Online Supplement for. Engineering Optimization

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1 Online Supplement for Constrained Optimization by Radial Basis Function Interpolation for High-Dimensional Expensive Black-Box Problems with Infeasible Initial Points Engineering Optimization Rommel G. Regis Department of Mathematics, Saint Joseph s University Philadelphia, Pennsylvania 19131, USA, rregis@sju.edu Appendix A. Information on Test Problems Table A1 provides some information on the test problems used in this study. In particular, it gives the number of variables, the number of inequality constraints, the region defined by the bound constraints, and the global minimum or best known feasible objective function value for each problem. Table A1: Constrained optimization test problems. d is the number of decision variables and m is the number of inequality constraints. (If A and B are sets, A B := {(a, b) : a A, b B}. Test Problem d m Region Defined by Global Minimum or Bound Constraints Best Known Value G6 2 2 [13, 100] [0, 100] G8 2 2 [0, 10] WB4 4 6 [0.125, 10] [0.1, 10] GTCD4 4 1 [20, 50] [1, 10] [20, 50] [0.1, 60] PVD4 4 3 [0, 1] 2 [0, 50] [0, 240] G5MOD 4 5 [0, 1200] 2 [ 0.55, 0.55] G4 5 6 [78, 102] [33, 45] [27, 45] Hesse 6 6 [0, 5] [0, 4] [1, 5] 310 [0, 6] [1, 5] [0, 10] SR [2.6, 3.6] [0.7, 0.8] [17, 28] [7.3, 8.3] 2 [2.9, 3.9] [5.0, 5.5] G9 7 4 [ 10, 10] G [10 2, 10 4 ] [10 3, 10 4 ] 2 [10, 10 3 ] G [0, 10] G [ 10, 10] G3MOD 20 1 [0, 1] G [0, 1] 9 [0, 100] 3 [0, 1] 15 G13MOD 5 3 [ 2.3, 2.3] 2 [ 3.2, 3.2] G [ , ] [68.6, ] [0, ] [193, ] [25, ] G [ 10, 10] 8 [0, 20] G [0, 10] G [0, 3] [0, 4] MOPTA [0, 1]

2 B. Additional Numerical Results B1. Best Feasible Objective Function Values on the MOPTA08 Automotive Problem Table B2: Best feasible objective function value obtained by the different algorithms on the MOPTA08 problem after 1000, 2000, 3000 and 4000 function evaluations evaluations evaluations evaluations evaluations COBRA-Local Trial COBRA-Global Trial ConstrLMSRBF ConstrLMSRBF-BCS Trial Trial Trial Trial Trial Trial Trial Trial SDPEN Trial NOMADm-DACE Trial GLOBALm-SOLNP GLOBALm-UNIRANDI Global Search Trials 1 to Trial Trial Trial Trial Trials 1 to

3 B2. Dot Plots of Algorithm Performance on the MOPTA08 Problem Results of COBRA Local on MOPTA08 0 violations 1 5 violations >5 violations Results of COBRA Global on MOPTA08 0 violations 1 5 violations >5 violations objective function value objective function value number of objective and constraint function evaluations number of objective and constraint function evaluations Results of SDPEN on MOPTA08 0 violations 1 5 violations >5 violations Results of the GLOBALm SOLNP Algorithm on MOPTA08 0 violations 1 5 violations >5 violations objective function value objective function value number of objective and constraint function evaluations number of objective and constraint function evaluations Results of the GLOBALm UNIRANDI Algorithm on MOPTA08 0 violations 1 5 violations >5 violations Results of the Global Search Algorithm (Matlab) on MOPTA violations >5 violations objective function value objective function value number of objective and constraint function evaluations number of objective and constraint function evaluations Figure B1: Objective function values and number of constraint violations of the points in the trajectory of algorithms on the MOPTA08 problem. The horizontal line represents an objective function value of

4 B3. Results on the Test Problems Table B3: Statistics on the best feasible objective value for 30 trials obtained by optimization algorithms after 100 function evaluations (50 simulations). SR7 WB4 GTCD4 PVD4 G3MOD G7 G2 G10 G9 best known value COBRA-Local COBRA-Global ConstrLMSRBF SDPEN NOMADm-DACE GLOBALm-SOLNP GLOBALm-UNIRANDI GlobalSearch best worst median mean std error best worst median mean std error best worst median mean std error best worst median mean std error best worst median mean std error best worst median mean std error best worst median mean std error best worst median mean std error

5 Table B3: (Continued) Statistics on the best feasible objective value for 30 trials obtained by optimization algorithms on the test problems after 100 function evaluations (50 simulations). Hesse G4 G5MOD G8 G6 best known value best worst COBRA-Local median mean std error best worst COBRA-Global median mean std error best worst ConstrLMSRBF median mean std error best worst SDPEN median mean std error best worst NOMADm-DACE median mean std error best worst GLOBALm-SOLNP median mean std error best worst GLOBALm-UNIRANDI median mean std error best worst GlobalSearch median mean std error B4. Average Running Times Table B4: Mean running times of the different algorithms on the MOPTA08 problem (excluding time spent on function evaluations) on an Intel(R) Core(TM) i7 CPU Ghz desktop machine. Algorithm After 2000 function evaluations After 4000 function evaluations COBRA-Local sec (11.05 hrs) sec (41.74 hrs) COBRA-Global sec (15.68 hrs) sec (47.92 hrs) SDPEN sec sec GLOBALm-SOLNP sec sec GLOBALm-UNIRANDI sec sec Global Search sec sec 5

6 C. Test Problems for Constrained Optimization Below are the test problems used in this study. Four of these are engineering design problems and these are the Welded Beam Design Problem (WB4) (Coello Coello and Mezura-Montes 2002, Hedar 2004), Pressure Vessel Design Problem (PVD4) (Coello Coello and Mezura-Montes 2002, Hedar 2004), Gas Transmission Compressor Design Problem (GTCD) (Beightler and Phillips 1976), and Speed Reducer Design for small aircraft engine (SR7) (Floudas and Pardalos 1990). Nine of the test problems are from the well-known constrained optimization test problems in Michalewicz and Schoenauer (1996). These are labeled G2, G3MOD, G4, G5MOD, G6, G7, G8, G9, and G10. The G3MOD and G5MOD problems are obtained from G3 and G5 by replacing all equality constraints with inequality constraints. The Hesse problem is from Hesse (1973). In addition, the four test problems G16, G18, G19 and G24 are found in (Mezura-Montes and Cetina-Dominguez 2012). As mentioned earlier, some of the constraint functions are modified by either dividing by a positive constant or by applying a logarithmic transformation without changing the feasible region. A similar modification of the constraint functions was performed by Jones (2008) on the MOPTA08 problem so that the constraints are well-normalized. The plog transformation used in some of the constraints was introduced in Regis and Shoemaker (2012) and it is defined by { log(1 + x) if x 0 plog(x) = log(1 x) if x < 0 where log is the natural logarithm. The mathematical properties of this transformation are discussed in Regis and Shoemaker (2012). In particular, it is strictly increasing, symmetric with respect to the origin, and it tones down extremely high or extremely negative function values. Welded Beam (Deb 2000, Coello Coello and Mezura-Montes 2002, Hedar 2004, Hedar and Fukushima 2006): f(x) = x 2 1x x 3 x 4 ( x 2 ) P = 6000, L = 14, E = , G = t max = 13600, s max = 30000, x max = 10, d max = 0.25 M = P (L + x 2 /2), R = 0.25(x (x 1 + x 3 ) 2 ), J = 2x 1 x 2 (x 2 2/ (x 1 + x 3 ) 2 ) ( ) P c = 4.013E 6L x 2 3 x 3 E/G x 3 L t 1 = P/( 2x 1 x 2 ), t 2 = MR/J t = t t 1t 2 x 2 /R + t 2 2 s = 6P L/(x 4 x 2 3), d = 4P L 3 /(Ex 4 x 3 3) g 1 (x) = (t t max )/t max g 2 (x) = (s s max )/s max g 3 (x) = (x 1 x 4 )/x max g 4 (x) = x x 3 x 4 ( x 2 ) g 5 (x) = (d d max )/d max g 6 (x) = (P P c )/P x 1 10, 0.1 x i 10 for i = 2, 3, 4 6

7 Pressure Vessel Design (Coello Coello and Mezura-Montes 2002, Hedar 2004, Hedar and Fukushima 2006): f(x) = x 1 x 3 x x 2 x x 2 1x x 2 1x 3 g 1 (x) = x x 3 g 2 (x) = x x 3 g 3 (x) = plog( πx 2 3x πx ) 0 x 1, x 2 1, 0 x 3 50, 0 x Speed Reducer (Floudas and Pardalos 1990): f(x) = x 1 x 2 2A 1.508x 1 B C D where A = x x , B = x x 2 7, C = x x 3 7, D = x 4 x x 5 x 2 7 g 1 (x) = (27 x 1 x 2 2x 3 )/27 g 2 (x) = (397.5 x 1 x 2 2x 2 3)/397.5 g 3 (x) = (1.93 (x 2 x 4 6x 3 )/x 3 4)/1.93 g 4 (x) = (1.93 (x 2 x 4 7x 3 )/x 3 5)/1.93 A1 = [ (745x 4 /(x 2 x 3 )) 2 + ( ) ] 0.5, B1 = 0.1x 3 6 g 5 (x) = ((A1/B1) 1100)/1100 A2 = [ (745x 5 /(x 2 x 3 )) 2 + ( ) ] 0.5, B2 = 0.1x 3 7 g 6 (x) = ((A2/B2) 850)/850 g 7 (x) = (x 2 x 3 40)/40 g 8 (x) = (5 (x 1 /x 2 ))/5 g 9 (x) = ((x 1 /x 2 ) 12)/12 g 10 (x) = ( x 6 x 4 )/1.9 g 11 (x) = ( x 7 x 5 )/ x 1 3.6, 0.7 x 2.8, 17 x 3 28, 7.3 x 4, x 5 8.3, 2.9 x 6 3.9, 5.0 x Gas Transmission Compressor Design (Beightler and Phillips 1976): f(x) = ( )x 1/2 1 x 2 x 2/3 3 x 1/2 4 + ( )x 3 + ( )x 1 1 x ( )x 1 1 g 1 (x) = x 4 x x x 1 50, 1 x 2 10, 20 x 3 50, 0.1 x 4 60 G2 (Michalewicz and Schoenauer 1996) (d = 10): d f(x) = i=1 cos4 (x i ) 2 d i=1 cos2 (x i ) d i=1 ix2 i ( ) d g 1 (x) = plog x i / plog(10 d ) ( i=1 d ) g 2 (x) = x i 7.5d / (2.5d) i=1 0 x i 10 for i = 1, 2,..., d 7

8 G3MOD (Michalewicz and Schoenauer 1996) (d = 20): ( f(x) = plog ( ) d d) d x i i=1 d g 1 (x) = x 2 i 1 i=1 0 x i 1 for i = 1, 2,..., d G4 (Michalewicz and Schoenauer 1996): f(x) = x x 1 x x u = x 2 x x 1 x x 3 x 5 g 1 (x) = u g 2 (x) = u 92 v = x 2 x x 1 x x 2 3 g 3 (x) = v + 90 g 4 (x) = v 110 w = x 3 x x 1 x x 3 x 4 g 5 (x) = w + 20 g 6 (x) = w x 1 102, 33 x 2 45, 27 x i 45 for i = 3, 4, 5 G5MOD (Michalewicz and Schoenauer 1996): f(x) = 3x x x 2 + ( /3)x 3 2 g 1 (x) = x 3 x g 2 (x) = x 4 x g 3 (x) = 1000 sin( x ) sin( x ) x 1 g 4 (x) = 1000 sin(x ) sin(x 3 x ) x 2 g 5 (x) = 1000 sin(x ) sin(x 4 x ) x 1, x , 0.55 x 3, x 4.55 G6 (Michalewicz and Schoenauer 1996): f(x) = (x 1 10) 3 + (x 2 20) 3 g 1 (x) = ( (x 1 5) 2 (x 2 5) )/100 g 2 (x) = ((x 1 6) 2 + (x 2 5) )/ x 1 100, 0 x

9 G7 (Michalewicz and Schoenauer 1996): f(x) = x x x 1 x 2 14x 1 16x 2 + (x 3 10) 2 + 4(x 4 5) 2 + (x 5 3) 2 +2(x 6 1) 2 + 5x (x 8 11) 2 + 2(x 9 10) 2 + (x 10 7) g 1 (x) = 4x 1 + 5x 2 3x 7 + 9x g 2 (x) = 10x 1 8x 2 17x 7 + 2x g 3 (x) = 8x 1 + 2x 2 + 5x 9 2x g 4 (x) = 3(x 1 2) 2 + 4(x 2 3) 2 + 2x 2 3 7x g 5 (x) = 5x x 2 + (x 3 6) 2 2x g 6 (x) = 0.5(x 1 8) 2 + 2(x 2 4) 2 + 3x 2 5 x g 7 (x) = x (x 2 2) 2 2x 1 x x 5 6x g 8 (x) = 3x 1 + 6x (x 9 8) 2 7x x i 10 for i = 1, 2,..., 10 G8 (Michalewicz and Schoenauer 1996): f(x) = sin3 (2πx 1 ) sin(2πx 2 ) x 3 1 (x 1 + x 2 ) g 1 (x) = x 2 1 x g 2 (x) = 1 x 1 + (x 2 4) 2 0 x 1, x 2 10 G9 (Michalewicz and Schoenauer 1996): f(x) = (x 1 10) 2 + 5(x 2 12) 2 + x (x 4 11) x x x 4 7 4x 6 x 7 10x 6 8x 7 g 1 (x) = (2x x x 3 + 4x x 5 127)/127 g 2 (x) = (7x 1 + 3x x x 4 x 5 282)/282 g 3 (x) = (23x 1 + x x 2 6 8x 7 196)/196 g 4 (x) = 4x x 2 2 3x 1 x 2 + 2x x 6 11x 7 10 x i 10 for i = 1,..., 7 9

10 G10 (Michalewicz and Schoenauer 1996): f(x) = x 1 + x 2 + x 3 g 1 (x) = (x 4 + x 6 ) g 2 (x) = ( x 4 + x 5 + x 7 ) g 3 (x) = ( x 5 + x 8 ) g 4 (x) = plog(100x 1 x 1 x x ) g 5 (x) = plog(x 2 x 4 x 2 x x x 5 ) g 6 (x) = plog(x 3 x 5 x 3 x x ) 10 2 x , 10 3 x 2, x , 10 x i 10 3 for i = 4, 5,..., 8 Hesse (1973): f(x) = 25(x 1 2) 2 (x 2 2) 2 (x 3 1) 2 (x 4 4) 2 (x 5 1) 2 (x 6 4) 2 g 1 (x) = (2 x 1 x 2 )/2 g 2 (x) = (x 1 + x 2 6)/6 g 3 (x) = ( x 1 + x 2 2)/2 g 4 (x) = (x 1 (3x 2 ) 2)/2 g 5 (x) = (4 (x 3 3) 2 x 4 )/4 g 6 (x) = (4 (x 5 3) 2 x 6 )/4 0 x 1 5, 0 x 2 4, 1 x 3 5, 0 x 4 6, 1 x 5 5, 0 x 6 10 G18 (Mezura-Montes and Cetina-Dominguez 2012): f(x) = 0.5(x 1 x 4 x 2 x 3 + x 3 x 9 x 5 x 9 + x 5 x 8 x 6 x 7 ) g 1 (x) = x x g 2 (x) = x g 3 (x) = x x g 4 (x) = x (x 2 x 9 ) 2 1 g 5 (x) = (x 1 x 5 ) 2 + (x 2 x 6 ) 2 1 g 6 (x) = (x 1 x 7 ) 2 + (x 2 x 8 ) 2 1 g 7 (x) = (x 3 x 5 ) 2 + (x 4 x 6 ) 2 1 g 8 (x) = (x 3 x 7 ) 2 + (x 4 x 8 ) 2 1 g 9 (x) = x (x 8 x 9 ) 2 1 g 10 (x) = x 2 x 3 x 1 x 4 g 11 (x) = x 3 x 9 g 12 (x) = x 5 x 9 g 13 (x) = x 6 x 7 x 5 x 8 10 x i 10 for i = 1,..., 8, 0 x 9 20 G19 (Mezura-Montes and Cetina-Dominguez 2012): f(x) = 5 5 j=1 i=1 c ijx (10+i) x (10+j) j=1 d jx 3 (10+j) 10 i=1 b ix i g j (x) = 2 5 i=1 c ijx (10+i) e j + 10 i=1 a ijx i, j = 1,..., 5 where 0 x i 10 for i = 1,..., 15 10

11 b = [ 40, 2, 0.25, 4, 4, 1, 40, 60, 5, 1] c = e = [ 15, 27, 36, 18, 12] a = d = [4, 8, 10, 6, 2] G24 (Mezura-Montes and Cetina-Dominguez 2012): f(x) = x 1 x 2 g 1 (x) = 2x x 3 1 8x x 2 2 g 2 (x) = 4x x x x 1 + x x 1 3, 0 x 2 4 References [1] Beightler, C. S., Phillips, D. T., Applied Geometric Programming. Wiley, New York. [2] Coello Coello, C. A., Mezura-Montes, E., Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Advanced Engineering Informatics, 16, [3] Deb, K., An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering, 186(2 4), [4] Floudas, C. A., Pardalos, P. M A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, Berlin. [5] Hedar, A., Studies on Metaheuristics for Continuous Global Optimization Problems. Ph.D. thesis. Kyoto University, Kyoto, Japan. [6] HedFuk06 Hedar, A., Fukushima, M., Derivative-free filter simulated annealing method for constrained continuous global optimization. Journal of Global Optimization, 35, [7] Hesse, R., A heuristic search procedure for estimating a global solution of nonconvex programming problems. Operations Research, 21,

12 [8] Jones, D.R., Large-scale multi-disciplinary mass optimization in the auto industry. Presented at the Modeling and Optimization: Theory and Applications (MOPTA) 2008 Conference. Ontario, Canada. [9] Michalewicz, Z., Schoenauer, M Evolutionary algorithms for constrained parameter optimization problems. Evolutionary Computation, 4(1):1 32, [10] Mezura-Montes, E., Cetina-Dominguez, O., Empirical analysis of a modified artificial bee colony for constrained numerical optimization. Applied Mathematics and Computation, 218(22), [11] Regis, R.G., Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions. Computers and Operations Research, 38(5), [12] Regis, R.G., Shoemaker, C. A., A quasi-multistart framework for global optimization of expensive functions using response surface models. Journal of Global Optimization, DOI: /s , in press. 12