1 0 165 2 44 O 0 1 1 1 1 1 33 45 2 5 3 33 38 1 3 The examples represent analysis results at parameters 1 11 2 22 with value of the objective function 6 1605 values of constraint functions 0 165 and 2 44 gradients of the objective function 2 22 4 44 gradient of the first constraint function 1 5 0 and gradient of the second constraint function 0 2 In the second example gradients of the objective and constraint functions could not be calculated although they were requested all request calculation flags in curly brackets are 1 Therefore the error code is 1 rather than 0 7 2 3 XML formats 7 2 3 1 XML format for analysis results analysis output file Example 1 shows an example of the analysis output file in XML format The structure of the file can be easily established form this example and precise rules are stated below 7 2 3 1 1 Format rules general Comments are ignored when processing the analysis output file and all of them can be omitted Comments can only be put before XML elements they may not be put within opening or closing tags Name of the outer most element is arbitrary but data is recommended All other names must be precisely as in the exampl
11. be used for checking the course of execution an ct control file for analysis program Control files should be deleted from time to time because output is appended to existent contents so that output from several successive runs can be checked Sample analysis program analysis c a C language source for a sample analysis program A function for reading the analysis input data in standard format Section 7 2 1 and for writing analysis results in the standard format Section 7 2 2 The program uses only standard C libraries and ANSI C syntax therefore it can be easily compiled on any platform provided that a C 7 4 2 Running the example and using custom analysis program In order to run the example optimization program Inverse must be installed on your computer Let s assume that invan is the command for executing Jnverse Then the example is run by typing the following command in the operating system s command shell invan opt cm Main definitions that one may want to change in order to modify the example i e to modfy the optimization method used or the definition of the problem response functions are in the file def cm By setting e g the calculator variable unconstrained to 1 the unconstrained simplex method is used instead of the constraint method and penalty terms are added to the constraint functions in order to prevent constraint violations Definitions in the file are commented comments are inside such th
12. data Section 7 2 1 anout dat analysis output file written by analysis program and read by optimization program contains analysis results response functions such as objective and constraint functions and their gradients and calculation flags see Section 7 2 2 Auxiliary files def cm command file that contains some basic definitions and initialization steps Its interpreted both from an cm and opt cm since these definitions are used commonly by the optimization and analysis program defuser cm additional definitions This file is included in deficm via the interpret command Location of inclusion is such that important definitions from defc are overridden but dependent portion of the definitions such as allocation of the 45 INVERSE 3 18 7 4 Uniform File Interface Between Optimization and Analysis Programs Demonstrative example meaningful variables are performed after interpretation of this file such that correct effects are ensured This file is convenient for automatic generation by the environment that manipulates the analysis and optimization programs analysis cm additional command file for analysis program contains definitions of functions for calculating the response and definition of analysis block It is interpreted form an cm Control files opt ct control file for optimization program here the optimization program writes its results if this is ordered in its command file The file can
13. ind 1 gt 4 287974793 lt vector_el gt lt vector_el type scalar ind 2 gt 105 38479 lt vector_el gt lt vector_el type scalar ind 3 gt 2 4558e 4 lt vector_el gt lt param gt lt obj type scalar gt 72 424979429783 lt obj gt lt constr type table eltype scalar dim 2 gt lt table_el type scalar ind 1 gt 1 48479e 3 lt table_el gt lt table_el type scalar ind 2 gt 2 8793872 lt table _el gt lt constr gt lt data gt 42 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats 7 2 3 2 XML format for analysis input file Example 3 shows an example of the analysis input file in XML format The complete structure of the file can be easily established form this example The general rules are similar to the general rules for the analysis output file stated in Section 7 2 3 1 1 Specific rules are stated below 7 2 3 2 1 Analysis input specific rules The outer most XML element must be of type analysispoint and its attribute named mode must have value analysis_input Its name can be arbitrary but data is recommended Elements regcalcobj reqcalcconstr reqcalcgradobj and reqcalcgradconstr of type counter are obligatory Their values must be 1 if calculation of the corresponding portions of analysis results are requested actually any non zero value is allowed and 0 if not They correspond to the following portions of analysis results respecti
14. is specified then gradient of the approximation is also calculated and stored to gradspec hgrad step size for numerical differentiation if 0 or not specified then approximate analytical differentiation is performed Example Setmatrix samp 100 3 setvector point 3 10 1 3 55 setvector rweight 3 0 5 0 5 0 5 sampling functions setcounter which 1 setcounter val 0 smoothapproxsimpbas 4 samp rweight point which val 6 5 1 2 smoothapproxsimp samples rweight point which valspec lt gradspec gt Similar to smoothapproxsimpbas except that only the Gaussian type of the exponential function can be used 6 5 1 3 smoothmeas meas rweightrel resdiv numit smoothspec j Calculates a smooth approximation of a table of measurements in time or related to any other one dimensional parameter The measurements must be specified by matrix argument meas The matrix meas must have two columns and each row of the matrix represents contains a time measurement pair representing one measured sample rweightrel is a scalar argument that represents a relative size of effective radius of sample influence with respect to interval length of the independent variable resdiv counter argument specifies the number of interval divisions i e number of sampling points for resulting smoothed approximation 28 INVERSE 3 18 6 5 Uniform File Interface Between Optimization and Analysis Programs
15. not relevant For example if the gradient of the objective function has not been calculated then the element named gradobjective can be omitted The order in which sub elements are listed is not important 7 2 3 1 2 Analysis output specific rules The above stated rules are general while the following additional rules apply for the analysis output file The outer most XML element must be of type analysispoint and its attribute named mode must have value analysis_output Element ret of type counter must be defined It must have the integer value 0 if no errors were detected in the analysis or a negative integer value if errors occurred Elements regcalcobj reqcalcconstr reqcalcgradobj and reqcalcgradconstr of type counter may be omitted If included then their values must be 1 if calculation of the corresponding portions of analysis results were requested actually any non zero value is allowed and 0 if not They correspond to the following portions of analysis results 40 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats respectively objective function constraint functions gradient of the objective function and gradients of constraint functions Elements calcobj calcconstr calcgradobj and calcgradconstr of type counter are obligatory Their values must be 1 if calculation of the corresponding portions of analysis results were requested actually any non zero
16. replaced by 2 o data about arguments and optimization results are printed o 2 basic information about iterations and more detailed information about results are also printed o 3 simplex co ordinates of apices and values of the objective function is also printed during iterations and at the o 4 Complete results are printed included values of the constraint functions o 5 at the end all results of all analyses are also printed Sets of results in all simplices over al iterations are also printed in the list form readable by Mathematica initial initial guess vector value parameter 11 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms e step step sizes in different directions used to create the initial simplex vector value parameter Optimal parameters are written to paramopt and optimal value of the objective function to objectiveopt Storage of other functions or gradients is not guaranteed Note Inequality constraints are stated as c x lt 0 jel 6 2 where c are the constraint functions whose value is expected from the analysis function Remarks See introductory section for how the problem should be defined You can also take a look at inquick2 pdf which can be obtained at http www c3m si inverse doc other index html 6 2 4 NLPSimpS nlpsimps numconstr tolx tolf tolconstr maxit printlevel initial step A variant of the constraint nonlinear simplex method of
17. to objectiveopt Storage of other functions or gradients is not guaranteed Warning Inequality constraints are stated as c x lt 0 jel 6 7 where c are the constraint functions whose value is expected from the analysis function Remarks See introductory section for how the problem should be defined You can also take a look at inquick2 pdf which can be obtained at 16 INVERSE 3 18 6 3 Optimization And Inverse Analyses Older functions for optimization http www c3m si inverse doc other index html A detailed description of the SolvOpt algorithm can be found at http www kfunigraz ac at imawww kuntsevich solvopt 6 3 Older functions for optimization 6 3 1 inverse methodspec params This function performs different types of optimization algorithm methodspec determines which optimization algorithm is used It is followed by parameter specifications params which are dependent on the type of algorithm used methodspec begins either with string d or nd indicating whether we will solve one dimensional one parameter or multi dimensional problems respectively The second part of methodspec is a string that specifies the method more precisely Method and parameter specifications for different methods are described below 6 3 1 1 inverse Id parabolic x0 step0 tol maxitbrac maxit Performs minimization of the objective function of one parameter Successive three points quadratic approximations of the
18. tolconstr tolerance for constraint residuum scalar argument if it is 0 then none of the constraints may be violated in the solution e maxit maximal number of iterations counter argument e printlevel the level of output produced counter argument 0 or less is replaced by 2 o data about arguments and optimization results are printed o 2 basic information about iterations and more detailed information about results are also printed o 3 simplex co ordinates of apices and values of the objective function is also printed during iterations and at the o 4 Complete results are printed included values of the constraint functions o 5 at the end all results of all analyses are also printed Sets of results in all simplices over al iterations are also printed in the list form readable by Mathematica e initial initial guess vector value parameter e step step sizes in different directions used to create the initial simplex vector value parameter e lowbounds vector of lower bounds on optimization parameters see explanation below vector value parameter e upbounds vector of upper bounds on optimization parameters see explanation below vector value parameter e bignum large positive value which is used for deciding whether components of lower and upper bound vectors actually define bound constraints see explanation below vector value parameter e kpen factor for penalty generating func
19. 4 INVERSE 3 18 7 4 Uniform File Interface Between Optimization and Analysis Programs Demonstrative example data flow and dotted arrows denote calling directions Horizontal arrows denote calling directions that are commonly present Other calls are performed when the optimization and simulation modules are integrated in a broader framework 7 4 Demonstrative example An example has been set up in order to demonstrate use of the uniform file interface The example files are contained in the directory named filean in IGHOME inverse ex An example contains a synthetic case for which analytical response functions are defined There are two optimization parameters and two constraints Both optimization and direct analysis are performed by Inverse with appropriate command files defined separately for optimization and analysis Usually this will not be the case and there will be a separate program for the direct analysis such as in Figure 1 where a special analysis program runs a finite element simulation and calculates the response function 7 4 1 List of files The main files are the following opt cm command file for optimization performed by Inverse an cm command file for direct analysis also performed by nverse anin dat analysis input file written by optimization program and read by analysis program contains current values optimization parameters calculation request flags and possibly an additional analysis definition
20. Solving Optimization Problems By the Optimization Program INVERSE FOR VERSION 3 18 Igor Gresovnik Ljubljana 13 March 2008 INVERSE 3 18 6 1 Optimization And Inverse Analyses Table of contents Contents 6 Optimization And Inverse Analyses sscccsssscssssscssssscssnnscssnscccsscccesssccessscesssscssencees 3 6 1 Definition of Optimization Problem and its Solution ccscccssccssccsecesssseesseeees 3 Gola l SBasice aT EEEE cares sak a Ae at ta des E E as hes ahaa tela rack ah sean EEE 3 6 1 2 Installing and running the optimization program Inverse sccsceseeseeeceseereetecneeetenceeeeeaeeatenes 4 6 1 3 Definition of the Problem in the Command file 0 cece cecesecseeeecneeeeceeeeeesecaeeeesaecaeeseenereres 5 6 1 4 Defining the Direct Analysis 20 0 0 ccceccssecssecsseessesseeeseeeceseeseceecsseeesecesecaecaecaecaaeceeeaeesneenss 5 6 1 5 Implicit Gradient Calculation sneeeseneesseeeesseseesesetsesseeressesessteresseseeseseestsseeresseeeesseseseseenes 6 6 2 Optimization al orithMS s sssecssssssssscssssessssssgssosseasosssossrissv costo ss osasi sss stoso soosse i sesoses ee 9 6 2 1 optfsqp numob numnonineq numlinineq numnoneq numlineq eps epseqn maxit grad initial lt lowbound upbowunid gt P aeons ne ts es EE RN E EE A E ne E cnehoe 9 6 2 2 minsimp tolx tolf maxit printlevel initial step cecccccccccesscessceeseeseceseceseesseeeeeseesseeseeeeseneeees 10 6 2 3 nlpsimp numconstr to
21. Such examples can be found in the directory of training examples subdirectory opt 6 1 5 Implicit Gradient Calculation Some optimization algorithms need gradients of the objective and constraint functions beside their values Most commonly these should be calculated in the analysis block and stored in the appropriate pre defined variables e g gradobjectivemom or gradconstraintmom see the manual on variables chapter on pre defined variables This essentially means that the algorithm for calculation of the objective and constraint functions must be differentiated with respect to the design parameters This is sometimes difficult to achieve especially when some numerical simulation is used as a black box and the user does not have access to its source code INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Problem and its Solution The derivatives can always be obtained numerically e g by sequentially perturbing values of individual parameters calculating the functions at perturbed parameters and dividing the difference with respect to the response at original parameters by the perturbation i e difference in parameter value or step size This can be eventually programmed within the analysis block of the interpreter Doing so however can significantly reduce the clearness and readability of the analysis block The tools have been providing for automatic implicit numerical calculation of th
22. Writes direct analysis input data to the file named filename string argument The data are obtained from the pre defined interpreter variables optimization parameters from vector variable parammom and request calculation flags from counter variables calcobj calcconstr calcgradobj and calcgradconstr Optional string argument cd can specify additional definition data that is passed to the direct analysis This function is usually used within the analysis block The following interpreter code illustrates the typical use that replaces the fileanalysis function analysis filewriteaninput anin dat 0 system analyse anin dat anout dat filereadanres anout dat In this example it is assumed that the direct analysis program is called analyse and it takes the input and output file as command line arguments Analysis input is written to the file in the format described in Section 7 2 1 7 1 1 4 filewriteaninput_oneline filename lt cd gt 7 P The same as filewriteaninput except that the analysis input data is printed in a single line which is usually less readable 32 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions 7 1 1 5 filereadanres filename 7 Reads analysis results from the analysis output file in the standard format named filename string argument The data that are defined are stored to the corres
23. ables This function is usually called form the analysis block of the Inverse command file If the analysis program performs complete calculation of the response functions and no additional processing is requred then this function can be all that is called in the analysis block Arguments e ancommand command passed to the system that executes the analysis program string argument e aninfile name of the input file for the analysis program This file is generated by the function prior to passing ancommand to the system for execution The format is described in Section 7 2 1 and it is assumed that the analysis command will read data from this file and will correctly interpret its data string argument e anoutfile name of the output file of the analysis program It is assumed that the analysis program will write the analysis results to this file after calculation of the response following the format described in 7 2 2 otherwise the function can not correctly interpret the resuts string argument e cd Optional argument that may be used to choose between several different kinds of analyses which the analysis program is able to perform i e analysis definition data If it is not specified then 0 is assumed This data can be used for passing any kind of additional instructions to the analysis program if it is designed in such a way that it can interpret the definition data string argument The format of the analysis i
24. absolute value of some component of either lower or upper bound is greater than bignum then it is also assumed that the corresponding bound is not defined which allows to define for a given component of the parameter vector only lower or only upper bound If there are components of the parameter vector for which only lower or only upper bound is defined then the large positive number bignum must be specified such that components of lower or upper bound vectors whose absolute vlue id larger than bignum are not taken into account bignum can be set to 0 In this case the default value is taken but this value can not fit the actual problem that is solved If Jowbounds and upbounds are not specified then the normal nonlinear constraint simplex algorithm is performed Optimal parameters are written to paramopt and optimal value of the objective function to objectiveopt Storage of other functions or gradients is not guaranteed Notes Inequality constraints are stated as c x lt 0 jel 6 3 where c are the constraint functions whose value is expected from the analysis function Bound constraints specify that Loe ST 4 where l is a vector of lower bounds argument owbounds ant r is a vector of upper bounds argument upbounds Each bound lower or upper therefore defines an additional constraint Corresponding to lower and upper bounds constraint functions can be assigned e g in the following way 14 INVERSE 3 18 6 2 Optimiz
25. alue argument scaling Additional scaling factor by which intervals are multiplied The factor can be used e g if we want the table extend a bit over some special point of interest which we set as endpoint Regardless of its size the table remains to be centered if centered is non zero or starting in point0 scalar value argument printtab if non zero then data is also printed in table form counter value argument 21 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools printparam if non zero then a table of parameters in sampled points is printed together with the corresponding table indices and factors defining relative position with respect to point0 and point1 counter value argument printlist if non zero then data is also printed in list form counter value argument 6 4 3 2 taban2d point0 point point2 numptl centered factor scaling numpt2 centered2 factor2 scaling2 lt printparam printlist printobj printconstr printgradobj printgradconstr gt Performs a two dimensional table of analyses with endpoints point0 and point and prints the results according to specifications Arguments point0 starting point of the table in the parameter space vector value argument point The first end point of the table defines the first table direction together with point0 vector value argument point2 The second end point of the table defines the second table direction to
26. at their meaning is obvious to the user The file defuser cm is included in def cm Definitions that should override the definitions in 46 INVERSE 3 18 7 4 Uniform File Interface Between Optimization and Analysis Programs Demonstrative example 7 4 3 Using a different analysis program Example can be easily re arranged in order to use a different analysis program In the file defcm the string variable ancommand must be modified such that it contains the command that runs that analysis program e g setstring ancommand my analysis inputfile outputfie if the analysis program is run by command my_analysis with arguments inputfile and outputfile that define its input and output file In this case inputfile and outputfile must be names of the files that are specified in def cm as analysis input and analysis output file and these are contained in the string variables aninfile and anoutfile defined in def cm Another possibility is that the analysis program uses input and output files with pre defined names that are always the same or that names of these files are defined in some kind of resources file It is only important that the files are synchronized between the optimization and the analysis program therefore names of the analysis input and output files must be updated accordingly for the optimization program which is done simply by appropriately setting the string variables aninfile and anoutfile in the definition file de
27. ata storage and exchange This section specifies the uniform file interface between Inverse and external analysis program This consists of file formats and procedures for exchange of data and analysis run The uniform file interface is designed to minimize and standardize the data exchanged between optimization and analysis program Its purpose is also being platform independent Cost for this is that analysis must be packed in a program that performs all extraction of the relevant results from simulation if simulation is involved and combination of these results to calculate the final values of response functions gt Mathematica the symbolic algebra system 29 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions 7 1 Interpreter functions 7 1 1 1 fileanalysis ancommand aninfile anoutfile lt cd gt 70P This file interpreter function is called in the analysis block in order to run the direct analysis implemented as stand alone program The command writes optimization parameters and request flags that define which response functions must be evaluated in the analysis input file argument aninfile runs the analysis program by passing to the operating system the command for running the program argument anommand and after the program exits it reads the results from the analysis output file anoutfile and writes them to the appropriate pre defined interpreter vari
28. ation And Inverse Analyses Optimization algorithms li 1 c x x 6 c x 7 In standard form for definition of bound constraints in Inverse neither lower nor upper bound on a given component of the parameter vector is considered defined if the corresponding component of the lower bound vector is larger than the corresponding component of the upper bound vector Le if 1 gt r In addition a lower or uper bound is considered unspecified if absolute value of the corresponding component of the lower or upper bound vector is larger than some specified large positive number argument bignum denoted by B If the corresponding components of the lower and upper bound vectors are the same then this defines an equality constraint To summarize lower and upper bounds on optimization variables parameters are defined conditially in the following way l ABR KES eT 6 lt Bal lt r gt l lt x r lt Bal r gt x l Handling of bound constraints Two ways of handling bound constraints can be combined and are governed by arguments kpen and kconstr if not specified the default values taken are 1 and 0 respectively These coefficients must be non negative A zero coefficient means that the corresponding method of handling bound constraints is not imposed Coefficient kpen corresponds to transformation of parameters with addition of penaty terms for violated bound constraints The original analysis is a
29. cate that the appropriate quantities should be printed In any case before the print out of the table values parameters are printed that correspond to the sampling points along the line Points are indexed by proportional factors from 0 for point to 1 for point2 6 4 3 5 2 linetab exp numpt pppar pmeas factor point point2 Runs numpt direct analyses with sampling points non equidistantly distributed along the straight line between point and point2 factor is the factor for which the distance between the length of the successive sampling interval is extended ppar and pmeas specify whether the parameters and measurements should be printed in table lines respectively values different than zero indicate that the appropriate quantities should be printed In any case before the print out of the table values 24 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools parameters are printed that correspond to the sampling points along the line Points are indexed by proportional factors from 0 for point to 1 for point2 Older tabulating functions 6 4 3 6 tab1d0 which vall val2 numpt pobjective pmeas Obsolete Use taban1d instead Runs a set of numpt direct analyses so that parameter which is varied between val and val2 with a constant step pobjective and pmeas specify if the objective function and measurements should be printed respectively values different than zero indicate that the appropriat
30. don t intend to store a given object into a variable then the variable specification should be NULLI Available commands are the following e stop instructs to stop parsing the file e in current position in the parsed file is moved inside the first curly brackets e out current position in the parsed file is moved out of the current curly brackets Example Allocation of variables for storage of the data read form the file newvector vecvar 5 newmatrix mat parsefilevarprint parsed dat count NULL scal NULL vec vecvar 1l mat mat str NULL str NULL in scal NULL vec vecvar 2 out scal NULL scal NULL anpt NULL scal NULL W stop W 35 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats printvector vecvar 1 printvector vecvar 2 printmatrix mat exit 7 2 File Formats In order to use the uniform file interface the analysis program must be able to read analysis input files that provide calculation flags and parameter values and must be able to generate the output file in a proper way such that analysis results can be correctly read by the optimization program nverse The file formats used in the uniform file interface are provided by the free optimization library IOptLib and are therefore free formats They can be freely used by any other software
31. dratic programming optimization algorithm of Craig Lawrence Jian L Zhou and Andre Tits which is the basic and most powerful nonlinear programming algorithm built in nverse Arguments e numob number of objective functions should normally be 1 counter value argument e numnonineg number of non linear inequality constraints counter value argument e numlinineg number of linear inequality constraints counter value argument e numnoneg number of non linear equality constraints counter value argument e numlineg number of linear equality constraints counter value argument e eps final norm requirement for the Newton direction scalar value argument e epseqn maximum violation of nonlinear equality constraints at an optimal point Both criteria must be satisfied to stop the algorithm the second one is in effect only if there are equality constraints scalar value argument e maxit maximum number of iterations counter value argument e grad specifies if gradients are provided by the direct analyses 1 or should be calculated numerically 0 counter value argument e initial initial guess vector value argument e lowbound lower bounds on parameters vector value argument e upbound upper bounds on parameters vector value argument If vector value arguments owbound and upbound are not specified then parameters are not bounded below or above If they are specified the
32. e Names of attributes of individual XML elements with specific meaning must exactly match names from the example and XML elements with specific meaning must have all the attributes defined as in the example Order of attributes is not important but attribute names must be unique for a given element Values of attributes must also match exactly except for those attributes that specify dimensions or indices Such attributes are only dim which specifies the dimension or number of elements of given data and ind which specify the index of component sub element of given data Values of such attributes must be strings that represent integer numbers in base 10 notation 39 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis Programs File Formats Each data element has the attribute type defined which specifies the type of the data represented by the XML element Types used in the analysis output file are counter representing an integer number scalar representing a real number vector representing a real vector string representing a string of characters table representing a table of elements of any kind all elements of the same kind and analysispoint representing a structure that carries analysis input and or output data Elements of type counter i e those whose attribute type has value counter must have contents that can be interpreted as an integer e g 1 425 10 e
33. e derivatives When implicit derivative calculation is switched on on any request for performing the analysis at given parameter values the non derivative analysis is actually performed with the original and perturbed parameter values Numerical approximation of gradients of the objective and constraint functions is calculated on the basis of the results and stored to the appropriate pre defined variables most commonly gradobjectivemom and gradconstraintmom together with function values at the original parameters of the request objectivemom and constraintmom are commonly used to store these The interpreter functions for providing implicit numerical gradient calculation are described below 6 1 5 1 analysisnumgradfdvec stepvec Installs the implicit numerical calculation of gradients of the objective and constraint functions if defined with respect to optimization parameters by the forward difference scheme This applies to the functions that are calculated by the direct analysis direct analysis which includes interpretation of the analysis block stepvec must be a vector value argument that specifies the step size for each parameter Its dimension must therefore be the same as the number of parameters i e the dimension of the pre defined vector parammom If stepvec is not specified then the default step size 10 is taken for derivation with respect to all parameters It is usually a very bad idea not to specify the step sizes beca
34. e quantities should be printed At the sampling points parameters other than which are taken from the pre defined vector parammom 6 4 3 7 tab2d0 whichx xl x2 numptx whichy yl y2 numpty pobjective pmeas Obsolete Use taban2d instead Runs a set of numptx numpty direct analyses organised in a two dimensional table so that parameters whichx and whichy are changed Parameter whichx is varied between xl and x2 with numptx equidistant sampling values and parameter whichy is varied between y and y2 with numpty equidistant sampling values pobjective and pmeas specify if the objective function and measurements should be printed respectively values different than zero indicate that the appropriate quantities should be printed At the sampling points parameters other than which are taken from the pre defined vector parammom 6 4 3 8 tablineO kindspec args Obsolete Use taban1d instead Runs a set of direct analyses along a line in the parameter space and prints the requested results to the programme s standard output and output file Aindspec is a string that specifies what kind of table of direct analyses should be made The remaining arguments args specify the line along which the table is made number of points etc 25 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools kindspec can be either lin or exp The meaning of the remaining arguments for both possibilities is explained below 6 4 3 8 1 tablin
35. e response according to a chosen internal test problem definition After calculation it stores the results to the appropriate pre defined interpreter variables If a string argument festname is specified then a particular test is performed Otherwise the default test with 2 parameters and 2 constraint functions is solved 6 4 2 Testing the Direct Analysis 6 4 2 1 analyse lt param calcobj calcconstr calcgradobj calcgradconstr gt Performs the direct analysis at the specified parameter param If param is not specified then the direct analysis is performed at parameters stored in the pre defined variable parammom The pre defined vector parammom must therefore be set in this case before the function is called The values of the pre defined global variables that hold analysis results are printed to the programme s standard output and output file If the vector value argument param is specified then the analysis is performed at the specified parameters In this case the scalar value arguments calcobj calcconstr calcgradobj and calcgradconstr are the evaluation flags that define which response functions should be evaluated they refer to the objective function constraint function s gradient of the objective function and gradient s of the constraint function s respectively 20 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools 6 4 2 2 analysenoprint lt param calcobj calcconstr calcgradobj ca
36. eO lin numpt pobjective pmeas point point21 Runs numpt direct analyses with sampling points equidistantly distributed along the straight line between point and point2 pobjective and pmeas specify if the objective function and measurements should be printed respectively values different than zero indicate that the appropriate quantities should be printed 6 4 3 8 2 tablineO exp numpt factor pobjective pmeas point point2 Runs numpt direct analyses with sampling points non equidistantly distributed along the straight line between point and point2 pobjective and pmeas specify if the objective function and measurements should be printed respectively values different than zero indicate that the appropriate quantities should be printed factor is the factor for which the distance between the length of the following sampling interval is extended 6 4 4 Test optimization problems 6 4 4 1 installtestanalysis insttestan idspec testname P Installs a test optimization problem from OptLib testname is the name that identifies the test problem to be installed and idspec is a scalar variable element specification that specifies the address where problem ID is stored The test problem that is installed by this function can be run by the testanalysis function e idspec specification of a scalar element in which problem ID is stored scalar variable element specification e testname name of the test problem string va
37. esults analysis point 7 3 Solution Scheme Integrated environment Prepare FEM analysis Run analysis Define optimization parameters Run optimization Inverse input KA fiele s i H a i t Additional data file s R opt cm i These data define precisely p te how to perform analysis te E to Inverse optimization program Ko anin dat e invanan cm Run optimization K Analysis input er file Inside optimization loop iteratively f l ee ip e write a d toa file e run Analysis program SPER e read f a d from a file Analysis program read a d from a file prepare FEM input FEM input file FEM output file perform analysis output results Output results write f c to a file Analysis output file i Cis o anout dat Figure 1 Typical software organization when using uniform file interface between optimization and analysis module which are implemented as stand alone programs Analysis module is a program that reads parameters and calculation flags from analysis input file performs direct analysis i e calculates the objective and constraint functions and eventually their gradients and outputs these results to the analysis output file response file In the scheme a d denote optimization parameters fis objective function and c constraint functions Solid arrows denote 4
38. f cm Beside re defining the command for running the analysis program and taking care that the optimization program and analysis program use the same analysis input and analysis output file some variables related to the optimization problem definition must be updated accordingly In def cm calculator variable numparam must be set to actual number of optimization parameters in the problem that is actually defined by the analysis program and numconstr must be set to the actual number of constraints for the new problem Dimensions of some other relevant variables that depend on these numbers are set automatically in def cm Then commands in opt cm that perform the optimization must be modified such that their arguments are consistent with the specific optimization problem that is solved and defined by the external analysis program Such commands are inverse and optfsqp In particular the dimensions of the starting guess must be corrected and arguments that are dependent or define the number of optimization parameters or the number of constraints Legend TOptLib 47 INVERSE 3 18 7 4 Uniform File Interface Between Optimization and Analysis Programs Demonstrative example This functionality is from TOptLib Investigative Optimization Library References 1 I GreSovnik Simplex Algorithms for Nonlinear Constraint Optimization Problems revision 0 technical report 2007 2 IL GreSovnik JoptLib User s Manual revision
39. f several programs by passing a single command e g by using a semicolon or a newline for separation of commands then the command can be composed in the appropriate way convertaninfile in dat n myanalysis in dat out dat n convertanoutfile out dat where convertaninfile is the name of the program that reads the analysis input file in the format described in Section 7 2 1 and writes it back in the format readable by the myanalysis program and convertanoutfile is the name of the program that reads the analysis output file in the format used by the myanalysis program and writes the data back to the file in the format specified in Section 7 2 2 Below there is an example of running an external analysis program by using converters setstring ancom convertaninfile in dat n myanalysis in dat out dat n convertanoutfile out dat setstring aninfile in dat setstring anoutfile out dat 31 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions analysis fileanalysis ancom aninfile anoutfile 7 1 1 2 fileanalysis_oneline ancommand aninfile anoutfile lt cd gt pes The same as fileanalysis except that input data for direct analysis is written in a single line in the analysis input file By specification this should not matter for parsers of the analysis input file 7 1 1 3 filewriteaninput filename lt cd gt 70P
40. formed its results will appear in the appropriate global shell variables User must take care of that in the analysis block by storing results in these variables For example value of the objective function must appear in scalar variable objectivemom values of constraint functions must appear in scalar variable constraintmom objective function gradient in vector variable gradobjectivemom INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Problem and its Solution gradients of constraint functions in vector variable gradconstraintmom simulated measurements in the case of inverse analyses in vector variable measmom etc These variables with a pre defined meaning are treated just like other user defined variables and the same functions can be used for their manipulation There are however some particularities in behaviour of variable manipulation functions in the case of variables with a pre defined meaning Rules are more or less the same there is only some additional intelligence incorporated which enables user not to specify dimensions that are already known to the shell For details see the Shell Variables with a Pre defined Meaning chapter of the User Defined Variables in the Optimization Shell Inverse manual Within the analysis block the user is expected to run a numerical simulation with parameters found in vector parammom combine its results to evaluate the requested function values ob
41. gether with point0 vector value argument numptl Number of analysis points divisions in the first direction counter value argument centered Flag for a centered table in the first direction If non zero then the table is centered around the starting point point0 If table is centered with geometrically growing intervals then the interval lengths first fall from point reflected over point0 until point0 and then grow from point0 to pointl counter value argument factor Factor of interval length growth in the first direction If 0 or 1 then intervals between table points are uniform If it is greater than 1 then intervals grow in such a way that each successive interval length is the previous length multiplied by factor If it is smaller than 1 then factors fall in the same way scalar value argument scaling Additional scaling factor by which intervals are multiplied in the first direction The factor can be used e g if we want the table extend a bit over some special point of interest which we set as endpoint Regardless of its size the table remains to be centered if centered is non zero or starting in point0 scalar value argument numpt2 Number of analysis points in the second direction counter value argument centered2 Flag for a centered table in the second direction counter value argument factor2 Factor of interval length growth in the second direction scalar value argument 22
42. gorithm e tolx tolerance on optimal parameters approximate It is a vector value argument a tolerance is specified for each co ordinate If vector dimension is less than the problem dimension then missing components are replaced by the first component For components that are 0 no tolerance is imposed e tolf tolerance on on optimal value of the objective function scalar argument If it is 0 then this tolerance is not imposed e maxit maximal number of iterations counter argument e printlevel the level of output produced counter argument 0 or less is replaced by 2 o data about arguments and optimization results are printed o 2 basic information about iterations and more detailed information about results are also printed o 3 simplex co ordinates of apices and values of the objective function is also printed during iterations and at the o 4 Complete results are printed included values of the constraint functions o 5 at the end all results of all analyses are also printed Sets of results in all simplices over al iterations are also printed in the list form readable by Mathematica e initial initial guess vector value parameter 10 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms step step sizes in different directions used to create the initial simplex vector value parameter Optimal parameters are written to paramopt and optimal value of the object
43. gt lt table_el type vector dim 2 ind 1 gt lt vector_el type scalar ind 1 gt 1 1 lt vector_el gt lt vector_el type scalar ind 2 gt 2 1 lt vector_el gt lt table_el gt lt table_el type vector dim 2 ind 2 gt lt vector_el type scalar ind 1 gt 0 lt vector_el gt lt vector_el type scalar ind 2 gt 1 lt vector_el gt lt table_el gt lt gradconstr gt lt Optional definition data gt lt cd type string gt Definition data lt cd gt lt data gt Another example below shows another possible analysis output file in which the values of calculation requests flags are skipped gradients are not calculated obviously they were also not requested since the return value is 0 indicating no errors and no additional data is passed between the analysis and optimization module Example 2 Another analysis output file with only partial output provided 3 parameters 2 constraint no gradients requested or calculated request flags not specified lt Analysis output file created by analysis wrapper gt lt data type analysispoint mode analysis_output ind 1 gt lt ret type counter gt 0 lt ret gt lt calcobj type counter gt 1 lt calcobj gt lt calcconstr type counter gt 1 lt calcconstr gt lt calcgradobj type counter gt 0 lt calcegradobj gt lt calcgradconstr type counter gt 0 lt calcgradconstr gt lt param type vector dim 3 gt lt vector_el type scalar
44. he initial step of the bracketing stage The second and the third point of the initial bracketing triple are obtained by adding step0 to x0 once and twice respectively tol is the tolerance for function minimum The algorithm terminates when the difference between the highest and the lowest value of the objective function in the current three bracketing points is below tol maxitbrac is the maximal allowed number of iterations at searching for bracketing triple If the algorithm fails to find the three points satisfying the bracketing condition in maxitbrac iterations it terminates and reports an error maxit is the maximal allowed number of iterations in the second stage 6 3 1 2 inverse nd simplex tol maxit startguess Obsolete Use other functions instead Performs minimization of the objective function by simplex method Apices of a simplex is successively moved in such a way that the simplex moves and shrinks toward function minimum Simplex is a geometrical body in an n dimensional space that has n 1 dimensions tol is tolerance for function minimum The algorithm terminates when the difference between the greatest and the least value of the objective function in simplex apices becomes less than tol maxit is the maximal allowed number of iterations If the minimum is no reached in maxit iterations the algorithm terminates and reports an error startguess is the starting guess containing the initial simplex This must be a matrix of dime
45. he optimum and the tolerance should be set rather conservatively in order to avoid failure of algorithms due to excessive noise 6 1 5 2 analysisnumgradfd stepsize Does the same as analysisnumgradfdvec except that the step size for all parameters are set equal to stepsize which is a scalar value argument If stepsize is not specified then a default step size 10 is taken However it is usually a very bad idea not to specify the step size because the accuracy of the derivatives depend essentially on it and the optimal step size may vary drastically from case to case since it depends on scaling of the design parameters and on the level of noise of the differentiated functions 6 1 5 3 analysisplain Cancels the implicit numerical differentiation of the objective function and constraint functions if defined and places instead the original analysis function which performs the direct analysis including interpretation of the analysis block at only one set of design parameters 6 1 5 4 analysisnumgradprn doprn If the counter value argument doprn is different than 0 then reporting on gradient calculation is switched on which can be used for control INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms 6 2 Optimization algorithms 6 2 1 optfsqp numob numnonineg numlinineq numnoneg numlineq eps epseqn maxit grad initial lt lowbound upbound gt Performs the fsqp feasible sequential qua
46. introductory section for how the problem should be defined You can also take a look at inquick2 pdf which can be obtained at http www c3m si inverse doc other index html 6 2 6 solvopt numconstr numconstreg tolx tolf tolconstr maxit lowgradstep initial Performs the SolvOpt optimization algorithm of Alexei Kuntsevich amp Franz Kappel This algorithm is particularly suited for non smooth differentiable functions e numconstr the number of constraints equality inequality counter argument e numconstregq the number of equality constraints If there are equality constraints these must be returned at the ens after inequality constraints counter argument e tolx relative tolerance on optimal parameters infinity norm scalar argument e tolf relative tolerance on on optimal objective function scalar argument e tolconstr tolerance for constraint residuum maximal violation of any constraint absolute value for equality constraints scalar argument e maxit maximal number of iterations counter argument e lowgradstep the smallest step size for numerical calculation of gradients If 0 then gradients provided by the analysis function are used otherwise the algorithm will perform numerical differentiation of the constraint functions scalar argument e initial initial guess vector value parameter vector argument Optimal parameters are written to paramopt and optimal value of the objective function
47. ion Problem and its Solution 6 1 1 Basic Terms We state the optimization problem quite generally as minimise subject to and where Function f is called the objective function c and c are called constraint functions and and ux are called upper and lower bounds The second and third line of the equation are referred to as inequality and equality constraints respectively with J and E being the corresponding inequality and equality index sets We will collectively refer to f c ie 1 and c icE as constraint functions Sometimes the algorithm can in addition take the advantage of explicitly stated eventual linear constraint functions such as in the case of fsqp The set of points in which all constraints are satisfied is called feasible region Gs t f x x R x lt 0 iel 6 1 c x 0 jek l SX Su a lel age Solution of the problem is contained in the feasibleregion INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Problem and its Solution 6 1 2 Installing and running the optimization program Inverse In order to run nverse you need an executable for your platform and a I G s software home directory referred to as ighome which is default name for this directory The executable is usually put to ighome Installation procedure is simple 1 Copy the I G s software home directory ighome somewhere on your hard disk e g in c on windows in this ca
48. ive function to objectiveopt Storage of other functions or gradients is not guaranteed Remarks See introductory section for how the problem should be defined You can also take a look at inquick2 pdf which can be obtained at http www c3m si inverse doc other index html 6 2 3 nlpsimp numconstr tolx tolf tolconstr maxit printlevel initial step IOptLib Performs the basic non linear constraint simplex optimization algorithm of Igor Gresovnik This is a non gradient algorithm suitable also for non differentiable and even non continuous functions that have a well defined constrained minimum The basic framework is similar to the Ne der Mead simplex algorithm numconstr the number of constraints equality inequality counter argument tolx tolerance on optimal parameters approximate It is a vector value argument a tolerance is specified for each co ordinate If vector dimension is less than the problem dimension then missing components are replaced by the first component For components that are 0 no tolerance is imposed tolf tolerance on on optimal value of the objective function scalar argument If it is 0 then this tolerance is not imposed tolconstr tolerance for constraint residuum scalar argument if it is 0 then none of the constraints may be violated in the solution maxit maximal number of iterations counter argument printlevel the level of output produced counter argument 0 or less is
49. jective and constraint functions and their derivatives and store these results in the appropriate variables with a pre defined meaning This can include a number of sub tasks for example parameter dependent domain transformation in the case of shape optimization problems this is reduced to finite element mesh transformation in some cases Interfacing the simulation programme i e changing input data according to parameter values running the programme and obtaining results is usually an important issue as well as combining of these partial results according to problem definition in order to derive final results Several modules of the shell provide tools for performing such sub task and the user can combine these tools using the file interpreter according to the character of problems that are being solved All tools and algorithms of the shell are accessed through the shell file interpreter This together with the expression evaluator the calculator and interpreter flow control functions gives the user a great flexibility at defining different optimization problems and also the solution procedures The shell is in the first place designed for use with simulation programmes For test purposes however the user can define optimization problems in such way that evaluation of objective and other functions do not include numerical simulation The functions are in this case defined analytically using shell variables and expression evaluator
50. lcgradconstr gt The same as analyse except that no output is generated This function is predominantly intended for use in interfaces e g in interface with Mathematica to define an analysis function that runs the analysis defined in nverse 6 4 3 Tabulating Functions 6 4 3 1 taban tabanld point0 point numpt centered factor scaling lt printtab printparam _ printlist printobj printconstr _ printgradobj printgradconstr gt Performs a one dimensional table of analyses with endpoints point0 and point and prints the results according to specifications Arguments pointO starting point of the table in the parameter space vector value argument point end point of the table in the parameter space vector value argument numpt Number of analysis points counter value argument centered Flag for a centered table If non zero then the table is centered around the starting point point0 If table is centered with geometrically growing intervals then the interval lengths first fall from point reflected over pointO until point0 and then grow from point0 to pointl counter value argument factor Factor of interval length growth If 0 or 1 then intervals between table points are uniform If it is greater than 1 then intervals grow in such a way that each successive interval length is the previous length multiplied by factor If it is smaller than 1 then factors fall in the same way scalar v
51. lue argument e The remaining parameter depend on the particular test problem installed and provide eventual parameters that are necessary for that particular problem A list of available test problems with necessary parameters is below 6 4 4 2 testanalysis testan lt probed gt 0P P Performs a direct analysis according to the particular test problem that has been installed by the installtestanalysis command The function retrieves analysis parameters 26 INVERSE 3 18 6 5 Optimization And Inverse Analyses Approximation tools and stores the results to pre defined global variables This function is typically called within the analysis block In this way the test problem run by the function can be used in optimization procedures e probid ID of the test problem to be run string value argument If probid is not specified then the last problem that has been installed is run This will work even if the installtestanalysis has not been called at all because the JOptLib library automatically installs some test problems a tinitialization 6 5 Approximation tools 6 5 1 Smooth approximation 6 5 1 1 smoothapproxsimpbas type samples rweight point which valspec lt gradspec gt lt hgrad gt 70P Calculates an approximation of a sampled function The moving least squares method is applied which approximates the function locally by low order square in this case polynomial Coefficients of approximation are not cons
52. lways performed at transformed parameters that satisfy all bound constraints original parameters taht do not satisfy bound constraints are simply shifted on bounds In addition penalty terms are added to the objective function for each bound constraint that is violated by non transformed parameters The penalty term is zero for nonviolated constraints and grows linearly with the magnitude of violation of a particular constraint with factor kpen Coefficient Aconstr corresponds to conversion of bound constraints to usual constraints that are added to the original problem Each bound constraint is represented by linear constraint function with coefficient kconstr whose argument is a function of the difference between the parameter component and the corresponding bound the sign is taken according to whether there is a lower or upper bound in question The solution of the modified problem therefore satisfies the original constraints plus the bound constraints Defining i e setting non zero both Aven and kconstr is currently considered the best practice Since bound constraints are convex it is recommendable that kconstr is set high enough that bound constraint functions grow more rapidly than other constraint 15 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms functions in the domain that is rougly defined as the domain between the starting guess and the closest point in the feasible region Remarks See
53. lx tolf tolconstr maxit printlevel initial step ccccccceccessesseesteeseeees 11 6 2 4 NLPSimpS nlpsimps numconstr tolx tolf tolconstr maxit printlevel initial step 12 6 2 5 nlpsimpbound0 numconstr tolx tolf tolconstr maxit printlevel initial step bignum lt lowbounds upbounds bignum lt kpen kconstr lt numviolations maxresid gt gt gt cccccccseesseseeseeees 12 6 2 6 solvopt numconstr numconstreq tolx tolf tolconstr maxit lowgradstep initial 0 0 0 0 16 6 3 Older functions for optimization sessoesooesooesooescoesooesooesooscoossocessecsocsssesssesssecssesssese 17 6 3 1 inverse methodspec params es sssessesesssesesseesereessessrsresersssreressesresreseestssesresseseesseserseseent 17 6 3 2 optfsqp1 numob numnonineq numlinineq numnoneq numlineq eps epseqn maxit grad initial lowbound upbound fp eecccccccccceescessceseceseceseccecseeeseeesecsseseesecesecesecaecaeceeceeeseeeaeenseeereeas 18 6 3 3 optsimplex tol maxit Startguess eeeceeccecccescesecesecessesscessceeecesecesecesecaecsaecseecaeeeaeeeseeeeeeerenrens 19 6 4 PNI BATIET AALO E E EE TEAT 20 6 4 1 Testing the Direct Analysis sseseseeesseeeessessesrssersesstsesetsessreressestestssetstssteresseseesseserseseent 20 6 42 Tabulating FUnctions 20 0 0 ecceseesseesceeseeeseeseceseeeneeeeeseenseceaeceecnaeceaecaaecaeecaeseneeeeeeeeesereserens 21 6 5 Approximation LOOMS s scisccissedesccossscenessencsdsonssscecs soncsssssenvsnssecss s
54. n successive sampling points are extended If factor is 1 then points will be equidistantly distributed if it is different than 1 then the distances between successive points will decrease or increase from point0 towards point printparam and printmeas specify whether parameters and measurements should be printed respectively values different than zero indicate that the appropriate quantities should be printed example tabld cent 4 0234 4 2043 81 00 tabld by Domen Cukjati 6 4 3 4 tab2d kindspec point0 point numptl factor point2 numpt2 factor2 printparam printmeas Obsolete Use taban2d instead Runs numptl x numpt2 direct analyses with sampling points being nodes of a planar grid of points lying on a parallelogram in the parameter space and prints the requested results to the programme s standard output and output file Aindspec is a string that specifies what kind of table of direct analyses should be made and can be either 23 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools noncent or cent noncent means that sampling points will lie in the parallelogram defined by two vectors which are defined by basic point point0 and final points point and point2 cent on the other hand means that parallelogram is also defined by the same points but basic point point0 lies in the middle of the parallelogram Parallelogram is four times bigger in this case All three points should be specified as vectors of
55. n those components for which the corresponding components of owbound are greater or equal to the corresponding components of upbound are not bounded Note Inequality constraints are stated as in 6 1 namely c x lt 0 jel 6 1 where c are the constraint functions whose value is expected from the analysis function 9 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms In variables which hold values or derivatives of constraint functions these must appear in the appropriate order the same as in the argument block of the function First must be non linear inequality constraints then linear inequality constraints then non linear equality constraints and finally linear equality constraints if any of these are specified of course Remarks See introductory section for how the problem should be defined You can also take a look at inquick2 pdf which can be obtained at http www c3m si inverse doc other index html A detailed description of the fsqp algorithm can be found at http www isr umd edu Labs CACSE FSQP fsqp html 6 2 2 minsimp tolx tolf maxit printlevel initial step Performs the non gradient unconstrained minimization algorithm based on the Nelder Mead simplex method This is a non gradient algorithm suitable also for non differentiable and even non continuous functions that have a well defined unconstrained minimum The basic principle is similar to the Ne der Mead simplex al
56. nput file that is generated by this command is described in Section 7 2 1 The format in which the analysis output file must be generated by the external analysis program is described in Section 7 2 2 30 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions It must be ensured that the analysis program will be able to read input data form aninfile and write the response in anoutfile in the correct file For example the analysis program can be designed in such a way that its input and output file must be stated as command line arguments Then the command would look something like myanalysis in dat out dat and the analysis function would be called from the command file in the following way analysis fileanalysis myanalysis in dat out dat in dat out dat In the above case it is assumed that myanalysis is the name of the analysis program program file named in dat is used as analysis input file and file out dat is uses as analysis output file If the analysis program writes its results in a different format or expect input in a different format then converters must be provided If a converter is implemented as a stand alone program it should be executed right after the analysis program Since the fileanalysis function anticipates execution of only a single system command this can be solved in two ways First if the system permits successive execution o
57. nsions numparam I x numparam Rows of this matrix represent apices of the initial simplex Warning Use optsimplex instead of this command inverse is becoming an obsolete command and will be replaced by some other commands in the future However the command will remain implemented in the programme and will behave in the same way through a lot of future versions 6 3 2 optfsqp1 numob numnonineg numlinineg numnoneg numlineq eps epseqn maxit grad initial lowbound upbound Obsolete Use optfsqp instead 18 INVERSE 3 18 6 3 Optimization And Inverse Analyses Older functions for optimization Performs the fsqp feasible sequential quadratic programming optimization algorithm of Craig Lawrence Jian L Zhou and Andre Tits which is the basic and most powerful nonlinear programming algorithm built in Jnverse numob is the number of objective functions usually one numnonineg the number of non linear inequality constraints numlinineq the number of linear inequality constraints numnoneq the number of non linear equality constraints and numlineq the number of linear equality constraints eps is the final norm requirement for the Newton direction and epseqn maximum violation of nonlinear equality constraints at an optimal point Both criteria must be satisfied to stop the algorithm the second one is in effect only if there are equality constraints maxit is the maximum number of iterations grad specifies if gradients are p
58. objective function are used where possible The minimization is performed in two steps In the first step the interval containing a local minimum is searched for This is achieved by searching for combination of three points such that the middle point has the lowest value of the objective function The first point is given by the user x0 and the second two points are obtained by adding the initial bracketing step step0 to that point once and twice respectively Then the three points are moved if necessary until the bracketing condition is reached i e the middle point has the lowest value of the objective function In the second step the bracketing interval that contains the three bracketing points is narrowed in such a way that the bracketing condition remains satisfied In each iteration a new point is added in the larger of the two intervals defined by the three bracketing points Among four points we obtain this way those three which satisfy the bracketing condition and define the smallest interval are kept for the next iteration The point that is added is usually chosen by finding the minimum of quadratic parabola that 17 INVERSE 3 18 6 3 Optimization And Inverse Analyses Older functions for optimization through the current bracketing points This is not done if one of the two intervals becomes much smaller since in such cases successive quadratic approximations can converge slowly x0 is the initial point and step0 is t
59. oblem and its Solution variables gradobjectivemom and gradonstraintmom See the manual on variables chapter on pre defined variables for more details regarding the meaning of specific pre defined variables and rules for their manipulation The accuracy of the numerically calculated derivatives crucially depends on the step size The derivative calculation is mathematically exact for linear functions and therefore there are two sources of error The first one is because the function is normally not linear and this contributes larger errors where the step size gets large and the function deviates more from the linear model The second source is due to the noise in the function value If there is no other source of noise at least the function values are inexact because of finite precision that is used for all computer operations Errors in calculated derivatives that come from this source are amplified when the step size is reduced and die away when the step size gets large compared to the amplitude of noise Therefore there exists an optimal step size which is large enough with respect to noise amplitude and yet small enough that the function is adequately approximated by a linear model within the step size The user should provide the step size that is not necessarily optimal but is a good compromise for both sources of error When it is hard to estimate the level of noise the step size should be taken that is a bit smaller than the tolerance for t
60. of the second constraint function with respect to the third parameter errorcode integer error code of analysis 0 for no error usually a negative number for errors values are function specific reqcalcob reqcalcconstr reqcalcgradobj and reqcalcgradconstr are request flags for calculation of the various values as have been passed to the analysis function The same as with parameter values these flags are requested only for verification In vast majority of cases these flags will not be used by the optimization program and they can simply be set to 1 Angle brackets lt gt contains portion of data that is optional and can be omitted in the file there are no angle brackets indl ind2 is a set of integer numbers that can be used to pass some supplemental data about a particular calculation e g the sequential number of the particular analysis that the analysis server performed In most cases the set will be empty Le coefl coef2 is a set of real numbers that can be used to pass some data about a particular calculation In most cases the set will be empty i e cd is the definition data for the analysis It can have different forms usually it is an integer and does not have any meaning Spaces tab characters and newlines are not important Empty brackets should be used for any vector or set of vectors that are not calculated 38 INVERSE 3 18 7 2 Uniform File Interface Between Optimization and Analysis
61. ontessecssetesssessecseccsensess 27 6 5 1 Smooth approximattones 20 8k ncn ei eRe RI ee ee a E 27 7 Uniform File Interface Between Optimization and Analysis Programs 29 7 1 Interpreter FUMCtHONS 56 3 ssiceccoscsccnsssonessesctssetssensesoseecoseesionaveescesseuesdsnntsoncecsesssseuvessoesses 30 7 2 Pile POriiats 03 csscosisetsnsdinsssstessossseesesasdecdcesnsecntoasdundsndsssnecsensesdinsssoncsdtoseeceesnsoestanednccenss 36 7 2 1 File format for analysis request analysis input file ceeeeseceseeseeeeeeeeeeeeceseeeeeeseeeerenrens 37 7 2 2 File format for analysis results analysis output file cceeceseeceeseeeeeeeeceeeseeeeeeeseeereerens 37 723 XML formats aiene rainn Ses ieee meee atin a aa a 39 7 3 Solution Scheme cscssssscssessscsccsccssesscssccssssscsscssssesesessssenesscssssssssesssessesssseseeseeees 44 7 4 Demonstrative example sscsscscsssssssssscssssccsscssssssssssesssscessnssssssesssssssssscsssnssssssssees 45 PAN Testor Mesaana aei Se aes Aa ibe Date wi A E T RE E iea 45 7 4 2 Running the example and using custom analysis program c cccsceeseeceseeeeeeeeeeseeseeeerens 46 74 3 Using a different analysis program cccccecsceescessceescesceseceecesecesecaecsaecseeeaeeeaeeeseeeeeeereeeens 47 INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Problem and its Solution 6 OPTIMIZATION AND INVERSE ANALYSES 6 1 Definition of Optimizat
62. parameters numptl specifies the number of sampling points in the first direction and numpt2 in the second one factor is the factor by which the distance between successive sampling points in the first direction is extended and factor2 is for the second direction If any factor is equal to 1 points will be equidistantly distributed printparam and printmeas specify whether parameters and measurements should be printed respectively values different than zero indicate that the appropriate quantities should be printed example tab2d noncent 4 003 4 4 1034352 4 0134 5111 tab2d by Domen Cukjati 6 4 3 5 linetab kindspec args Obsolete Use taban1d instead Runs a set of direct analyses along a line in the parameter space and prints the requested results to the programme s standard output and output file Aindspec is a string that specifies what kind of table of direct analyses should be made The remaining arguments args specify the line along which the table is made number of points etc kindspec can be either lin or exp The meaning of the remaining arguments for different types of table is explained below 6 4 3 5 1 linetab lin numpt ppar pmeas point point21 Runs numpt direct analyses with sampling points equidistantly distributed along the straight line between point and point2 ppar and pmeas specify whether the parameters and measurements should be printed in table lines respectively values different than zero indi
63. ponding standard interpreter variables i e objective function to objectivemom constraint functions to constraintmom gradient of the objective function to gradobjectivemom and constraint gradients to gradconstraintmom Calculation flags are also stored to counter variables calcobj calcconstr calcgradobj and calcgradconstr The file must contain analysis results in the standard form described in Section Vad 7 1 1 6 filewriteanres filename Writes current analysis results extracted form the standard interpreter variables to the file named filename in the format described in Section 7 2 2 filename is a string argument This function is used for writing analysis results e g in the example described in Section 7 4 where Jnverse itself acts as analysis program In this example reading analysis input is performed by the parsefilevar function Section 7 1 1 13 7 1 1 7 filereadaninput filename fo Reads analysis input i e parameter values and request calculation flags from the file named filename Analysis input must be written to the file in the format described in Section 7 2 1 filename is a string argument This function can be used for reading input data for direct analysis when Jnverse itself acts as the direct analysis program It can be used instead of using the function parsefilevar Section 7 1 1 13 such as in the example described in Section 7 4 7 1 1 8 fileanalysis_xml ancommand aninfile anoutfile lt cd
64. rbitrary sequence of commands string arguments and pairs type varspec where type is a string argument specifying the type of the next argument and varspec is the variable element specification that defines the element of the file interpreter variable into which the object read from the file is stored Possible type specifications are e scal scalar real number e g 6 02e26 or 1 55 99 66 e count scalar integer number e g 324 29 66 e vector vec vector in a list format e g 7 1 2 1 3 34 INVERSE 3 18 7 1 Uniform File Interface Between Optimization and Analysis Programs Interpreter functions e marix mat matrix in a list format e g 1 1 1 2 2 1 2 2 e string str string which can be embedded in double quotes e g string element or in curly brackets e g string element or can be a single word specified without the quotes or brackets e g string_element e analysispoint anpt analysis results in the format specified in Section 7 2 2 Analysis results can not be stored in a variable because there is no corresponding type defined so the only variable elemet specification stated with this type is NULL Variable element specifications stated after types are usual specifications e g v 1 2 or str The corresponding elements they refer to must exist and variables must be of the correct type If we
65. rovided by direct analyses 1 or should be calculated numerically 0 initial is the initial guess and Jowbound and upbound are vectors of lower and upper bounds on parameters All three vectors must be in curly brackets The components which are not specified in the Jowbound or upbound vectors are not bounded below or above respectively Dimensions must be specified for all three vectors and all components must be specified for initial 6 3 3 optsimplex tol maxit startguess Obsolete Use minsimp instead Performs unconstrained minimization by the Nelder Mead simplex method Scalar argument tol is a tolerance counter argument maxit is maximal number of iterations and matrix argument startguess is a matrix whose rows are co ordinates of apices of the initial simplex One should take care that startguess represents a simplex with non zero volume which means that all vectors along the edges of the simplex joining in a given common apex are linearly independent 19 INVERSE 3 18 6 4 Optimization And Inverse Analyses Auxiliary tools 6 4 Auxiliary tools 6 4 1 Built in test analysis problems 6 4 1 1 testanalysis lt testname gt Performs a direct analysis for one of the built in test problems Normally this function should be run within the analysis block The function extracts analysis parameters optimization parameters and calculation request flags from pre defined interpreter variables and performs calculation of th
66. se the I G s software home would be c ighome The location must be such that all users have read amp write access to files in the directory 2 Set the value of environment variable IGHOME to the location of the I G s software directory ighome Note that the case matters on some platforms The environment variable must be created if it does not yet exist otherwise its value must be changed such that it contains the absolute path of ighome 3 Add the bin subdirectory of the software home directory ighome to the path environment variable You can usually use the previously defined variable IGHOME e g IGHOME bin on Windows or IGHOME bin on Unix like systems to refer to this directory 4 Copy the executable for your platform to the bin subdirectory of ighome 5 Now you can run the program in a terminal window Usually you will have to re open the terminal window so that the new environment variables will take effect You run nverse by typing the name of its executable followed by command line arguments Usually the first and often the only argument is the name of the command file or path if the file is not contained in the current directory Command file must contain instructions that are executed by nverse On Windows for example provided that the file name of Inverse executable is inverse exe and there is a command file named opt cm in the current directory you would run the program in the following way
67. tant but depend on the position of the point This is so because weights assigned to sampling points for calculating the least squares approximation depend on the relative position of the point of approximation with respect to these sampling points Arguments type Counter argument specification of type of weighting function used for approximation 2 2 0 Gaussian w x w x x Mad B 2 s ni vafi AREA sJ 1 2 samples Matrix argument sampled data Each matrix row corresponds to a sampling point and columns of a row contain the co ordinates of the sampling points followed by sampled values more than one functions may be sampled n 27 INVERSE 3 18 6 5 Optimization And Inverse Analyses Approximation tools rweight Vector argument that contains effective radii of the weights in individual co ordinate directions Weight corresponding to a sampling point fall from 1 size of the weight exactly in the sampling point to 1 e at the distance r from a sampling point in the co ordinate direction i where 7 is the component i of rweight point Vector argument point of approximation which counter argument specifies which sampled function should be approximated usually it is 1 meaning the first function valspec Scalar element specification specifies an element of a scalar variable to which the approximated value is stored gradspec Vector element specification If gradspec
68. tc Elements of type scalar must have contents that can be interpreted as a real number e g 06 1 45 10 4e 5 etc Elements of type vector must have an attribute named dim whose value must be a string representation of vector dimension e g 12 if the vector is of dimension 12 All components of the vector must be listed as sub elements of type scalar with element names vector_el Beside the attribute type whose value must be scalar these elements must have the attribute ind which must be a string representation of an integer that is equal to the index of specific vector component Elements of type table must have an attribute named dim whose value must be a string representation of the number of elements of the table e g 5 if the table has 5 elements Element of type table must have the attribute named e type whise value must be equal to the type of the table elements All components of the table must be listed as sub elements of a given type arbitrary but the same for all elements of the table and the type must match the value of the eltype attribute of the table element with element names table el Beside the attribute type these elements must have the attribute ind which must be a string representation of an integer that is equal to the index of specific table element Sub elements of the element whose type is analysispoint can be omitted if they are
69. terpreted What concerns tasks that do not need to be performed in every analysis it is better if they are invoked outside the analysis block so that they are performed only once 6 1 4 Defining the Direct Analysis The term direct analysis refers to the evaluation of the objective and constraint functions and possibly their gradients at a given set of optimization parameters User defines how the direct analysis is performed in the analysis block of the shell command file This is the block of code in the argument block of the analysis command i e within the curly brackets that follow this command The analysis block is interpreted by the shell interpreter every time the direct analysis is performed Direct analysis can be called by an optimization algorithm or by some other function invoked by the interpreter Typical examples are tabulating functions or the analyse function for performing test direct analyses Data transfer between the direct analyses and the functions that invoke them is implemented through global shell variables with a pre defined meaning The shell takes care that the current set of optimization parameters is always in the vector variable parammom when the direct analysis is invoked In the analysis block the user can therefore obtain parameter values from this variable using the interpreter and expression evaluator functions for accessing variables In the similar way it is expected that after the direct analysis is per
70. tion default 1 0 must be non negative if non zero then parameter bounds are handled by simultaneous parameter transformation such that bound constraints are always satisfied in all points in which the original analysis function is called and addition of penalty terms according to bound violations of untransformed parameters scalar value parameter e kconstr factor for constraint generating function default 0 0 must be non negative if non zero then parameter bounds are converted to normal constraints that are added to problem definition scalar value parameter e numviolated if non zero then the number of violated constraints is used as the first criterion in comparison of analysis results counter value parameter 13 INVERSE 3 18 6 2 Optimization And Inverse Analyses Optimization algorithms e maxresid if non zero then the maximal residuum positive constraint function is used in comparison of results instead of the sum of residua either of these criteria is used right before comparison of the objective function values counter value parameter Bound constraints are specified by vector arguments Jowbounds and upbounds whose components specify lower and upper bounds respectively for individual components of the parameter vector If for some index the specified lower bound is larger than the corresponding upper bound then it is understood that no bounds are defined for this component of the parameter vector If
71. undant digits i e digits that exceed the actual precision of the floating point number type long double are not printed 7 2 1 File format for analysis request analysis input file pl p2 reqcalcobj reqcalcconstr reqcalcgradobj reqcalcgradconstr cd Meaning of symbols used above is as follows pl pl p3 optimization parameters at which analysis was performed Flags that tell whether something has actually been calculated 0 yes 1 no reqcalcobj flag for the objective function reqcalcconstr flag for constraint functions reqcalcgradobj gradient of the objective function reqcalcgradconstr gradients of constraint functions cd a free parameter that can be used to transfer additional information to the direct analysis In principle cd can be anything embedded in curly brackets If only the eventual embedded curly brackets are properly closed Most commonly it will not be used at all and therefore empty brackets will be put in place of cd Otherwise interpretation of what stands in curly bracket is entirely in the domain of the analysis program therefore the documentation of the analysis program should provide information on how to compose cd 7 2 2 File format for analysis results analysis output file pl p2 calcobj obj calcconstr constrl1 constr2 calcgradobj dobjdpl dobjdp2 calcgradconstr dconstridp1l dconstrldp2
72. use the accuracy of the derivatives depend essentially on it and the optimal step size may vary drastically from case to case since it depends on scaling of the design parameters and on the level of noise of the differentiated functions After the call to the function every direct analysis at a given set of parameters is replaced by a number of plain analyses The first one is performed at the requested parameters and n others are performed at the parameter sets in which one parameter is perturbed by the appropriate step size as specified by stepvec n being the number of parameters After this the function values calculated with the requested parameter values are stored as usual e g in the pre defined variables objectivemom or and constraintmom In addition numerical approximations to the parameter gradients of these values are calculated and stored at the appropriate place e g in the pre defined This scheme is called the finite difference method There are also more complex schemes for numerical derivative calculation all of which include repeating calculation of function values at a number of perturbed parameters but differ significantly in sampling strategies and underlying mathematics Description of these schemes exceeds the scope of this manual This would normally be done explicitly by the appropriate interpreter code in the analysis block 7 INVERSE 3 18 6 1 Optimization And Inverse Analyses Definition of Optimization Pr
73. value is allowed and 0 if not They correspond to the following portions of analysis results respectively objective function constraint functions gradient of the objective function and gradients of constraint functions Element param of type vector must contain values of optimization parameters for which analysis results were calculated In Example 1 vector of parameters has dimension 2 and therefore 2 sub elements of type scalar that carry its components Element obj of type scalar contains if defined the value of the objective function Element constr of type table contains if defined the values of the constraint functions It must have as many sub elements as there are constraints Its elements must be of type scalar and must carry values of individual constraints Element gradobj of type vector contains if defined the gradient of the objective function It must have as many sub elements as the number of parameters Element gradconstr of type table contains if defined gradients of the constraint functions It must have as many sub elements as there are constraints Sub elements must be of type vector and their dimension must he equal to the number of parameters Element cd is an optional element of type string It can contain additional data such as analysis definition data that might be exchanged between the optimization and analysis routines In many cases this field is not used Example 1 Analysis output file in XML format
74. vely objective function constraint functions gradient of the objective function and gradients of constraint functions Element param of type vector must contain values of optimization parameters for which analysis results are to be calculated In Example 3 vector of parameters has dimension 2 and therefore 2 sub elements of type scalar that carry its components Element cd is an optional element of type string It can contain additional data such as analysis definition data that might be exchanged between the optimization and analysis routines In most cases this field is not used Example 3 Analysis input file in XML format lt Analysis input file created by IOptLib gt lt data type analysispoint mode analysis_input ind 1 gt lt reqcalcobj type counter gt 1 lt reqcalcobj gt lt reqcalcconstr type counter gt 1 lt reqcalcconstr gt lt reqcalcgradobj type counter gt 1 lt reqcalcgradobj gt lt reqcalcgradconstr type counter gt 1 lt reqcalcgradconstr gt lt param type vector dim 2 gt lt vector_el type scalar ind 1 gt 1 6000000000000001 lt vector_el gt lt vector_el type scalar ind 2 gt 1 lt vector_el gt lt param gt lt Optional definition data gt lt cd type string gt Definition data lt cd gt lt data gt 43 INVERSE 3 18 7 3 Uniform File Interface Between Optimization and Analysis Programs Solution Scheme 7 2 3 3 XML format for storage of analysis r

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