Uniqueness of KKT multipliersin multiobjective optimization

K e y w o r d s M u l t i o b j e c t i v e optimization, Mangasarian-l'~romovitz type conditions, Second-order optimality conditions. 1. I N T R O D U C T I O N We consider the following constrained multiobjective program: min f(x), subject to x E X, (1) int R~ where the feasible region is described by inequalities and equalities X := {x • R": g(x) <_ O, h(x) = 0}, with f : R" ---* R t, g : R n ~ R m, and h : R n ---* R p differentiable functions; let K := {1 , . . . , g} , I : { 1 , . . . , m } , and J := { 1 , . . . , p } denote the index sets of the involved vector functions. Moreover, let I (~) := {i • I : gi(~) = 0} denote the index set of the active inequality constraints at a given £, • X. 0893-9659/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.aml.2003.10.011 Typeset by .AA~-TEX 1286 G. BIGI AND M. CASTELLANI The notat ion minint R~_ marks vector minimum with respect to the cone int R~_ : 5: E X is called a local vector minimum point of (1) if and only if there exists a neighbourhood N of 2, such that no x E X M g satisfies f (2) f ( x ) E int R~_, that is, fk(x) > fk(x) for all k E K. It is well known (see [1]) that a necessary condition for • E X to be a local vector minimum point of (1) is that there exist 0 E ]~, A E R m, and # E ]~v not all zero such that ekVfk(~) + ~ ~Vg,(~) + ~ ~jWh(~) = 0, (2) k 6 K iEI j E J Aigi(~) = 0, i E I , (3) 0k_>0, k E K , A i k 0 , i E I . (4) The vectors (8, A,/z) satisfying the above system axe known as John multipliers; all the feasible points • for which John multipliers exist are called stationary. In the case of scalar optimization (~ -1), John multipliers are called K K T multipliers when -1. The structure of the set of KKT multipliers has been analysed by some authors (see, for instance, [2-6]). In particular, it has been proved in [2] that this set is nonempty and bounded if and only if the well-known Mangasarian-Fromovitz constraint qualification (MFCQ) holds. To achieve uniqueness, the following modification of (MFCQ) has been introduced in [3]. DEFINITION 1. The strict Mangasarian-Fromovitz constraint qualification (SMFCQ) holds at a stationary point • 6 X when there exJst John multipliers such that for the index sets I+(~) := {i e x(~): ~ > 0}, Io(~) := I(~) \ I+(~), the foflowing conditions hold: • Vgi(~), i E I+(~), Vhj(~), j 6 J, axe linearly independent; • there exists w E R n such that V g ~ ( ~ ) ~ < 0, i e Io(~) ,