Most Probable Point Based Constraint Handling (Amit Saha) SEIT Seminar, UNSW, Canberra, Australia http://echorand.me June 7, 2012 

Constrained 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. 

Constraint Types ◮ Equality Constraints: h(x) = 0 (Difficult) ◮ Inequality Constraints: g(x) ≥ 0 (Easy) ◮ Active Inequality Constraints: g(x) = 0 (Easy) 

Past Work ◮ Constraint handling very actively researched into ◮ Studies on equality constrained Multi-objective Optimization problems are few ◮ Recent equality constraint handling work focuses on SO optimization problems ◮ Need for a study on Equality constrained MO optimization problems 

Past Work ◮ Constraint handling very actively researched into ◮ Studies on equality constrained Multi-objective Optimization problems are few ◮ Recent equality constraint handling work focuses on SO optimization problems ◮ Need for a study on Equality constrained MO optimization problems 

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. 

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 

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 

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. 

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 

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 

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 ) 

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 

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. 

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. 

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 

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. 

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. 

Other Applications ◮ Single-objective Constrained Optimization ◮ Active inequality constraint handling 

Other Applications ◮ Single-objective Constrained Optimization ◮ Active inequality constraint handling 

Relevant Publications ◮ Saha, A. and Ray, T. (2012).Equality constrained multi-objective optimization, in IEEE Congress of Evolutionary Computation (CEC), (Brisbane, Australia) ◮ Saha, A. and Ray, T. (2012).A repair mechanism for active inequality constraint handling, in IEEE Congress of Evolutionary Computation (CEC),(Brisbane,Australia) ◮ Saha,A. and Ray, T.(2011), Attaining Feasible State in Equality Constrained Optimization Using Genetic Algorithms (Technical Report) 

Summary ◮ Proposed a set of Equality Constrained MO optimization problems ◮ Clustered Repair approach based on MPP ◮ Need for a more rigorous set of test problems ◮ Effect of introducing artificial constraints?