Sensitivity in multi‐objective programming under homogeneity assumptions

Abstract

Duality theory is applied to measure the sensitivity of a multi-objective programming problem. Since the dual problem does not always measure primal sensitivity, the paper states necessary and sufficient conditions (homogeneity conditions) to guarantee the existence and uniqueness of a pair of solutions (primal and dual) such that the dual solution provides the primal sensitivity.

Sensitivity of Pareto Solutions in Multiobjective Optimization

“…When this surface passes through stationary points satisfying the KKT conditions (13)- (21), it shrinks continuously into a less-dimensional surface, due to the complementarity relations (18)- (20). For the Pareto points, the surface (10) becomes a (1 + N p + m + k)-dimensional surface given by (11), with directions l restricted to intersect the balance set. We call this surface (11) the optimal Lagrangian L 0 (·).…”

“…This necessitated studies on the sensitivity of multiobjective programs (Refs. [6][7][8][9][10][11], with respect to perturbations of objective functions, or utility functions, or the ordering cone (Ref. 8) to analyze the stability of solutions under certain assumptions like convexity (Ref.…”

“…In order to overcome these difficulties, the recent literature (see for instance Refs. [3][4][5][6][7][8][9][10][11][22][23][24][25][26] has developed new ideas and methods; we try to show here that the balance space approach may be a useful alternative and can broaden possible techniques. Since the adjoint Pareto set equals the ideal point J plus the balance set [P * = f (P ) = J + B (see Refs.…”

Measuring the Balance Space Sensitivity in Vector Optimization

“…In fact, statements are far more complicated than the corresponding results for scalar problems, and their practical applications present larger difficulties too. Linear vector problems are mucb easier but, as pointed out in [1] and [4], even this case becomes complex when using flexible criteria to choose an optimal solu tion in the efficient set 2 In order 1,0 overcome these difficulties the recent literature (see for instance [2], [3], [11] or [12]) has developed new ieleas anel methods, anel we will try to show here that the balance space approach may be an useful alternative anel can broaden possible techniques. Most of the classical caveats disappear…”

“…Henceforth 'T p 2 O wiU represent the optimal value of (3), and x(p) will represent the corresponcling PareLo solution of (1).…”