Constrained Optimization by the  Constrained Differential Evolution with an Archive and Gradient-Based Mutation Tetsuyuki TAKAHAMA （ Hiroshima City University ） Setsuko SAKAI （ Hiroshima Shudo University)

2010/07/21T.Takahama and S.Sakai in CEC20102 Outline n Constrained optimization problems The  constrained method  Constraint violation and  -level comparisons The  constrained differential evolution (  DEag) u differential evolution (DE) with an archive u gradient-based mutation  control of the  -level n Experimental results n Conclusions

2010/07/21T.Takahama and S.Sakai in CEC20103 Constrained Optimization Problems objective function f, decision variables x i inequality constraints g j, equality constraints h j lower bound l i, upper bound u i

2010/07/21T.Takahama and S.Sakai in CEC20104  constrained method n Algorithm transformation method  algorithm for unconstrained optimization → algorithm for constrained optimization   -level comparison F comparison between pairs of objective value and constraint violation  by replacing ordinary comparisons to  -level comparisons in unconstrained optimization algorithm

2010/07/21T.Takahama and S.Sakai in CEC20105 Constraint Violation Constraint Violation  ( x ) n max n sum

2010/07/21T.Takahama and S.Sakai in CEC20106  -level comparison Function value and constraint violation （ f,  ） u precedes constraint violation usually u precedes function value if violation is small

2010/07/21T.Takahama and S.Sakai in CEC20107 Definition of  constrained method Constrained problems can be solved by replacing ordinary comparisons with  level comparisons in unconstrained optimization algorithm  ＜ → ＜  ， →  ∥

2010/07/21T.Takahama and S.Sakai in CEC20108 Differential Evolution (DE) n simple operation avoiding step size control n trial vector (child) will survive if the child is better n robust to non-convex, multi-modal problems population difference vector - F base vector parent crossover (CR) + trial vector

2010/07/21T.Takahama and S.Sakai in CEC20109  DEa:  DE with an archive (1) n A small population and a large archive are adopted u Small population is good for search efficiency but is bad for diversity n Generate M initial individuals  A={ x k | k=1,2,...,M } (M=100 n ) n Select top N individuals from A as an initial population  P={ x i | i=1,2,...,N } (N=4 n ) u A=A-P A PN M-N

2010/07/21T.Takahama and S.Sakai in CEC  DE with an archive (2) n DE/rand/1/exp operation u mutant vector: u and are selected from P u is selected from P A w.p or P w.p u exponential crossover n Uniform convergence of individuals u When a parent generates a child and the child is not better than the parent, the parent can generate another child correction of Fig.2

2010/07/21T.Takahama and S.Sakai in CEC  DE with an archive (3) n Direct replacement for efficiency u Continuous generation model  If the child is better than the parent, the parent is directly replaced by the child ( f ( x trial ),  ( x trial )) ＜  ( f ( x i ),  ( x i )) n Perturb scaling factor F in small probability u to escape from local minima u F is a fixed value (0.5) w.p 0.95 u F=1+|C(0,0.05)| truncated to 1.1 w.p. 0.05

2010/07/21T.Takahama and S.Sakai in CEC Gradient-based mutation (1) n adopts the gradient of constraints to reach feasible region n Constraint vector and constraint violation vector n Gradient of constraint vector

2010/07/21T.Takahama and S.Sakai in CEC Gradient-based mutation (2) n inverse cannot be defined generally n Moore-Penrose inverse (pseudoinverse) u approximate or best (LSE) solution n Modifications  Numerical gradient (costs n+1 FEs)  Mutation is applied only in every n generations u Skipped w.p. 0.5, if num. of violated constraints is one

2010/07/21T.Takahama and S.Sakai in CEC Control of  -level Small feasible region and  -level

2010/07/21T.Takahama and S.Sakai in CEC Control scheme of  -level   -level should converge to 0 gradually

2010/07/21T.Takahama and S.Sakai in CEC control of cp instead of specifying cp, specify  -level at T u, n To search better objective value  generation from T  to Tc  enlarge  -level and scaling factor F

2010/07/21T.Takahama and S.Sakai in CEC Effectiveness of  constrained method The  level comparison does not need objective values if one of the constraint violations is larger than  -level n Lazy evaluation u objective function is evaluated only when needed u evaluation of objective function can be often omitted when feasible region is small

2010/07/21T.Takahama and S.Sakai in CEC Conditions of experiments n 18 constrained problems, 25 trials per a problem  DEag/rand/1/exp  Max. FEs: 20,000 n  M=100 n, N=4 n, F=0.5, CR=0.9   level control:  =0.9, T c =1,000, T =0.95T c n Gradient-based mutation u mutation rate: P g =0.1, max. iterations: R g =3  applied only in every n generations

2010/07/21T.Takahama and S.Sakai in CEC Summary of Results Feasible and stable solutions in all runs  10D: C01-C07, C09, C10, C12-C14, C18 (13)  30D: C01, C02, C05-C08, C10, C13-C16 (11) Feasible solutions in all runs  10D: C08, C11, C15, C16, C17 (5)  30D: C03, C04, C09, C11, C17, C18 (6) Often infeasible solutions  30D: C12 (1)

2010/07/21T.Takahama and S.Sakai in CEC D (C01-C06)

2010/07/21T.Takahama and S.Sakai in CEC D (C07-C012)

2010/07/21T.Takahama and S.Sakai in CEC D (C13-C018)

2010/07/21T.Takahama and S.Sakai in CEC D (C01-C06)

2010/07/21T.Takahama and S.Sakai in CEC D (C07-C12)

2010/07/21T.Takahama and S.Sakai in CEC D (C13-C18)

2010/07/21T.Takahama and S.Sakai in CEC Computational Complexity n T 1 : Time (seconds) of 10,000 function evaluations for a problem on average n T 2 : Time (seconds) of 10,000 function evaluation with algorithm for a problem

2010/07/21T.Takahama and S.Sakai in CEC Conclusions  DE with a large archive and gradient-based mutation u can find feasible solutions in all run and all problems except for C12 of 30D u can often omit evaluation of objective values and find solutions efficiently and very fast

2010/07/21T.Takahama and S.Sakai in CEC Future works n To find better objective values  dynamic control of  level  changing  level according to the number of feasible points u mechanism for maintaining diversity F subpopulations or species to search various regions F adaptive control of F and CR

2010/07/21T.Takahama and S.Sakai in CEC n Thank you for your kind attention

2010/07/21T.Takahama and S.Sakai in CEC D problems

2010/07/21T.Takahama and S.Sakai in CEC D problems

2010/07/21T.Takahama and S.Sakai in CEC D problems

2010/07/21T.Takahama and S.Sakai in CEC D problems

2010/07/21T.Takahama and S.Sakai in CEC Moore-Penrose inverse