LLBo Applets User Guide - Lehrstuhl für Leichtbau
Contents
1. The Kuhn Tucker conditions in Two Dimensions Tal ER ce ke hee RK ERE REE RHO hE a Bw Gan na 4 2 The Graphical User Interface 2 2 0 0 0 2 00020 eee 4 3 Optimization of the Fiber Reinforced Plate 431 Load Cases copas ehh ow Oe Se ee a Bes 4 3 2 Coordinate System Transformation EME c secaraa adi See So he eKEE Ee ES we 4 4 Calculating the Lagrangian Multipliers 2 2 22 2 oo none Augmented Lagrangian Procedure BI EE A ee ea ee 5 2 The Graphical User Interface 53 Ve Del Method o o sor o d d a Aa AN 5 4 The Augmented Lagrangian Procedure e Bibliography List of Figures 2 fuk 31 4 1 4 2 5 1 5 2 SA Applet window 1D unconstrained o o o 6 One iteration of the iterative minimization procedure 7 Applet window 1D constrained 2 2 2 rn 12 Applet window 2D Kuhn Tucker o rn 16 Fiber reinforced plate the two load cases 0 0 18 Applet window The Augmented Lagrangian Procedure 22 The Dual Method 2 ke dana hae a RAS ER ARE SHS 24 The Augmented Lagrangian Procedure 200 26 1 Introduction The following sections describe a suite of interactive Java applets designed to support the course Multidisciplinary Design Optimization The sample problems were chosen to make them easy to follow Thus unconstrained optimization is illustrated with an objective func2. 0 00 g2 0 00 x 101 x2 0 78 Kuhn Tucker conditions met Load cases SACH Lo load case NAS Plate design Current design d d 0 78 mm a 101 deg FR Stresses MPa A of 686 of 36 2 Objective function TF Constraint functions 2 2 Legend q of 139 of 111 6 of 39 4 gt Objective function gradient gt Constraint function gradient Figure 4 1 Applet window 2D Kuhn Tucker 16 CHAPTER 4 THE KUHN TUCKER CONDITIONS IN TWO DIMENSIONS Load cases contains a pair of mnemonic images that represent load cases considered in the problem see Section 4 3 When the design point is fixed the active load case s are black otherwise grayed out Plate design shows a 3D sketch of a fragment of the plate the values of design variables and also six values for in plane stresses resulting from the design choice A triple of stresses is defined for each of the two load cases The subscript signs have the following meanings ou normal stress parallel to the direction of the fibers K i normal stress perpendicular to the direction of the fibers O in plane shear stress 4 3 Optimization of the Fiber Reinforced Plate A classical design optimization problem is considered min w d g a d lt 0 gala d lt 0 Here the weight w to be minimized depends only on the thickness of the plate d The constraints g and ga correspond to load cases descr
3. Verst The gradient of the selected Lagrangian at the current point allows judging whether the point can be an optimum 1Note that the multipliers Au 2 are calculated to be the best in the least squares sense The values thus obtained differ over the neighbourhood of a given point in the design space It should be noted that both the value and the gradient of the Lagrangian depend on these multipliers Essentially a different function L is considered in every design point Therefore the habitual relation between function derivatives and how the function itself varies cannot be expected to hold in this particular case 15 CHAPTER 4 THE KUHN TUCKER CONDITIONS IN TWO DIMENSIONS 91 2 The values of the constraint functions allow judging whether the point lies within the feasibility region 212 The values of the design variables x and 2 correspond to o and d respectively The Calculate multipliers button makes the current point fixed and triggers caluclation of the values enumerated above The Reset button produces the reverse effect the values disappear and the selected point again traces the mouse pointer freely thus updating the Plate design pane TUT TECHNISCHE Lehrstuhl f r Leichtbau In UNIVERSIT T i LIN MUNCHEN Institute of Lightweight Structures 8 Lagrangian multipliers Check the Kuhn Tucker criteria Calculate mutipiers_ Reset zum ann L 0 78 A 0 27 A2 0 58 VL 0 00 Y L 0 00 gi
4. 2 backward y 1 ya ui 2 3 Step size accepts a value for the parameter a explained in section 2 3 Termination criteria offers a selection of termination criteria explained in Section 2 4 Optimization ends as soon as at least one of the criteria that are currently checked on is fulfilled TUT TECHNISCHE Lehrstuhl fiir Leichtbau f UNIVERSITAT S 4 MUNCHEN Institute of Lightweight Structures LB P _ Procedure control _ Run the minimization procedure _Minimize Start over Minimized in 4 iterations total of 4 evaluations Initial point Objective function Select the function to minimize y x sin x y x x 2 x e _ Gradient calculation Set up differentiation mode 8 Analytical Finite differences Ax Step size Adjust the step size factor Termination criteria Set up the termination criteria y by relative change in Y by absolute change in X Legend Objective function gt Antigradients D Optimum by number of evaluations Figure 2 1 Applet window 1D unconstrained CHAPTER 2 UNCONSTRAINED OPTIMIZATION IN ONE DIMENSION 2 3 The Iterative Minimization Procedure The algorithm described below can be viewed as a simplified special case of the steepest descent in 1D see 1 86 4 For numerical optimization an initial point xy must be selected as a rough approximation A seque
5. follows Relative change in y E Yi Yi lt Absolute change in x Ziyi lt Predefined limit number of evaluations d 2 imaz CHAPTER 2 UNCONSTRAINED OPTIMIZATION IN ONE DIMENSION The parameters e and ima can be set independently for every criterion in the user interface When more than one criteria are checked a change in a parameter sometimes has no effect on the course of execution This indicates that the respective criterion does not cause termination of the algorithm in this particular setting but rather one of the other activated criteria does Also note that the effect of imaz on the number of iterations changes depending whether the Finite differences or the Analytical mode is selected in the Gradient calculation tab In the former case the counter i in line 60 of Algorithm 1 1 is incremented by two units instead of one The difference in the number of evaluations per iteration should be apparent from formulas 2 1 2 3 Of course with finite differences function evaluations are counted not evaluations of the analytical derivative 3 Constrained Optimization in One Dimension 3 1 Overview The following sections describe the applet that illustrates two approaches to constrained optimization The sample problem is identical to the one considered previously see Section 2 just the constraints are added Therefore the user interface contains some additional controls which are describ
6. aints to an uncon strained auxiliary function Here suitability means that the solutions of the original and the transformed problems must coincide The lecture script dwells in quite some detail on how this is achievable with penalty functions see 1 6 2 for the mathematical and the algorithmical details Just two more points are to be made in connection with this specific implementation 1 For illustrative purposes the algorithm is left to be semi automatic i e steps 4 and 5 in 1 p 76 are left to be triggered by the user with the button labeled 10 found in the Methods pane 2 As mentioned in the script the conditioning of the replacement function becomes increasingly worse as r becomes smaller 1 p 76 Indeed this may manifest itself as an error message reading The procedure led into an infeasible region in the Procedure control pane This would indicate that the initial approximation is unsuitable given the current value for In order to keep to the feasibility region must be decreased using the Step size pane However with decreased a the number of steps until convergence may become unreasonable This in turn can be helped through somewhat relaxing the temination criteria see Section 2 4 As can be noticed in practice this more sophisticated constraint handling strategy may involve certain subtleties Experimenting with the sample problem can help revealing them For thorough und
7. ed in the next section Sections 3 3 and 3 4 provide references concerning two methods based on the iterative pro cedure Section2 3 e The method of usable feasible direction is quite a straightforward extension just one more termination criterion is added An outline of this principle can be found in 1 86 6 e Sequential unconstrained minimization technique SUMT constructs an auxiliary func tion that automatically takes the constraints into account for details please consult D 6 2 3 2 The Graphical User Interface The applet window is shown in Figure 3 1 with the additional control pane expanded The rest of the panes are the same as in LLBoAppletl please refer to Section 2 2 for details The Methods pane contains two radio buttons and also the means to fine tune the factor r e Usable feasible direction activates the simple way of constraint handling it has no parameters see Section 3 3 e SUMT activates Sequential Unconstrained Minimization Technique which depends on parameter r see Section 3 4 e The text field labeled r accepts the value for r The new value is applied upon the next minimization run invoked via the Minimize button in the Procedure control pane 11 CHAPTER 3 CONSTRAINED OPTIMIZATION IN ONE DIMENSION e The buttons labeled 10 and 10 serve to automatically scale r by an order of magnitude with every click Besides they simultaneously produce a convenient side e
8. er the stresses must be transformed to the coordinate system that is aligned to the direction of the fibers in other words the system is rotated by angle a The needed transformation involves two rotations First it is necessary to calculate horizontal and vertical components of stresses at the edges of a rotated element of the plate Second these components must be expressed relative to the rotated axes of the new coordinate system This explains why the transformation matrix in 4 1 has products of trigonometric functions as its elements Uniting all that was said so far the following formula allows arriving from distributed forces defined by the load cases to fiber aligned stresses used for defining the constraints ou 1 cos o sin a 2 sina cosa Fii a as sin a costa 2 sina cosa x Fa 4 1 0 sina cosa sina cosa cos a sin a Fa As can be seen all derived quantities are obtained analytically However the expressions became quite involved already More complications are added at the stage of defining the constraints which is discussed next 4 3 3 Constraints The requirement that the fibers must not suffer damage is formulated in terms of the bound ary allowable stress on as follows 18 CHAPTER 4 THE KUHN TUCKER CONDITIONS IN TWO DIMENSIONS d 2 Med sisi 4 2 ES Moreover fibers may not become detached from the plate 2 2 2 Op OLB O B From algebraic considerations it is clear that whene
9. erstanding based on theory a mathematically rigorous treatment of SUMT may be found in 3 lAlso called replacement function in 1 13 4 The Kuhn Tucker conditions in Two Dimensions 4 1 Overview The applet described below illustrates some fundamental aspects of optimization in multiple dimensions To keep the example comprehensible a problem in just two dimensions is considered The very next section describes the user interface which displays the 2D space of design variables the objective function the constraints and some supplementary visual clues Further the design problem to be considered is introduced Section 4 3 Evaluation and analysis of the Kuhn Tucker conditions see 1 83 6 which form the essence of this applet is discussed last Section 4 4 4 2 The Graphical User Interface The overall view of the applet window is shown in Figure 4 1 A 2D plot on the left repre sents the design space in variables o and d see Section 4 3 for a reference to the problem statement A click in the plot selects the point to analyse On the right there are three panes that display characteristics of the currently selected design space point Lagrangian multipliers displays the following values once they are calculated A12 The values of the Lagrangian multipliers selected through application of the least squares method see Section 4 4 for details L The value of the selected Lagrangian at the current point
10. ffect the currently achieved minimum is set as the initial approximation x9 for the next run see Section 2 3 TECHNISCHE Lehrstuhl f r Leichtbau Da UNIVERSITAT i F LANN Institute of Lightweight Structures LB MUNCHEN Procedure control D Run the minimization procedure m Minimize Start over Minimized in 5 iterations total of 5 evaluations Objective function Select method to handle constraints Usable feasible direction SUMT r o Enjo Gradient calculation Objective function gt Antigradients Optimum egend Ss Constraints ae Auxiliary function Figure 3 1 Applet window 1D constrained 12 CHAPTER 3 CONSTRAINED OPTIMIZATION IN ONE DIMENSION 3 3 The Method of Usable Feasible Direction For the 1D setting the method under consideration is trivial Another termination criterion that is added internally to the algorithm is all that changes compared to the basic steepest descent see Section 2 3 This additional criterion is simply as soon as x 4 lies outside the feasibility region return x as an optimum One example of how the same principle is applied in a more sophisticated context is Sequential Quadratic Programming SQP see 1 86 6 3 4 Sequential Unconstrained Minimization Technique A more advanced way to hanle constraints is to eliminate them altogether through a suitable transformation of the problem from the objective function with constr
11. ibed in the next section These constraints heavily depend on the angle a of the fibers as becomes clear from sections 4 3 2 and 4 3 3 Statement and solution of the problem under consideration is taken from work 2 Before delving into the specific details of this statement expounded below it is advisable to review a somewhat more accessible formulation The lecture script 1 2 1 3 can be helpful in gaining an overall picture 4 3 1 Load Cases The specific load values that were used as an example in this implementation are given in Figure 4 2 Regarding load case g gt it can be observed that due to the choice of values the principal stresses are compressive only This observation is relevant for what will be discussed further since in the definition of this problem s constraints compression and tension are distinguished in a special way discussed in Section 4 3 3 17 CHAPTER 4 THE KUHN TUCKER CONDITIONS IN TWO DIMENSIONS Figure 4 2 Fiber reinforced plate the two load cases 4 3 2 Coordinate System Transformation The loads shown in Figure 4 2 are expressed as forces distributed over the length of the edges of a plate element In other words integration of stresses over plate thickness has been performed The constraints that will be discussed in a moment are formulated in terms of stresses a oi and oy Therefore the values of distributed forces are to be divided by thickness d in order to obtain stresses Moreov
12. jective function TECHNISCHE Lehrstuhl f r Leichtbau In UNIVERSITAT r A S LL MUNCHEN Institute of Lightweight Structures Ce Constrained 1D optimization problem Unconstrained 2D dual problem f A TE Y Objective function Be Iteration values Legend Nk Optimum 3 2 2 0 0 7 0 5 1 8 31 43 56 65 a Constraint bw Direction to next iteration value Procedure control Initial point Objective function Run the minimization procedure Adjust the starting point Select the function to minimize yoo sino y x x 2 x Minimized in 8 evaluations Figure 5 1 Applet window The Augmented Lagrangian Procedure 5 3 The Dual Method Along the lines of a more general case 5 817 1 let us consider the Lagrangian function that corresponds to the 1D constrained optimization problem with one equality constraint 22 CHAPTER 5 AUGMENTED LAGRANGIAN PROCEDURE L x A f x Aglz where f x is the objective function g x the constraint and the Lagrangian multiplier Now we can define the dual function D A which is the minimum of L x A along x when A is fixed to a certain value as the independent variable D A minz L x A The optimum of the constrained problem is defined as the saddle point of the Lagrangian Using the definitions introduced above at the optimum it holds moar D A min L z A Conventional descent methods are not applicable for finding saddle points Stil
13. l an iterative procedure called the dual method can be constructed as a sequence of repeated minimizations while X is adjusted 1 Given L x f x Ag x assume some initial A AF k 0 2 Obtain the intermediate optimal solution x A min L a A 3 Modify A 4 Repeat steps 2 and 3 until convergence An open question here is how to modify A One possibility is to change it by the value of the constraint g at current point x multiplied by certain factor 1 u Ab A arta The reasoning behind this formula goes like this If constraint g is still violated at the saddle point this means that the lagrangian multiplier is not large enogh In case the iterative process does not converge this indicates that factor u should be decreased The resulting algorithm is presented as a flowchart in Figure 5 2 To make the update rule for A mathematically sound we need to augment the Lagrangian function with an extra term which is discussed in the next section 23 CHAPTER 5 AUGMENTED LAGRANGIAN PROCEDURE pe fF He A Ly Lo a A H Ea a LL A DT Ly a a HA Figure 5 2 The Dual Method 24 CHAPTER 5 AUGMENTED LAGRANGIAN PROCEDURE 5 4 The Augmented Lagrangian Procedure The ALP see 5 17 5 is similar to the penalty method in that a penalty term is added see Section 3 4 But to avoid an ill conditioned problem in the end it is added not to the objective function itself but ra
14. nce of increasingly better approximations is generated via the following update rule La TQ Vy zx The value for x when 0 and the value of a are parameters to the procedure The deriva tive Vy can be evaluated either numerically or analytically as was mentioned in section 2 2 Gradient calculation The update rule is illustrated in Figure 1 2 tan 0 Mme Figure 2 2 One iteration of the iterative minimization procedure The overall procedure flow may be summarized in pseudo code as follows assuming that the functions y y_prime and converged have been suitably defined CHAPTER 2 UNCONSTRAINED OPTIMIZATION IN ONE DIMENSION Algorithm 2 1 10 minimize_iterative x0 alpha 20 i 0 30 xi_1 x0 40 yi_1 y xi_1 50 do 60 isai t 1 70 xi xi_l 80 yi yi_l 100 xi_1 xi alpha y_prime xi 110 yi_1 y xi_1 120 while not converged xi yi xi_1 yi_1 i 130 return xi_1 As can be seen the call to converged in line 120 accepts both the old and the updated values of x and y The next section describes how these values are used to halt the procedure in due time 2 4 Termination Criteria Three different criteria are implemented and can be applied either individually or in com bination In every case the purpose is to halt the optimization procedure when no further progress is envisioned i e when x would stay almost constant in further iterations The criteria are as
15. niques Philadelphia SIAM 1990 Haftka Raphael T abd G rdal Zafer Elements of structural optimization Dordrecht Kluwer Academic Publishers 1992 Nocedal Jorge and Wright Stephen Numerical Optimization New York Springer Verlag 2000 29
16. nn Technische Universit t M nchen Aerospace Department Ki Institute of Lightweight Structures en Univ Prof Dr Ing Horst Baier LLB age LLBo Applets User Guide Author Alexei Perro B Sc Supervisor Erich Wehrle M Sc Date September 9 2010 Abstract Application of concepts introduced in the course Multidisciplinary Design Optimization often requires skills that cannot be acquired from theoretical studies only This set of interactive applets is meant to illustrate the most fundamental notions in a hands on way This guide can serve as a reference while using those applets and also provides some theoretical background on relevant topics Keywords Minimization constrained optimization finite differencing feasible directions SUMT Kuhn Tucker conditions fiber reinforced plate Lagrangian dual method ALP Technische Universitat M nchen Lehrstuhl f r Leichtbau Boltzmannstr 15 85748 Garching Contents List of Figures 1 2 Introduction Unconstrained Optimization in One Dimension 21 OyervieWw II 2 2 The Graphical User Interface 2 2222 Coon 2 3 The Iterative Minimization Procedure 2 4 Termination Criteria cc cocina aaa a aaa Constrained Optimization in One Dimension A a ee we Be eg eee Be ee oe ee 3 2 The Graphical User Interface 2 2 22 Co oo mon 3 3 The Method of Usable Feasible Direction 3 4 Sequential Unconstrained Minimization Technique
17. on is obtaining the La grangian multipliers regardless of whether the point x under consideration is an optimum 19 CHAPTER 4 THE KUHN TUCKER CONDITIONS IN TWO DIMENSIONS or not The vector A is calculated under an a priory assumption that the primary condition holds OL Og Ox SE u gt j Ox Or in matrix form Vf NA 0 4 7 Here Vf is the gradient of the objective function at point x and N is defined by ni 2 The following formula from 4 approximates A as a solution to 4 7 in the least squares sense NTN NTVf 4 8 Once the multipliers are calculated the gradient of the Lagrangian VL at point x can be obtained One of the following things then happens 1 WL is zero This indicates that the formula 4 8 produced an exact solution to equation 4 7 and x may be an optimum The rest of the conditions must be checked 2 WL is nonzero This means that the initial assumption was wrong and x cannot be an optimum 20 5 Augmented Lagrangian Procedure 5 1 Overview The following sections describe the applet illustrating the dual method of constrained opti mization The sample problem is similar to the one considered previously see Section 3 but there is only one constraint instead of two The user interface controls are identical to the first three panes in the unconstrained applet see Section 2 2 and the plots are described in the next section Sections 5 3 and 5 4 outline two const
18. rained optimization strategies based on the concept of the Lagrangian The corresponding mathematical theory can be found in 5 Ch 17 e The dual method is used in this guide for introductory purposes The applet itself is implemented in a more sophisticated way e The Augmented Lagrangian Procedure ALP is a further development aiming at better stability and faster convergence The implemented algorithm is given in full detail as a flow chart in Figure 5 3 However only the most fundamental aspects are thoroughly commented upon For a detailed treatment of the topic please refer to 5 5 2 The Graphical User Interface The applet window is shown in Figure 5 1 The left part shows a plot of the objective function and the constraint On the right there is a 2D plot of level lines representing the corresponding Lagrangian function see Section 5 3 At the bottom there are three panes for controlling the optimization procedure Procedure control contains two buttons e Minimize runs the procedure with the parameters currently set and displays the inter mediate steps in the plot e Start over clears the plot to the initial state 21 CHAPTER 5 AUGMENTED LAGRANGIAN PROCEDURE Initial point provides one button and displays a message string Upon pressing Set start you will be offered to specify the initial approximation with a mouse click in either of the two plots Objective function offers a choice among the two available ob
19. ther to the Lagrangian If we take g to be again an equality for a start the Lagrangian is augmented as follows Lala dsp L a d ze f x dg 5 Po From the Kuhn Tucker condition by certain transformations it follows that near a critical point this holds yO ee The above suggests an update rule for converging to the correct value for lambda From this the following algorithmic framework results 1 Start with initial approximation o An and a chosen factor iy gt 0 Request final tolerance tol For k 0 1 2 2 Starting from zg minimize La x Az u until UL A 2 Aj 10 lt Tk The result of minimization is xj 3 If Tr lt tol we are done Else Apt An iG Ze Tk 1 T The flowchart for the resulting algorithm is shown in Figure 5 3 25 zTol zTol 3 a lt aTol amp amp Ak Ak T To xTol xTolo s maz g x 0 k 1 Q ao PE Dale cus AR SC La a sk Xe PULL FA D 2 L AF E k 2 x o0 L gk gktl st max g 0 Akl A Ak Ik y ts m AF IA lt AT ol amp amp LU il Tol xTol gt xTolMin JAR AF 1 lt ATol amp amp ILL lt tal CHAPTER 5 AUGMENTED LAGRANGIAN PROCEDURE It can be seen that the flowchart contains a number of algorithmic features in addition to those outlined in the framework They were added as improvements and are sometimes heuristic in nature The purpose and main ideas of
20. these enhancements are briefly outlined below 1 Slack variable s is used to account for the fact that g is in fact inequality constraint An expression g x s is used in place of g x and s is kept equal to max 0 g x Adjustment of a is needed to ensure stability of the search along x Armijo condition see 5 3 1 is used to determine whether o is small enough to produce sufficient decrease in the objective function value and o is a parameter to that condition Multi step optimization along x for k 1 allows to reach good approximation of the intermediate optimum during the first run along x regardless of the initial xo This allows making just one optimization step along x per every adjustment of A Adjustment of xT ol can result in this parameter changing by as much as three orders of magnitude during an optimization run It improves efficiency to opt for less precise but faster solutions while the final optimum is far away As the method converges the precision is gradually increased 27 6 Bibliography 1 2 3 4 5 Baier Horst Huber Martin Petersson gmundur and Wehrle Erich Multidisciplinary Design Optimization Lecture Notes Baier Horst Mathematische Programmierung zur Optimierung von Tragwerken ins besondere bei mehrfachen Zielen Dissertation Darmstadt 1978 Fiacco Anthony V and McCormick Garth P Nonlinear programming sequential un constrained minimization tech
21. tion in one variable Section 2 Then constraints are added to the same problem Section 3 For optimization in multiple dimensions a 2D case is taken as an example Section 4 In order to better understand the concept of the Lagrangian a method that uses it directly is also implemented Section 5 The illustrative examples enumerated above allow making many important connections to the material of the lectures 1 Hopefully experimenting with the applets can be helpful in grasping the practical implications of the theory involved Concerning the theory in both its mathematical and its algorithmical aspects the necessary remarks can be found in the text that follows Special attention is devoted to those crucial details that are covered only at an abstract level in the lecture script 1 Of course this User s Guide cannot provide the overall picture of all the relevant topics For that the lecture notes must be consulted Both detailed and comprehensive treatment of optimization theory aspects can be found in books 3 4 5 References to those are made accordingly 2 Unconstrained Optimization in One Dimension 2 1 Overview As an introductory example the problem of unconstrained minimization in 1D is considered In the following the relevant theoretical aspects are outlined and the workings of this partic ular iteractive implementation are explained The very next section describes available GUI controls and how they help to e
22. ver 4 3 holds 4 2 holds as well Therefore only 4 3 is taken into account As for the boundary allowable stresses it must be noted that their magnitudes are changed whenever the sign of the respective fiber aligned stress changes This reflects the fact that materials withstand tension and compression not equally well In this particular example the boundary values are as follows in MPa 200 for a zU O B ll 4 4 75 for ou lt 0 OB 6 1 4 5 48 fora gt 0 gt 4 6 SE foro lt 0 ee To summarize three sources of nonlinearity are present coordinate system transformation 4 1 requirement 4 3 and parameters 4 4 4 6 When acting in sequence these result in the complicated shape of the feasibility region which can be observed in Figure 4 1 4 4 Calculating the Lagrangian Multipliers The Kuhn Tucker conditions are introduced in 1 3 6 as a generalization of the Lagrange multiplier theory that is true for extrema of functions with equality restrictions The con struction of this generalization is explicated in 4 Chapter 5 There additional mathematical apparatus is used e g slack variables the notion of regular points etc With that it is possible to formulate the conditions also for the case of inequalities Besides they can be made meaningful anywhere in the design space and not just at the bounds of the feasibility region Another thing that becomes possible with the extended formulati
23. xplore the effects of various decisions in the construction of the minimization procedure Further the algorithm of root finding with the method of itera tions is explained In this applet the minimum is identified as the root of the first derivative for the details see 1 6 4 Finally the three available termination criteria are explicated with formulas 2 2 The Graphical User Interface The overall view of the applet window is shown in Figure 2 1 The left part shows a plot of the objective function and on the right there are a number of panes for controlling the optimization procedure Procedure control contains two buttons e Minimize runs the procedure with the parameters currently set and displays the inter mediate steps in the plot e Start over clears the plot to the initial state Initial point provides one button and displays a message string Upon pressing Set start you will be offered to specify the initial approximation with a mouse click in the plot Objective function offers a choice among the two available objective function Gradient calculation allows adjustment of the way in which the derivative of the objective function is evaluated CHAPTER 2 UNCONSTRAINED OPTIMIZATION IN ONE DIMENSION e Analytical the formula that results from differentiating the objective function by hand is used e Finite differences y z Az y x forward y 1 Ce 2 1 central y x yla 2i U Si 2
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