design variable x1

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1 Multipoint linear approximations or stochastic chance constrained optimization problems with integer design variables L.F.P. Etman, S.J. Abspoel, J. Vervoort, R.A. van Rooij, J.J.M Rijpkema and J.E. Rooda Department o Mechanical Engineering, Eindhoven University o Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 23rd May Introduction In multidisciplinary analysis and optimization response surace approximations are requently applied. An important reason is that response surace techniques provide a convenient representation o data rom one discipline to other disciplines [1]. In each discipline usually one or more computationally expensive computer simulation models are involved. The response surace approximations are used to alleviate the computational burden o the optimization and circumvent discontinuities or noisy responses that are encountered. They also enable to deal in a practical way with discrete design variables: the discete optimization problem can be solved on the approximate optimization level which avoids a combinatorial increase o the number o numerical analyses (see e.g. [2]). The computational burden becomes even larger i the system behaves stochastic instead o deterministic. One may think o analysis or reliability or discrete-event simulation, where stochastic distributions are part o the modeling. In that case the stochasticity has to be accounted or during the optimization. Response surace techniques have their origin in physical experimentation and are thereore especially suitable to build deterministic approximate optimization subproblems rom stochastic data. The optimum o the deterministic approximate subproblem then has to be evaluated in a statistical sense with respect to easibility and change o objective unction value. I no urther inormation is available on the actual distributions, a statistical evaluation usually requires several replications o the numerical analysis o the proposed approximate optimum design. This paper addresses integer optimization problems with stochastic objective unction and constraints. Starting point is a sequential approximate optimization approach, where response-surace techniques are employed to build multipoint linear approximations o objective unction and constraints in search subregions o the design space. The research question is whether such an approach is able to solve a stochastic integer optimization problem in a practical valuable way within a manageable number o computer simulations. The ocus o this paper lies on optimization problems with so-called chance constraints. This type o constraints typically arise or structures, such as aircrat, where conditions on, or example, reliability are included. 1

2 2 Optimization problem ormulation The ollowing optimization problem is considered: Problem P Minimize: E[F (x;!)] subject to: P G j (x;!) > c j g 6 j = 1; : : : ; m; x l i x i x u i ; i = 1; : : : ; n; x i 2 Z; i = 1; : : : ; n: (1) Objective is to minimize the expected value o the stochastic unction F (x;!) subject to chance constraints on the stochastic unctions G j (x;!). Chance constraint P G j (x;!) > c j g 6 requires that the chance o a single realization o G j violating the constraint bound c j is smaller than (or reversely, the chance o satiying higher than 1? ). It is assumed that in problem P a unction evaluation o F and G j is computationally expensive and that design sensitivities are not available. Due to the stochastic eects (denoted by!) a replication o a simulation experiment or the same design x generally gives dierent values or F and G j. The chance constraints imply that the optimum design should be positioned at some distance d j rom the constraint boundaries c j. The smaller the larger this distance should be. Furthermore, this distance has to take into account any uncertainty with respect to the distributions o G j. In many practical applications with simulation models the actual distributions o G j are unknown and may dependent on the design variable values x. In some cases they even may ollow some non-normal distribution, which complicates probability calculations. 3 Approximate optimization approach We have developed a sequential approximate optimization strategy based on linear response-surace approximations o objective unction and constraints. Each linear approximate subproblem is built rom N experiments that are planned according to a D-optimal experimental design [3] within the search subregion under consideration. This approximate subproblem is then solved by a branch and bound integer linear programming solver. The calculated approximate design is evaluated on the basis o M replications o the simulation experiment. I the design is accepted, the design serves as starting point o a new approximate optimization cycle with a repositioned search subregion. The approximate subproblem is o the orm: Problem ~P L Minimize: ~ L subject to: ~g jl 6 c j? d j j = 1; : : : ; m; z l i 6 x i 6 z l u i = 1; : : : ; n (2) Herein, ~ L and ~g jl denote the linear approximations o objective unction and constraints obtained rom regression o the N data points. z l i 6 x i 6 z l u represents the search subregion. The distance d j is taken equal to: d j = spec g s gj (3) where s gj is the estimated standard deviation o G j. Since this standard deviation is not yet known at the approximate optimum, this value is taken equal to the standard deviation computed rom the M replications at the starting point o the approximate optimization cycle. The parameter value o spec g has to be chosen in accordance with the required condence. Typically, a small value requires a large spec g value. 2

3 4 Evaluation o designs The starting design x 0 and the optimum design x o each approximate subproblem is evaluated by M replications o the simulation experiment. Feasibility o a design is determined via the saety index dened by: gj = g j? c j s gj (4) I j <? spec g or all j = 1; : : : ; m then x is easible, otherwise x is ineasible. Such an index is commonly employed in reliability based design (see e.g. [4]). Evaluation o the change o objective unction requires a comparison o the mean objective unction values 0 and at x 0 and x. This is a standard statistical procedure, giving: =? 0 q(s 2 + s20 )=M (5) with s and s 0 being the estimated standard devations o the objective unction at the cycle optimum design x and cycle start design x 0, respectively. I? spec 6 6 spec then the objective unction values o x s and x o are considered as equal, which is one o the preconditions or convergence. Again spec has to be chosen according to the required condence level. In case o a normal distribution, values can be obtained rom a student t table. 5 Investigation We are investigating the perormance o this approximate optimization approach or the chance constrained optimization problem (1). An important aspect is the inuence o the number o experiments N and the number o replications M on the quality o the optimization outcome. To save computational cost one tries to keep N and M as low as possible while still getting satisactory optimization results. Two test problems are used to illustrate the eect o N and M. 6 Test problems 6.1 non-convex problem The rst test problem originates rom [5]. It is a 2-D analytical problem with non-convex constraints. The optimization problem is ormulated here as ollows: minimize E(F ) subject to P G 1 > 3:718g 6 with P G 2 > 15:854g 6 x 1 ; x 2 2 N F = + G j = g j + gj ; j = 1; 2 (6) =?9x x 1 x 2? 50x 1 + 8x g 1 = x 1? (0:2768x 2 2? 0:235x 2) + g1 g 2 = x 1? (?0:019x :446x 2 2? 3:98x 2) + g2 2 N(0; j0:05 j) gj 2 N(0; j0:05g j j); j = 1; 2 3

4 design variable x easible = global discrete optimum = local discrete optimum design variable x1 Figure 1: The constraints and objective unction or the non convex problem. The deterministic problem has two discrete optima: the global one at (5,3), and a local one at (3,0). The stochastic problem with = 2:5% gives a tightening o the constraint bound with two times the standard deviation. This is visualized in Figure 1. For a tightening o two times the standard deviation the two optima remain at (5,3) and (3,0). The ollowing test has been carried out. For several values o N and M stochastic problem (6) has been solved twenty times starting rom each discrete design point within the range o 0 6 x 1 ; x Parameters spec and spec g are set to 2. The outcome o the optimization runs is summarized in Table 1. For increasing M and N the requency with which (5,3) is ound increases, while the number o times a discrete neighbourhood point (DN) is ound decreases. N and M have approximately an equal eect. Also the deterministic problem has been solved or the constraints tightened with two times the standard deviation. Starting rom each grid point gives in 100% o the cases point (5,3). For N and M very large the perormance or the stochastic problem approaches the outcome o the deterministic problem. 6.2 Cantilever beam problem The second analytical test problem is a cantilever beam problem with discrete heights. The determinstic and continuous version originates rom [6]. The optimization problem is mathematically ormulated as: minimize = 0:0624(x 1 + x 2 + x 3 + x 4 + x 5 ) subject to P G > 1:0g 6 with x i 2 Z + ; i = 1; : : : ; 5 G = g + g g = x 3 1 x 3 2 x 3 3 x 3 4 x 3 5 g 2 N(0; j0:05gj) (7) The perormance o the optimization tool is investigated by starting 200 optimization runs rom initial design (10; 10; 10; 10; 10) or several values o N and M. Parameters spec and spec g are set to: spec = 0 and spec g = 2. The calculated optimum designs are compared with the optimum solutions o the deterministic problem with 2 = 10% tightened constraint bounds. These deterministic optima 4

5 No. exp. Number o exp. or and g linear Design M = 5 M = 10 M = 25 M = 2000 approx. [%] [%] [%] [%] N = 4 (5,3) DN(5,3) (3,0) DN(3,0) other N = 8 (5,3) DN(5,3) (3,0) DN(3,0) other N = 32 (5,3) DN(5,3) (3,0) DN(3,0) other N = 1024 (5,3) DN(5,3) (3,0) DN(3,0) other Table 1: Optimization results or the non-convex test problem. DN(5,3) denotes the discrete neighboorhood o (5,3). 5

6 group local optimum I II III Table 2: Local optima with corresponding objective unction value ( ) or the deterministic cantilever beam with corrected constraint. The optima have been determined by evaluation o each grid point in the space 1 6 x i 6 10, i = 1; : : : ; 5. are given in Table 2. Three groups o local deterministic optima can be identied, sharing the same objective unction value. Table 3 shows the results o the experiment. The majority o optimization runs yields a group I point or a neighbor o group I. Group II and III points are hardly ound, probably because it is unlikely to end up in these points starting rom (10; 10; 10; 10; 10). Increasing N improves the quality o the linear approximation. Table 3 shows that increasing N decreases the number o other points ound, as well as the number o neigbor points. For suciently large M (M 10) this increase is mainly due to N, and is hardly aected by M. The main inuence o N is conrmed by Table 4 where the number o dierent solutions or each combination o N and M is compared. But this Table 4 also shows that urther increasing M urther decreases the number o dierent optimum solutions ound. 7 Conclusion The proposed approximate optimization approach shows promising results to deal with stochastic chance constrained optimization problems with integer design variables. Even or small number o experiments N to build the linear approximations and a small number o replications M to evaluate candidate optimum designs, in the majority o cases a discrete optimum or one o its neighboors is ound. The variation o dierent solutions ound becomes smaller or increasing N and M. Reerences [1] Sobieszczanski-Sobieski, J.; and Hatka, R.T. 1997: Multidisciplinary aerospace design optimization: survey o recent developments. Structural Optimization 14, 1{23. [2] Korngold, J.C.; Gabriele, G.A. 1997: Multidisciplinary analysis and optimization o discrete problems using response surace methods. Journal o Mechanical Design 119, 427{433. [3] Myers, R.H.; Montgomery, D.C. 1995: Response Surace Methodology - process and product optimization using designed experiments. New York: John Wiley & Sons. 6

7 No. No. replications M points Design N [%] [%] [%] [%] 12 group I DN(group I) other group I DN(group I) other group I DN(group I) other group I DN(group I) other Table 3: Calculated optimum solutions o the stochastic cantilever beam problem starting 200 optimization runs rom (10,10,10,10,10). A solution is categorized as a group-i point o Table 2, a discrete neighbor o group I, or an other point. No. points No. replications M N Table 4: Number o dierent solutions ound or the 200 runs starting rom (10,10,10,10,10). 7

8 [4] Melchers, R.E. 1987: Structural reliability - analysis and prediction. New York: John Wiley & Sons. [5] Loh, H.T.; Papalambros, P.Y. 1991: Computational implementation and tests o a sequential linearization algorithm or mixed-discrete nonlinear design optimization. Journal o Mechanical Design 113, 335{345. [6] Svanberg, K. 1987: The method o moving asymptotes - a new method or structural optimization International Journal o Numerical Methods in Engineering 24, 359{373. 8