ALAMO user manual and installation guide v
Contents
1. If the directory exists it is erased in the beginning of the run Fitness metric to be used for model building Possible values are 1 2 3 4 5 and 6 corresponding respectively to Bayesian information criterion Mallow s Cp the cor rected Akaike s information criterion the Hannan Quinn information criterion mean square error and a convex penalty consisting of the sum of square errors and a term penalizing model size When MODELER is set to 6 a convex penalty consisting of the sum of square errors and a term penalizing model size is used for model building In this case the size of the model is weighted by CONVPEN Regularization method used to reduce the number of po tential basis functions before optimization of the selected fitness metric Possible values are 0 and 1 correspond ing to no regularization and regularization with the lasso respectively Technique to be used for adaptive sampling Setting this option to 0 turns off the adaptive sampling step thus forc ing ALAMO to resort to model building using the INI TIALPOINTS provided by the user A value of 1 directs ALAMO to use a random sampler that requires the pres ence of a user provided SIMULATOR A value of 2 in vokes the SNOBFIT code for sampling the SIMULATOR SNOBFIT requires a MATLAB license to run 1 0 almscr 10 0 ALAMO user manual and installation guide v 2015 9 18 SIMULATOR SIMULATOR is the name of the executable that ALAMO mysim can cal2. One per output variable space separated This vector must be specified if CONREG equals 1 EXTRAPXMIN Minimum values for safe extrapolation region One per input variable space separated If this vector is specified ZMIN and ZMAX are enforced over EXTRAPXMIN to EXTRAPXMAX ALAMO user manual and installation guide v 2015 9 18 11 EXTRAPXMAX Maximum values for safe extrapolation region One per input variable space separated If this vector is specified ZMIN and ZMAX are enforced over EXTRAPXMIN to EXTRAPXMAX Custom constrained regression i e constrained regression for enforcing conditions other than simple bounds can be done by setting CRNCUSTOM equal to the number of custom constrained regression conditions to be enforced Once this number has been specified the custom constraints themselves are specified through a related section BEGIN_CUSTOMCON END_CUSTOMCON where in each of CRNCUSTOM lines of this section one would need to specify the output variable index j associated with a custom constraint followed by white space followed by a function g x z expressed in terms of input and output variable labels ALAMO will then enforce the constraint g lt 0 when building a model for output variable 7 Finally some algorithmic options that control implementation aspects of constrained regression may be optionally set Option Description Default CRNINITIAL Number of random bounding points at which constraints 0 are sampled
3. We will sample no more data points at this stage BECO ARIA I I A I GK KK RK AK a ICA A I kkk ACA kk kkk kkk k k kk Iteration 1 Approx elapsed time 0 30E 02 s Step 1 Model building using BIC Model building for variable Z1 Z1 1 0 X1 2 0 Calculating quality metrics on observed data set Quality metrics for output Z1 RMSE 0 00 R2 1 00 Model size 1 BIC 2 40 Cp 9 00 AICC Infinity HQC Infinity MSE 0 00 Convex penalty 10 0 Total execution time 0 40E 02 s Times breakdown OLR time 0 0 MIP time 0 0 Simulation time 0 0 All other time 0 4 s in 17 ordinary linear regression problem s s in 0 quadratic integer problem s s to simulate 0 point s E Normal termination FRR ORO IO ROK RA AR RK E E E E EE E E E E E E E E k k k kkk 2K 2k a 2k 2k a ALAMO user manual and installation guide v 2015 9 18 17 The software first reports the version platform and compile date of the executable followed by credits Then after reading the input data a consistency check is run on the problem data and if passed the data structures are initialized In this specific example a warning is issued that logarithmic basis functions are not considered as the input variable is allowed to take negative values Subsequently information is provided for all algorithmic steps During initialization Step 0 it is reported that 11 data points are used for sampling and that no simulator is called in addition to using the preexist
4. initially CRNINITIAL must be a nonnega tive integer CRMAXITER Maximum allowed constrained regressions iterations 10 Constraints are enforced on additional points during each iteration CRMAXITER must be a positive integer CRNVIOL Number of bounding points added per round per bound 100 in each iteration CRNVIOL must be a positive integer CRNTRIALS Number of random trial bounding points per round of con 100 strained regression CRNTRIALS must be a positive in teger Compatibility with previous versions of ALAMO Starting with ALAMO v 2013 10 0 ALAMO s input format was changed Compatibility with input requirements of earlier versions was maintained with two exceptions e Previous versions required that ALAMO options be specified in a separate file than preex isting data all ALAMO input must now be entered in a single file 12 ALAMO user manual and installation guide v 2015 9 18 e Preexisting data can now be entered in a format that combines input and output measure ments in a column wise fashion For compatibility with early versions of ALAMO the following keywords are also acceptable in ALAMO v 2013 10 0 and beyond 3 5 Parameter Description NVARS This is equivalent to NINPUTS BEGIN XDATA Can be used in conjunction with BEGIN ZDATA to pass xz values separately from z values Only one of BE GIN_XDATA and BEGIN_DATA is permitted BEGIN_ZDATA Can be used in conjunction with BEGIN_XDATA to pass x values sepa
5. ALAMO user manual and installation guide v 2015 9 18 September 18 2015 For information about this software contact Nick Sahinidis at niksah minlp com Contents 1 Introduction s ss s s act as kacr kdos adis a Pok Gos a So E ew ws 1 it Licensing and software requirements 0000 2 1 2 Install e a ee y eg ee eh a dw E ee 2 Algorithms implemented 2 0 00 0c eee eevee Using ALAMO lt 2 465444840086 8 Bee ee ee ew Se eS ee 31 Calling ALAMO from the command line ooa a a 3 oe Exampe put DIE eoe eos anu ck pe Oe ee ee we eS 4 ao loput tle grammar og ee et a ee ee ee eR eG A 4 3 4 ALAMO data and options specification statements 5 3 5 Termination conditions and error messages 2 4 12 4 ALCAMO OMIpUt a s scc coace ee Re h e a he a ee a E eS 15 4 1 ALAMO screen output ooo ee 15 5 Bibliography lt o ses ec opad o Be ee i moit Se eS Ee SH GO 17 1 Introduction The purpose of ALAMO Automated Learning of Algebraic Models for Optimization is to generate algebraic surrogate models of black box systems for which a simulator or experimental setup is available Consider a system for which the outputs z are an unknown function f of the system inputs x The software identifies a function f i e a relationship between the inputs and outputs of the system that best matches data pairs of x and corresponding z values that are collected via simulation or experimentation ALAMO can e build
6. ATIOS are set the corresponding powers must also be specified as row vectors of corresponding length in the following way Parameter Description MONOMIALPOWER Row vector of monomial powers considered in basis func tions This vector must be of length MONO MULTIZ2POWER Row vector of powers to be considered for pairwise combi nations in basis functions This vector must be of length MULTIT2 MULTI3POWER Row vector of powers to be considered for triplet combi nations in basis functions This vector must be of length MULITITS RATIOPOWER Row vector of powers to be considered for ratios in basis functions This vector must be of length RATIOS Options and parameters for constrained regression This section contains optional data relating to constrained regression The primary options that control application of constrained regression are Option Description Default CONREG A 0 1 indicator to specify whether constraint regression 0 is used or not By default no constraint regression is used If CONREG is set to 1 bounds on output variables are enforced CRNCUSTOM Number of custom constraints other than bounds CRN 0 CUSTOM must be a nonnegative integer If CONREG is set equal to 1 some of the following vectors need to be specified Parameter Description ZMIN Minimum values for output variables One per output variable space separated This vector must be specified if CONREG equals 1 ZMAX Maximum values for output variables
7. OR val ues ALAMO user manual and installation guide v 2015 9 18 13 12 13 14 15 16 I7 18 19 20 21 22 23 24 25 26 21 28 20 30 3l 32 Number of input variables NINPUT must be specified before specifying XLABELS Number of output variables NOUTPUT must be specified before specifying ZLABELS Number of monomial powers MONO must be specified before specifying MONOMI ALPOWER values Number of input variables NINPUT must be specified before the DATA section of the input file Number of output variables NOUTPUT must be specified before the DATA section of the input file Number of data points NDATA must be specified before the DATA section of the input file Number of input variables NINPUT must be specified before the XDATA section of the input file Number of data points NDATA must be specified before the XDATA section of the input file Number of output variables NOUTPUT must be specified before the ZDATA section of the input file Number of data points NDATA must be specified before the ZDATA section of the input file Input data file missing required keyword s END_DATA missing or incomplete DATA section END_XDATA missing or incomplete XDATA section END_ZDATA missing or incomplete ZDATA section Only one of XDATA and DATA sections is allowed Only one of ZDATA and DATA sections is allowed Number of multi2 powers MULTI2 must be
8. OS is used DATA section must be specified when NDATA is nonzero Insufficient memory to allocate data structures Number of validation data points NVALDATA must be specified before the VALDATA section of the input file VALDATA section must be specified when NVALDATA is nonzero VALDATA section must be specified when NVALSECTIONS is nonzero Premature end of input file Number of custom constraints CRNCUSTOM must be specified before specifying CUS TOMCON section END_ZMIN missing or incomplete ZMIN section END_ZMAX missing or incomplete ZMAX section ALAMO user manual and installation guide v 2015 9 18 15 59 60 61 62 63 64 65 66 67 68 69 70 71 T2 73 T4 T5 76 4 4 1 Number of input variables NINPUT must be specified before specifying EXTRAPXMIN values Number of input variables NINPUT must be specified before specifying EXTRAPXMAX values END CUSTOMCON missing or incomplete CUSTOMCON section Number of output variables NOUTPUT must be specified before specifying ZMIN values Unable to open trace file TRACEFNAME No keyword may be specified more than once Variable index is out of range Error while trying to run SNOBFIT Error while trying to run ordinary least squares regression Maximum CPU time MAXTIME exceeded Error while trying to write in the ALAMO scratch directory Number of output variables NOUTPUT must be specified before specifying TOLMEAN ERROR
9. Option NDATA NVALDATA MINPOINTS XFACTOR SCALEZ XLABELS ZLABELS MONO MULTI2 MULTIS RATIOS EXPFCNS LOGFCNS Description Number of data points in a preexisting data set specified by the user NDATA must be a nonnegative integer that is no more than INITIALPOINTS Number of data points in a preexisting validation data set specified by the user This data set is not used to develop the model but only to compute model errors at the vali dation data points NVALDATA must be a nonnegative integer At any stage of the adaptive sampling process conver gence is assessed only if the simulator is able to compute the output variables for at least MINPOINTS out of the data points requested by ALAMO MINPOINTS must be a positive integer Row vector of scaling factors used to scale the input vari ables One per input variable space separated A 0 1 indicator If 1 outputs are scaled when solving mixed integer optimization problems otherwise they are not scaled Row vector of labels to denote the input variables One per input variable space separated Each label can be no more than 128 characters long Row vector of labels to denote the output variables One per input variable space separated Each label can be no more than 128 characters long Number of monomial powers to be considered as basis functions MONO must be a nonnegative integer Number of pairwise combinations of powers to be consid ered as
10. an algebraic model of a simulation or experimental black box system 2 ALAMO user manual and installation guide v 2015 9 18 e use previously collected data for model building e call a user specified simulation function to collect measurements e enforce response variable bounds physical limits and boundary conditions e use a preexisting data set for model validation e output models in simple algebraic form The problems addressed by the software have long been studied in the fields of statistics design of experiments and machine learning Whereas existing techniques from this literature can be used to fit data to models the main challenges addressed by the software are in determining where to run the simulations or experiments what models to fit and how to determine if a model thus produced is accurate and as simple as possible A distinguishing feature of the software is that it provides models that are as simple as possible and still accurate Moreover ALAMO is capable of utilizing theory driven insights alongside data The ALAMO models can be used to facilitate subsequent system analysis optimization and decision making 1 1 Licensing and software requirements The code is available for download at http minlp com alamo The same URL provides infor mation about licensing the software ALAMO makes calls to the third party software GAMS A separate install and license are required for GAMS and at least one of the mixed integer quadr
11. atic programming solvers under GAMS preferably GAMS BARON Use of the ALAMO constrained regression capability requires availability of GAMS BARON for the solution of general nonconvex optimization problems More information about GAMS can be found at http www gams com In the absence of a GAMS license ALAMO attempts to use enumerative approaches that may be more time consuming or impractical for large problems For constrained regression only ALAMO makes calls to MATLAB for which a separate install and license are required If con strained regression is not used MATLAB is not required by ALAMO 1 2 Installation Install ALAMO and the ALAMO license file in any directory of your choice and add it to your path Do the same for GAMS Installation of GAMS is optional but recommended Install MATLAB if the constrained regression capabilities of ALAMO are needed Octave does not work in the place of MATLAB 2 Algorithms implemented ALAMO seeks to identify low complexity surrogate models using a minimal amount of data for a system that is described by a simulator or experiment Surrogate models are constructed using a three step process In the first step an initial design of experiments is generated and the simulation is queried at these points In the second step an algebraic model is built using this ALAMO user manual and installation guide v 2015 9 18 3 initial training set The model is built using integer optimization techniques to sel
12. basis functions MULTI2 must be a nonnegative integer Number of three variable combinations of powers to be considered as basis functions MULTI3 must be a non negative integer Number of ratio combinations of powers to be considered as basis functions RATIOS must be a nonnegative inte ger A 0 1 indicator Exponential functions are considered as basis functions if 1 otherwise they are not considered A 0 1 indicator Logarithimic functions are considered as basis functions if 1 otherwise they are not considered Default 0 NINPUTS X1 X2 X3 Z1 Z2 Z3 ALAMO user manual and installation guide v 2015 9 18 SINFCNS COSFCNS RBF RBFPARAM SCRATCH MODELER CONVPEN REGULARIZER SAMPLER A 0 1 indicator Sine functions are considered as basis functions if 1 otherwise they are not considered A 0 1 indicator Cosine functions are considered as basis functions if 1 otherwise they are not considered A 0 1 indicator Radial basis functions centered around the set of the user specified NDATA points are considered as basis functions if 1 otherwise they are not consid ered These functions are Gaussian and are deactivated if their textual representation requires more than 128 char acters in the case of too many input variables and or data points Multiplicative constant used in the Gaussian radial basis functions ALAMO creates this directory and uses it to store tempo rary files and results
13. ect the best subset from a collection of potential basis functions that can be used to build up the model In the third step an adaptive sampling methodology based on derivative free optimization techniques is used to identify points where the model is inaccurate Once these points are added to the training set execution returns to the second step of the algorithm The process continues until the third step confirms accuracy of a previously built model In contrast to commonly used techniques such as forward or backward regression that in vestigate model sensitivities with respect to one basis function at a time ALAMO s best subset selection techniques ensure that synergistic effects between different basis functions are accounted for in its model building step Before ALAMO best subset selection techniques were considered too time consuming to apply to realistic data sets While developing ALAMO nonlinear integer programming techniques were devised that rely on the BARON software to solve these models in realistic computing times for many industrially relevant systems ALAMO is also unique in that it utilizes derivative free optimization techniques in its adaptive sampling step These techniques offer a systematic approach to interrogate models identify weaknesses and guide experimental design towards parts of the space requiring more attention Another distinctive feature of ALAMO is its constrained regression feature which is capable of enfo
14. ied and is comprised of 11 preexisting data points The user options do not call for adaptive sampling to be used effectively requesting the best possible model that can be derived from the preexisting data set Finally the following functions are permitted in the model logarithmic exponential sine cosine and monomials with powers 1 2 and 3 Example 1 with data from z x72 ninputs 1 noutputs 1 xmin 5 xmax 5 initialpoints 11 ndata 11 logfcns expfcns sinfcns a el cosfcns mono 3 monomialpower 1 2 3 BEGIN_DATA 5 25 4 16 3 9 2 4 1 1 0 0 1 1 2 4 3 9 4 16 5 25 END_DATA Several additional examples of ALAMO input files accompany the distributed code 3 3 Input file grammar The following rules should be adhered to while preparing an ALAMO input file ALAMO user manual and installation guide v 2015 9 18 5 e The name of the input file should include its exact path location if the file is not present in the execute directory e The name of the input file should not exceed 1000 characters in length e Input is not case sensitive e Blank lines white space and lines beginning with or are skipped e Most options are entered one per line in the form of keyword followed by value Certain vector options are entered in multiple lines starting with begin lt keyword gt followed by the vector input followed by end_ lt keyword gt e Certain options must appear firs
15. ing data set In Step 1 the model is built in stages The initial model is z 0 and after the first mixed integer quadratic program is solved the perfect model z x is identified Since there is no simulator provided there is no adaptive sampling and execution terminates here after reporting a detailed breakdown of CPU times for the different algorithmic steps including the number of calls to the optimizer 0 in this example and the simulator 0 in this example There are no calls to an optimizer in this example because the problem is small enough to be solved faster by complete enumeration 5 Bibliography The following is a partial list of ALAMO related publications that describe the algorithms im plemented in the software the theory behind them and some related applications 1 A Cozad N V Sahinidis and D C Miller Learning surrogate models for simulation based optimization AIChE Journal 60 2211 2227 2014 2 A Cozad N V Sahinidis and D C Miller A combined first principles and data driven approach to model building Computers amp Chemical Engineering 73 116 127 2015
16. l in order to obtain function evaluations of the black box The simulator must be capable of reading the num ber of requested data points k followed by k lines for each of the data points where function evaluations are re quested The simulator must return a number of lines each containing a point in the input variable space where a simulation was performed along with the corresponding output variable values ALAMO allows for the number of these points to be different than k and for these points to be different than the points where simulations were re quested If more than k points are provided only the first k are used If the simulation fails or is impossible for cer tain output variables partial simulation results may be returned and the non available output variables must be set equal to PRESET The simulator must be in the user s path or its complete path must be specified through this option ALAMO will execute the simulator in a scratch directory it generates during its run hence the simula tor should not rely on any relative paths to access other programs or files PRESET A value indicating that the simulator was not able to com 111111 pute a specific output variable at a specific point This value must be carefully chosen to be an otherwise not re alizable value for the output variables MAXTIME Maximum total execution time allowed in seconds This 1000 time includes all steps of the algorithm including time to read p
17. ll Complete path of GAMS executable or name if GAMS is in the user path On some systems GAMS does not run if there is white space in the current path Name of preferred GAMS solver for solving ALAMO s mixed integer quadratic subproblems Special facilities have been implemented in ALAMO and BARON that make BARON the preferred selection for this option However any mixed integer quadratic programming solver available under GAMS can be used A 0 1 indicator Output is directed to the listing file if this option is set to 1 if set to 0 no output is sent to the listing file PRINT_TO_SCREEN A 0 1 indicator Output is directed to the screen if this option is set to 1 if set to 0 no output is sent to the screen 0 0001 input txt output txt gams BARON In deciding whether to deactivate printing to the screen or file users should consider that model coefficients are printed with two significant digits to the screen and with 23 digits to the listing file If the parameter NDATA is set then a data section must follow subsequently in the input file with precisely NDATA rows one for each data point pair of z and z values specified in the following form BEGIN_DATA END_DATA If the parameter NVALDATA is set a similar data section must be provided using a similar construct BEGIN_VALDATA 10 ALAMO user manual and installation guide v 2015 9 18 END_VALDATA If the parameters MONO MULTI2 MULTI8 or R
18. rately from z values Only one of BE GIN_ZDATA and BEGIN_DATA is permitted Termination conditions and error messages Errors in the input file are reported on the screen and or the listing file in the form of warnings and errors ALAMO attempts to continue execution despite warnings If the errors are severe the program execution is stopped and the line where the fatal error occurred is displayed The input file should be checked even if the warnings are not severe as the problem might have been parsed in a way other than it was intended to be Detailed error messages are provided in that case If execution terminates normally ALAMO prints Normal termination If there is an error the message on the screen or file is ALAMO terminated with termination code followed by one of the following error codes all of which are self explanatory J 2 10 11 ALAMO must be called with exactly one command line argument ALAMO input file name must be no longer than 1000 characters ALAMO input file not found ALAMO input file cannot be opened Keyword not recognized in input file Keyword too long in input file Incomplete input file Input value in error in input file Number of input variables NINPUT must be specified before specifying XMIN values Number of input variables NINPUT must be specified before specifying XMAX values Number of input variables NINPUT must be specified before specifying XFACT
19. rcing theory driven requirements on response variables including response variable bounds thermodynamic limitations and boundary conditions To enforce these requirements over the entire domain of input variables ALAMO relies on BARON to solve semi infinite nonconvex optimization prob lems The bibliography at the end of this document offers more details of the methodology implemented in ALAMO and demonstrates the advantages of this methodology in comparison to currently utilized approaches including classical regression and the lasso 3 Using ALAMO 3 1 Calling ALAMO from the command line ALAMO reads model data and algorithmic options from a text file in a relatively simple format Even though not required it is strongly recommended that all ALAMO input files have the extension alm If the input file is named test bar and the ALAMO executable is named alamo issuing the command alamo test or alamo test alm results in ALAMO parsing the file and solving the problem In addition to screen displays ALAMO can optionally provide results in the listing file test lst that is generated during the run 4 ALAMO user manual and installation guide v 2015 9 18 3 2 Example input file The following file is referred to as el alm and pertains to learning the simple function z 2 There is one input and one output in the model The input is restricted between 5 and 5 An initial sampling data set is specif
20. roblem preprocess data solve optimization sub problems and print results MAXTERMS Row vector of maximum terms allowed in the modeling of eae output variables One per input variable space separated A 1 signals that no limit is imposed IGNORE Row vector of 0 1 flags that specify which output vari 000 ables if any ALAMO should ignore All output variables must be present in the data but ALAMO does not model output variables for which IGNORE equals 1 le 3 le 3 TOLMEANERROR Row vector of convergence tolerances for mean errors in 1e 3 the modeling of output variables One per output variable space separated TOLRELMETRIC Convergence tolerance for the chosen fitness metric for the le 4 modeling of output variables This must be a nonnegative scalar MIPOPTCA Absolute convergence tolerance for mixed integer opti 0 05 mization problems This must be a nonnegative scalar ALAMO user manual and installation guide v 2015 9 18 MIPOPTCR LINEARERROR SIMIN SIMOUT GAMS GAMSSOLVER PRINT_TO_FILE Relative convergence tolerance for mixed integer optimiza tion problems This must be a nonnegative scalar A 0 1 indicator If 1 a linear objective is used when solving mixed integer optimization problems otherwise a squared error will be employed Name of input file for the simulator ALAMO generates this file Name of output file for the simulator ALAMO expects the simulator to provide this file after each ca
21. specified before specifying MULTIZPOWER values Number of multi3 powers MULTIS3 must be specified before specifying MULTI3POWER values Unable to open output file Maximum number of iterations reached Number of ratio powers RATIOS must be specified before specifying RATIOPOWER values 14 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5l 52 53 54 55 56 57 58 ALAMO user manual and installation guide v 2015 9 18 Error while trying to use GAMS to solve the MIP for best subset Error while attempting to access the ALAMO execution directory Error while attempting to access the ALAMO scratch directory Error while attempting to access the external simulator Error while attempting to write the external simulator input file Error while attempting to read the external simulator output file Scaling by zero is not allowed XMAX XMIN for all input variables must be positive XDATA must be in the range XMIN XMAX Simulator should not return NaN for input variable values Simulator should not return NaN for output variable values For any variable that the simulator cannot compute return the value of PRESET Input file is missing XMIN values Input file is missing XMAX values MONOMIALPOWERS must be specified if MONO is used MULTI2POWER must be specified if MULTI2 is used MULTISPOWER must be specified if MULTI3 is used RATIOPOWER must be specified if RATI
22. t in the input file This requirement is discussed explicitly in option descriptions provided below e Character valued options such as paths and file names that contain spaces and forward slashes must be provided in quotes 3 4 ALAMO data and options specification statements Required scalar parameters The following parameters must be specified in the input file in the order listed below Parameter Description NINPUTS Number of model input variables NINPUTS must be a positive integer and defines the dimension of the vector zx NOUTPUTS Number of model output variables NOUTPUTS must be a positive integer and defines the dimension of the vector z INITIALPOINTS Number of data points in the initial sample set INITIAL POINTS must be a nonnegative integer Required vector parameters The following parameters must be specified in the input file in the order listed below and only after the scalar required parameters have already been specified Parameter Description XMIN Row vector specifying minimum values for each of the in put variables This should contain exactly NINPUTS en tries that are space delimited XMAX Row vector specifying maximum values for each of the input variables This should contain exactly NINPUTS entries that are space delimited ALAMO user manual and installation guide v 2015 9 18 Optional data specifications This section contains optional data relating to the particular problem to be solved
23. values A least squares subproblem failed during enumeration and no optimizer is available Licensing error A valid license is required in order to run this software Error while trying to use GAMS to solve the constrained regression model Error while trying to copy file to disk CUSTOMCON section must be specified when CRNCUSTOM is nonzero All output variables ignored by user No point in calling ALAMO ALAMO output ALAMO screen output The screen output below is obtained for problem el alm 2A A A k k A k kk k k 2k 2A A A k k 2k 2k A k k k 2k A k 2k 2 2K 2A 2A 2A K k 21 k k k 2 KA k 2A A 2K 2A 2A 2A K k 2k 2k 2K 2k 2k 2k 2K 2K K K K ALAMO version 2015 9 7 Built LNX 64 Mon Sep 7 18 12 41 EDT 2015 If you use this software please cite Cozad A N V Sahinidis and D C Miller 16 ALAMO user manual and installation guide v 2015 9 18 Automatic Learning of Algebraic Models for Optimization AIChE Journal 60 2211 2227 2014 ALAMO is powered by the BARON software from http www minlp com BEAR AAR I IOI AIRE ARI A A A 1 21121 21 21 21 2 A A 1 211 kk kkk AA A kkk kk kkk kkk Licensee Nick Sahinidis at The Optimization Firm LLC niksah gmail com BEA ARR AGRA I I I I OK KK KEK a A A ACA IKK KKK A CA ACA A A kk KK kkk kkk Reading input data Checking input consistency and initializing data structures Warning eliminating basis log X1 Step 0 Initializing data set User provided an initial data set of 11 data points
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