Journal of Global Optimization 16: 197–217, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. 197 An Inner Approximation Method for Optimization over the Weakly Efficient Set SYUUJI YAMADA, TETSUZO TANINO and MASAHIRO INUIGUCHI Department of Electronics and Information Systems, Graduate School of Engineering, Osaka University, Osaka 565-0871, Japan (e-mail: yamada@eie.eng.osaka-u.ac.jp (Received 7 August 1998; accepted in revised form 6 September 1999) Abstract. In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. To solve such a problem, we propose an inner approximation algorithm, in which two kinds of convex subproblems are solved successively. These convex subproblems are fairly easy to solve and therefore the pro- posed algorithm is practically useful. The algorithm always terminates after finitely many iterations by compromising the weak efficiency to a multiobjective programming problem. Moreover, for a subproblem which is solved at each iteration of the algorithm, we suggest a procedure for eliminating redundant constraints. Key words: Weakly Efficient Set, Global Optimization, Dual Problem, Inner Approximation Method 1. Introduction We consider the following multiobjective programming problem: (MOP)  maximize 〈c i ,x 〉, i = 1,...,K, subject to x ∈ X ⊂ R n , where X is a compact convex set and 〈· , ·〉 denotes the Euclidean inner product in R n . The objective functions 〈c i ,x 〉, i = 1,... ,K , express the criteria which the decision-maker wants to maximize. A feasible vector x ∈ X is said to be weakly efficient if there is no feasible vector y such that 〈c i ,x 〉 < 〈c i ,y 〉 for every i ∈{1,... ,K }. The set X e of all feasible weakly efficient vectors is called the weakly efficient set. From the compactness of X, the weakly efficient set X e is not empty. For problem (MOP), we shall assume the following throughout this paper: (A1) X ={x ∈ R n : p j (x)  0,j = 1,... ,t } where p j : R n → R, j = 1,... ,t , are differentiable convex functions satisfying p j (0)< 0 (whence 0 ∈ int X), (A2) {x ∈ R n :〈c i ,x 〉 < 0 for all i ∈{1,... ,K }}=∅.