New Constraint-handling Method for Multi-objective Multi-constraint Evolutionary Optimization and Its Application to Space Plane Design

A new constraint-handling method based on Pareto-optimality concept for multiobjective multi-constraint design optimization problems has been proposed. The proposed method does not need any constants to be tuned for constraint handling. In addition, the present method does not use weighted-sum of constraints and thus does not need tuning of weight coefficients and is efficient even when the amount of violation of each constraint is significantly different. The proposed approach is demonstrated to be remarkably robust than the dynamic penalty approach and other dominance-based approaches through the optimal design of a welded beam and conceptual design optimization of a two-stage-to-orbit space plane. Akira Oyama, Koji Shimoyama, and Kozo Fujii. 2 1 PROBLEM STATEMENT Without losing generality, constrained real-number optimization problems are written as: Find x r that minimize )) ( ),..., ( ),..., ( ( ) ( max 1 x f x f x f x f m m r r r r r = (1) subject to 0 )) ( ),..., ( ),..., ( ( ) ( max 1 ≤ = x g x g x g x g n n r r r r r (2) where ( ) max ,..., ,..., 1 l l x x x x = r is the vector of solution that minimizes objective function(s) ) (x f r r while satisfying the constraint(s) 0 ) ( ≤ x g r r . lmax , mmax and nmax are numbers of design parameter(s), objective function(s) and constraint(s), respectively. 2 INTRODUCTION Evolutionary algorithms (EAs, see [1] for example) are robust and efficient design optimization algorithms based on the Theory of Evolution proposed by Charles Darwin, where a biological population evolves over generations to adapt to an environment by selection, recombination and mutation. One of the key features of EAs is that they search from multiple points in the design space, instead of moving from a single point like gradientbased methods do. Furthermore, these methods work on function evaluations alone (fitness) and do not require derivatives or gradients of the objective functions. These features lead to advantages over deterministic optimization approaches such as robustness, capability to uniformly capture Pareto-optimal solutions, suitability to parallel computing, and simplicity in coupling the EA code and evaluation codes. As a result, EAs have been applied to many realworld design problems in various fields (for example, see [2-3]). However, EAs do not have any explicit mechanism to handle constraints while most of real-world design optimization problems have multiple constraints. A considerable amount of researches on constraint handing techniques that incorporate objective function(s) and constraint(s) into the fitness function of design candidates has been carried out (a good summary is given in [4]). Traditional approach for handling design constraints of single-objective design optimization problems for evolutionary optimization is the penalty function method [1] where fitness of a design candidate is determined based on a scale function F, which is weighted sum of the objective function value and the amount of design constraint violations as: ( ) ) 0 ), ( max( ) ( max