Optimization of Design. Lecturer:Dung-An Wang Lecture 12

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1 Optimization of Design Lecturer:Dung-An Wang Lecture 12

2 Lecture outline Reading: Ch12 of text Today s lecture 2

3 Constrained nonlinear programming problem Find x=(x1,..., xn), a design variable vector of dimension n, to minimize f=f(x) subject to 3

4 The method is based on linearization of the problem about the current estimate of the optimum design. Therefore, linearization of the problem is quite important and is discussed in detail. Once the problem has been linearized, it is natural to ask if it can be solved using linear programming methods. The answer is yes, and we first describe a method that is a simple extension of the Simplex method of linear programming. Then we describe the constrained steepest-descent method. 4

5 Definitions of the status of a constraint at a design point Active constraint Inactive constraint Violated constraint ε-active inequality constraint 5

6 Constraint Normalization Usually one value for ε (say 0.01) is used in checking the status of all of the constraints to check for the ε- active constraint condition. Since different constraints involve different orders of magnitude, it is not proper to use the same ε for all of the constraints unless they are normalized 6

7 In some cases, it may be better to use the constraints in their original form, especially the equality constraints. Thus, in numerical calculations some experimentation with normalization of constraints may be needed for some constraint forms 7

8 The Descent Function A function used to monitor progress toward the minimum is called the descent, or merit, function The descent function also has the property that its minimum value is the same as that of the original cost function 8

9 Convergence of an Algorithm An algorithm is said to be convergent if it reaches a local minimum point starting from an arbitrary point. The feasible set is closed if all of the boundary points are included in the set; that is, there are no strict inequalities in the problem formulation 9

10 12.2 LINEARIZATION OF THE CONSTRAINED PROBLEM At each iteration, most numerical methods for constrained optimization compute design change by solving a subproblem that is obtained by writing linear Taylor s expansions for the cost and constraint functions. Writing Taylor s expansion of the cost and constraint functions about the point x (k), we obtain the linearized subproblem 10

11 NOTATION FOR THE LINEARIZED SUBPROBLEM 11

12 DEFINITION OF THE LINEARIZED SUBPROBLEM Drop f k in the linearized cost function, the approximate subproblem, Note that f k is a constant, which does not affect solution of the linearized subproblem 12

13 EXAMPLE 12.2 DEFINITION OF A LINEARIZED SUBPROBLEM Linearize the cost and constraint functions about the point x (0) =(1,1) and write the approximate problem 13

14 Function gradients 14

15 Linearized subproblem Using the Taylor s expansion, the linearized cost function at the point (1,1) linearizing the constraint functions 15

16 16

17 Linearization in Terms of Original Variables the linearized subproblem is in terms of the design changes d 1 and d 2. We may also write the subproblem in terms of the original variables x 1 and x 2. To do this we substitute d=(x-x(0)) in all of the foregoing expressions 17

18 Since the linearized cost function is parallel to the linearized first constraint g1, the optimum solution for the linearized subproblem is any point on the line DE 18

19 EXAMPLE 12.3 Linearize the rectangular beam design problem formulated that is in Section 3.8 at the point (50,200) mm 19

20 Evaluation of problem functions At the given point 20

21 Evaluation of gradients In the following calculations, we will ignore constraints g4 and g5, assuming that they will remain satisfied; that is, the design will remain in the first quadrant gradients of the functions 21

22 Linearized subproblem 22

23 The linearized constraint functions the linearized cost function is parallel to constraint. The optimum solution lies at the point H, which is at the intersection of constraints For this point, the original constraints g1 and g2 are still violated. Apparently, for nonlinear constraints, iterations are needed to correct constraint violations and reach the feasible set 23

24 12.3 THE SEQUENTIAL LINEAR PROGRAMMING ALGORITHM Linearized functions in the variables d i can be solved by linear programming methods. Such procedures where linear programming is used to compute design change are referred to as sequential linear programming, or SLP for short The sequential linear programming algorithm is a simple and straightforward approach to solving constrained optimization problems 24

25 Move Limits in SLP To solve the LP by the standard Simplex method, the right-side parameters ei and bj must be nonnegative the problem may not have a bounded solution, or the changes in design may become too large, thus invalidating the linear approximations. Therefore, limits must be imposed on changes in design. Such constraints are usually called move limits, 25

26 Selection of Proper Move Limits Selecting proper move limits is of critical importance because it can mean success or failure of the SLP algorithm. the user should try different move limits if one specification leads to failure or improper design. Usually Δil (k) and Δiu (k) are selected as some fraction of the current design variable values (this may vary from 1 to 100 percent). If the resulting LP problem turns out to be infeasible, the move limits will need to be relaxed (i.e., larger changes in the design must be allowed 26

27 the current values of the design variables can increase or decrease. To allow for such a change, we must treat the LP variables di as free in sign. This can be done as 27

28 stopping criteria 28

29 Sequential linear programming algorithm 29

30 EXAMPLE 12.4 Consider the problem given in Example Define the linearized subproblem at the point (3, 3) and discuss its solution after imposing the proper move limits 30

31 functions and their gradients are calculated at the given point (3, 3) The given point is in the infeasible region, as the first constraint is violated 31

32 Linearized subproblem 32

33 The subproblem has only two variables, so it can be solved using the graphical solution We may choose d1=-1 and d2=-1 as the solution that satisfies all of the linearized constraints (note that the linearized change in cost is 6). If 100 percent move limits are selected then the solution to the LP subproblem must lie in the region ADEF. If the move limits are set as 20 percent of the current value of design variables, the solution must satisfy 33

34 EXAMPLE 12.5 Consider the problem given in Example Perform one iteration of the SLP algorithm, choose move limits such that a 15 percent design change is permissible 34

35 Solution The given point represents a feasible solution 35

36 linearized subproblem The linearized subproblem with 15 percent move limits on design changes d1 and d2 at the point x(0) is obtained in Example

37 solve the problem using the graphical solution Move limits of 15 percent define the solution region as DEFG. The optimum solution for the problem is at point F where d1=0.15 and d2=

38 solve the problem using the Simplex method in the linearized subproblem, the design changes d1 and d2 are free in sign. If we wish to solve the problem by the Simplex method, we must define new variables, A, B, C, and D such that 38

39 39

40 The solution to the foregoing LP problem with the Simplex method is obtained as: A=0.15,B=0, C=0.15, and D=0. Therefore, is larger than the permissible tolerance (0.001), we need to go through more iterations to satisfy the stopping criterion 40

41 The SLP Algorithm: Some Observations The method should not be used as a black box approach for engineering design problems. The selection of move limits is one of trial and error and can be best achieved in an interactive mode. The method may not converge to the precise minimum since no descent function is defined, and line search is not performed along the search direction to compute a step size. The method can cycle between two points if the optimum solution is not a vertex of the feasible set. The method is quite simple conceptually as well as numerically. Although it may not be possible to reach the precise local minimum point with it, it may be used to obtain improved designs in practice. 41

42 12.4 SEQUENTIAL QUADRATIC PROGRAMMING the SLP has some limitations, the major one being its lack of robustness. To overcome SLP s drawbacks, several other derivativebased methods have been developed to solve smooth nonlinear programming problems. gradient projection method the feasible directions method the generalized reduced gradient method Sequential quadratic programming (SQP) methods are relatively new and have become quite popular as a result of their generality, robustness, and efficiency. 42

43 SQP methods Step 1. A search direction in the design space is calculated by utilizing the values and the gradients of the problem functions; a quadratic programming subproblem is defined and solved. Step 2. A step size along the search direction is calculated to minimize a descent function; a step size calculation subproblem is defined and solved 43

44 12.5 SEARCH DIRECTION CALCULATION 1. The QP subproblem is strictly convex and therefore its minimum (if one exists) is global and unique. 2. The cost function represents a equation of a hypersphere with its center at -c (circle in two dimensions, sphere in three dimensions). 44

45 EXAMPLE 12.6 DEFINITION OF A QP SUBPROBLEM Linearize the cost and constraint functions about a point (1, 1) and define the QP subproblem 45

46 Solution constraints for the problem are already written in the normalized form graph 46

47 Evaluation of functions cost and the constraint functions are evaluated at the point (1, 1) 47

48 Gradient evaluation gradients of the cost and constraint functions 48

49 Linearized subproblem Substituting Eqs. (f) and (i) into Eq. (12.9), linearized cost function, linearized forms of the constraint functions can be written and the linearized subproblem the constant 5 has been dropped from the linearized cost function because it does not affect the solution to the subproblem. Also, the constants 3 and in the linearized constraints have been transferred to the right side 49

50 QP subproblem 50

51 Compart LP with QP subproblems Solution to LP subproblem Solution to QP subproblem 51

52 Solving the QP Subproblem EXAMPLE 12.7 SOLUTION TO THE QP SUBPROBLEM 52

53 Solution linearized cost function is modified to a quadratic function 53

54 Graphical solution 54

55 Analytical solution A numerical method must generally be used to solve the subproblem. However, since the present problem is quite simple, it can be solved by writing the KKT necessary conditions of Theorem

56 12.6 THE STEP SIZE CALCULATION SUBPROBLEM The Descent Function Recall that in unconstrained optimization methods the cost function is used as the descent function to monitor the progress of the algorithms toward the optimum point. Although the cost function can be used as a descent function with some constrained optimization methods, it cannot be used for general SQP-type methods. For most methods, the descent function is constructed by adding a penalty for constraint violations to the current value of the cost function. to determine the step size, we minimize the descent function. 56

57 EXAMPLE 12.8 CALCULATION OF DESCENT FUNCTION Taking the penalty parameter R as 10,000, calculate the value of the descent function at the point 57

58 Solution The cost and constraint functions at the given point x(0)=(40, 0.5) maximum constraint violation is determined 58

59 the descent function is calculated 59

60 Step Size Calculation: Line Search Once the search direction d(k)is determined at the current point x(k) the descent function becomes Thus the step size calculation subproblem becomes finding α to Minimize 60

61 EXAMPLE 12.9 CALCULATION OF THE DESCENT FUNCTION, follow Example 12.8 Descent function value at the initial point, α=0 The necessary condition of Eq. (12.33) is satisfied if we select the penalty parameter R as the descent function value at the starting point 61

62 Descent function value at the first trial point the maximum constraint violation 62

63 the descent function at the trial step size of α0=0.1 is given (note that the value of the penalty parameter R is not changed during step size calculation): need to continue the process of initial bracketing of the optimum step size. 63

64 Descent function value at the second trial point In the golden section search procedure, the next trial step size has an increment of (1.618xthe previous increment) next trial design point At the point (46.70, 0.618) 64

65 the minimum for the descent function has not been surpassed yet. Therefore we need to continue the initial bracketing process. The next trial step size with an increment of (1.618xthe previous increment) Value of the penalty parameter R is calculated at the beginning of the line search along the search direction and then kept fixed during all subsequent calculations for step size determination. 65

66 12.7 THE CONSTRAINED STEEPEST-DESCENT METHOD constrained steepest-descent (CSD) method has been proved to be convergent to a local minimum point starting from any point. This is considered a model algorithm that illustrates how most optimization algorithms work. Since the search direction is a modification of the steepest-descent direction to satisfy constraints, it is called the constrained steepest-descent direction. It is actually a direction obtained by projecting the steepestdescent direction on to the constraint hyperplane 66

67 The CSD Algorithm 67

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