Optimality criteria in mathematical programming involving generalized invexity. (English) Zbl 0647.90076

Constrained optimization problems of the form (1) minimize f(x) subject to \(x\in X\subseteq R^ n\), g(x)\(\leq 0\), with differentiable functions f, g f type I or type II are considered: The functions f, g are called of type I with respect to a vector function \(\eta\) (x) at \(x_ 0\) if the relations \[ f(x)-f(x_ 0)\geq [\nabla_ xf(x_ o)]' \eta (x),\quad - g(x_ 0)\geq [\nabla_ xg(x_ o)] \eta (x) \] hold for all feasible x of the problem (1).
Similarly f, g are called of type II with respect to x at \(x_ 0\), if \[ f(x_ 0)-f(x)\geq [\nabla_ xf(x)]' \eta (x),\quad and\quad -g(x)\geq \nabla_ xg(x) \eta (x) \] are satisfied for all feasible solutions of the problem (1). Various sufficient conditions, under which the functions f, g are of type I or II are given. Sufficient optimality conditions for the problem (1), in which f, g are of type I or II are proved.
Reviewer: K.Zimmermann


90C30 Nonlinear programming
49K05 Optimality conditions for free problems in one independent variable
Full Text: DOI


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