Equality Constrained Multi-object

Transcript

1. Equality Constrained Multi-Objective Optimization Amit Saha, Tapabrata Ray MDO Group, University of New South Wales, Australia IEEE CEC 2012, Brisbane, Australia May 22, 2012
2. Outline Introduction Proposed Work Results Conclusion
3. Equality Constrained Multi-objective Optimization Problem Generic Problem Structure: Min. [f1 (x), f2 (x)..fM (x)] Subject to gj (x) ≥ 0, j = 1, .., J, hk (x) = 0, k = 1, .., K, xl i ≤ xi ≤ xu i , i = 1, .., n. (1) where gj is an inequality constraint and hk is an equality constraint.
4. Past Work ◮ Lukewarm Interest so far ◮ Studies on equality constrained MO problems are few ◮ Recent equality constraint handling work focuses on SO optimization problems ◮ Need for a study on Equality constrained MO optimization problems
5. Past Work ◮ Lukewarm Interest so far ◮ Studies on equality constrained MO problems are few ◮ Recent equality constraint handling work focuses on SO optimization problems ◮ Need for a study on Equality constrained MO optimization problems
6. Most-Probable-Point (of Failure) Originally proposed in the context of Reliability Based Design Optimization X−space O A C D B X* h(x) = 0 ÁÒ × Ð Ö ÓÒ¸ h(x) = 0 U1 ÁÒ × Ð Ö ÓÒ¸ h(x) = 0 X U2 Figure: RIA approach: Finding the nearest feasible point, X∗, which is a candidate solution for repairing the infeasible solution, X.
7. RIA approach: Constrained Optimization Problem ◮ Objective function for RIA: ||U|| = ||X∗u − Xu || (2) ◮ Constraint function for RIA: The constraint for the RIA optimization exercise is the constraint of the original problem ◮ Solved using MATLAB’s fmincon function
8. NSGA-II with MPP based Repair NSGA-II_MPP: ◮ Initialize population of solutions, popp ◮ Eval. popp ◮ Evolve popp to get childpop, popc ◮ Eval. popc ◮ popc ⇐ MPP_repair(popc ) ◮ poppool ⇐ ND_sort(popp + popc ) ◮ popn ⇐ E_select(poppool ) ◮ popp = popn
9. Clustered Repair Procedure All infeasible solutions Solutions selected for repair 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F2 F1 The MPP repair operation repairs only 20% of all the infeasible solutions selected by the k-means clustering algorithm in the objective space.
10. Repair Operation Infeasible Feasible 0 0.5 1 1.5 2 3 3.5 4 4.5 5 0 1 2 3 4 5 6 7 8 9 10 x1 x2 ¾º The repair operation takes place in the X-space and an infeasible solution is repaired into a feasible one
11. Test Problems: cZDT series Based on the popular unconstrained ZDT problems ◮ Generic Structure: Minimize τ(x) = (f1 (x1 ), f2 (x)) Where f2 (x) = G(x2 , .., xn )H(f1 (x1 ), G(x2 , ..., xn )) Subject to h(x) = 0 Where xl i ≤ xi ≤ xu i , i = 1, .., n. (3) ◮ h(x) = 0 is defined for each of the problems such that the constrained global Pareto-front remains the same as the unconstrained one
12. cZDT Constraint Function Definitions ◮ czdt1: h(x) = N i=2 sin(xi ) ◮ czdt2: h(x) = N i=2 sin(xi ) ◮ czdt3: h(x) = N i=2 sin(xi ) ◮ czdt4: h(x) = N i=2 xi 2 ◮ czdt6: h(x) = N i=2 sin(xi )
13. Experimental Parameters ◮ Population size, P = 40 ◮ Max. Func. Evals: 30, 000 function evaluations ◮ Max. of 100 function evaluations is allowed for the inner optimization procedure ◮ SBX crossover with crossover probability, Pc = 0.9 and crossover index, ηc = 15 ◮ Polynomial mutation with mutation probability, Pm = 0.1 and mutation index, ηm = 20 ◮ Reference point: (1, 1). ◮ h(x) ≡ ǫ − |h(x)| ≥ 0, ǫ = 1e−05. ◮ 30 independent runs
14. Result: czdt1 NSGA−II NSGA−II NSGA−II_MPP 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F2 F1 NSGA−II_MPP NSGA−II 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5000 10000 15000 20000 25000 30000 35000 Hyper−area Function Eval. Figure: czdt1: Final Non-dominated set obtained after 30, 000 function evaluations (median run w.r.t. hyper-area) and progressive increase in Hyper-area of the feasible Non-dominated set. ¯ G( ) x = 1.0000 for both algorithms.
15. Result: czdt6 NSGA−II NSGA−II NSGA−II_MPP 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 F2 F1 NSGA−II_MPP NSGA−II 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 5000 10000 15000 20000 25000 30000 35000 Hyper−area Function Eval. Figure: czdt6: Final Non-dominated set obtained after 30, 000 function evaluations (median run w.r.t. hyper-area) and progressive increase in Hyper-area of the feasible Non-dominated solutions. ¯ G( ) x = 1.0000 for both algorithms.
16. Result: Attainment of Feasible State NSGA−II_MPP NSGA−II Feasible State 0 5 10 15 20 25 30 35 40 0 2000 4000 6000 8000 10000 12000 #Feas. Solns. Function Eval. 0 5 10 15 20 25 30 35 40 0 2000 4000 6000 8000 10000 12000 #Feas. Solns. Function Eval. ÈÖÓ Ð Ñ Þ Ø½ Figure: czdt1: Number of Feasible solutions found by NSGA-II and NSGA-II_MPP w.r.t function evaluations NSGA−II_MPP NSGA−II Feasible State 0 5 10 15 20 25 30 35 40 0 500 1000 1500 2000 2500 3000 3500 #Feas. Solns. Function Eval. 0 5 10 15 20 25 30 35 40 0 500 1000 1500 2000 2500 3000 3500 #Feas. Solns. Function Eval. ÈÖÓ Ð Ñ Þ Ø Figure: czdt6: Number of Feasible solutions found by NSGA-II and NSGA-II_MPP w.r.t function evaluations
17. Summarized Results: Feasible State Prob. Algo. #Succ. #GenID #Func. eval. Hyper-area (Mean, Median) (Mean) czdt1 NSGA-II 30 293 1.1723e+04, 1.1700e+04 0.3517 NSGA-II_MPP 30 7 5.6849e+03, 5.8520e+03 0.6090 czdt2 NSGA-II 30 293 1.1721e+04, 1.1720e+04 0.1259 NSGA-II_MPP 30 6 5.7188e+03, 5.8345e+03 0.2804 czdt3 NSGA-II 30 292 1.1711e+04, 1.1700e+04 0.4626 NSGA-II_MPP 30 6 5.6077e+03, 5.4365e+03 0.8501 czdt4 NSGA-II 30 418 1.6731e+04, 1.5520e+04 0.3317 NSGA-II_MPP 30 18 1.6007e+04, 1.6096e+04 0.4657 czdt6 NSGA-II 30 87 3.4973e+03, 3.4800e+03 0.0369 NSGA-II_MPP 30 4 1.2789e+03, 1.2830e+03 0.2622 NSGA-II_MPP reaches the feasible state faster than NSGA-II in all cases.
18. Summarized Results: Final Hyperarea and Convergence Prob. Algo. Hyper-area Convergence (Mean, Median) (Mean, Median) czdt1 NSGA-II 0.6514, 0.6515 0.0118, 0.0117 NSGA-II_MPP 0.6497, 0.6510 0.0161, 0.0153 czdt2 NSGA-II 0.3186, 0.3187 0.0119, 0.0117 NSGA-II_MPP 0.3174, 0.3178 0.0133, 0.0129 czdt3 NSGA-II 0.9629, 0.9863 0.0554, 0.0369 NSGA-II_MPP 0.9914, 0.9901 0.0393, 0.0253 czdt4 NSGA-II 0.6412, 0.6485 0.0227, 0.0130 NSGA-II_MPP 0.5784, 0.6077 0.1183, 0.0949 czdt6 NSGA-II 0.3124, 0.3124 0.0117, 0.0116 NSGA-II_MPP 0.3133, 0.3133 0.0105, 0.0102 The quality of the final non-dominated set obtained by the two algorithms is almost the same.
19. Summary ◮ Proposed a set of Equality Constrained MO optimization problems ◮ Clustered Repair approach based on MPP ◮ Results encouraging ◮ Need for a more rigorous set of test problems ◮ Effect of introducing artificial constraints?