Approximate necessary conditions for locally weak Pareto optimality

Necessary conditions for a given pointx⁰ to be a locally weak solution to the Pareto minimization problem of a vector-valued functionF=(f₁,...,f_m),F:XR^m,XR^m, are presented. As noted in Ref. 1, the classical necessary condition-conv {Df₁(x⁰)|i=1,...,m}T^*(X, x⁰) need not hold when the contingent coneT is used. We have proven, however, that a properly adjusted approximate version of this classical condition always holds. Strangely enough, the approximation form>2 must be weaker than form=2.The authors would like to thank the anonymous referee for the suggestions which led to an improved presentation of the paper.