Tetsuyuki TAKAHAMA （ Hiroshima City University ） Setsuko SAKAI （ Hiroshima Shudo University)

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

2. Outline • Constrained optimization problems • The e constrained method • Constraint violation and e-level comparisons • The e constrained differential evolution (eDEag) • differential evolution (DE) with an archive • gradient-based mutation • control of the e-level • Experimental results • Conclusions T.Takahama and S.Sakai in CEC2010

3. Constrained Optimization Problems • objective function f ,decision variables xi • inequality constraints gj,equality constraints hj • lower bound li, upper bound ui T.Takahama and S.Sakai in CEC2010

4. e constrained method • Algorithm transformation method • algorithm for unconstrained optimization→ algorithm for constrained optimization • e-level comparison • comparison between pairs of objective value and constraint violation • by replacing ordinary comparisons to e-level comparisons in unconstrained optimization algorithm T.Takahama and S.Sakai in CEC2010

5. Constraint Violation • Constraint Violation f (x) • max • sum T.Takahama and S.Sakai in CEC2010

6. e-level comparison • Function value and constraint violation（f ,f） • precedes constraint violation usually • precedes function value if violation is small T.Takahama and S.Sakai in CEC2010

7. Definition of e constrained method • Constrained problems can be solved by replacing ordinary comparisons with e level comparisons in unconstrained optimization algorithm • ＜→＜e， → e ∥ T.Takahama and S.Sakai in CEC2010

8. population crossover (CR) parent + base vector trial vector - F difference vector Differential Evolution (DE) • simple operation avoiding step size control • trial vector (child) will survive if the child is better • robust to non-convex, multi-modal problems T.Takahama and S.Sakai in CEC2010

9. N P M-N A eDEa: eDE with an archive (1) • A small population and a large archive are adopted • Small population is good for search efficiencybut is bad for diversity • Generate M initial individuals • A={ xk | k=1,2,...,M } (M=100n) • Select top N individuals from Aas an initial population • P={ xi | i=1,2,...,N } (N=4n) • A=A-P T.Takahama and S.Sakai in CEC2010

10. eDE with an archive (2) correction of Fig.2 • DE/rand/1/exp operation • mutant vector: • and are selected from P • is selected from P A w.p. 0.95 or P w.p. 0.05 • exponential crossover • Uniform convergence of individuals • When a parent generates a child andthe child is not better than the parent,the parent can generate another child T.Takahama and S.Sakai in CEC2010

11. eDE with an archive (3) • Direct replacement for efficiency • Continuous generation model • If the child is better than the parent,the parent is directly replaced by the child (f(xtrial),f(xtrial)) ＜e (f(xi), f(xi)) • Perturb scaling factor F in small probability • to escape from local minima • F is a fixed value (0.5) w.p 0.95 • F=1+|C(0,0.05)| truncated to 1.1 w.p. 0.05 T.Takahama and S.Sakai in CEC2010

12. Gradient-based mutation (1) • adopts the gradient of constraints to reach feasible region • Constraint vector and constraint violation vector • Gradient of constraint vector T.Takahama and S.Sakai in CEC2010

13. Gradient-based mutation (2) • inverse cannot be defined generally • Moore-Penrose inverse (pseudoinverse) • approximate or best (LSE) solution • Modifications • Numerical gradient (costs n+1 FEs) • Mutation is applied only in every n generations • Skipped w.p. 0.5, if num. of violated constraints is one T.Takahama and S.Sakai in CEC2010

14. Control of e-level • Small feasible region and e-level T.Takahama and S.Sakai in CEC2010

15. Control scheme of e-level • e-level should converge to 0 gradually T.Takahama and S.Sakai in CEC2010

16. control of cp • instead of specifying cp, specify e-level at Tl • , • To search better objective value • generation from Tl to Tc • enlarge e-level and scaling factor F T.Takahama and S.Sakai in CEC2010

17. Effectiveness of e constrained method • The e level comparison does not need objective values if one of the constraint violations is larger than e-level • Lazy evaluation • objective function is evaluated only when needed • evaluation of objective function can be often omitted when feasible region is small T.Takahama and S.Sakai in CEC2010

18. Conditions of experiments • 18 constrained problems, 25 trials per a problem • eDEag/rand/1/exp • Max. FEs: 20,000n • M=100n, N=4n, F=0.5, CR=0.9 • e level control: q=0.9, Tc=1,000, Tl=0.95Tc • Gradient-based mutation • mutation rate: Pg=0.1, max. iterations: Rg=3 • applied only in every n generations T.Takahama and S.Sakai in CEC2010

19. 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) T.Takahama and S.Sakai in CEC2010

20. 10D (C01-C06) T.Takahama and S.Sakai in CEC2010

21. 10D (C07-C012) T.Takahama and S.Sakai in CEC2010

22. 10D (C13-C018) T.Takahama and S.Sakai in CEC2010

23. 30D (C01-C06) T.Takahama and S.Sakai in CEC2010

24. 30D (C07-C12) T.Takahama and S.Sakai in CEC2010

25. 30D (C13-C18) T.Takahama and S.Sakai in CEC2010

26. Computational Complexity • T1: Time (seconds) of 10,000 function evaluations for a problem on average • T2: Time (seconds) of 10,000 function evaluation with algorithm for a problem T.Takahama and S.Sakai in CEC2010

27. Conclusions • eDE with a large archive and gradient-based mutation • can find feasible solutions in all run and all problems except for C12 of 30D • can often omit evaluation of objective values and find solutions efficiently and very fast T.Takahama and S.Sakai in CEC2010

28. Future works • To find better objective values • dynamic control of e level • changing e level according to the number of feasible points • mechanism for maintaining diversity • subpopulations or species to search various regions • adaptive control of F and CR T.Takahama and S.Sakai in CEC2010

29. Thank you for your kind attention T.Takahama and S.Sakai in CEC2010

30. 10D problems T.Takahama and S.Sakai in CEC2010

31. 10D problems T.Takahama and S.Sakai in CEC2010

32. 30D problems T.Takahama and S.Sakai in CEC2010

33. 30D problems T.Takahama and S.Sakai in CEC2010

34. Moore-Penrose inverse T.Takahama and S.Sakai in CEC2010