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Optimality criteria in mathematical programming involving generalized invexity. (English) Zbl 0647.90076
Constrained optimization problems of the form (1) minimize f(x) subject to $x\in X\subseteq R\sp n$, g(x)$\le 0$, 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 $x\sb 0$ if the relations $$ f(x)-f(x\sb 0)\ge [\nabla\sb xf(x\sb o)]' \eta (x),\quad - g(x\sb 0)\ge [\nabla\sb xg(x\sb o)] \eta (x) $$ hold for all feasible x of the problem (1). Similarly f, g are called of type II with respect to x at $x\sb 0$, if $$ f(x\sb 0)-f(x)\ge [\nabla\sb xf(x)]' \eta (x),\quad and\quad -g(x)\ge \nabla\sb xg(x) \eta (x) $$ 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

90C30Nonlinear programming
49K05Free problems in one independent variable (optimality conditions)
Full Text: DOI
[1] A. Ben-Israel and B. Mond, What is invexity? unpublished, Department of Mathematics, La Trobe University, Bundoora, Victoria, 3083, Australia.
[2] Hanson, M. A.: On sufficiency of the Kuhn-Tucker conditions. J. math. Anal. appl. 80, 545-550 (1981) · Zbl 0463.90080
[3] Hanson, M. A.; Mond, B.: Necessary and sufficient conditions in constrained optimization. FSU statistics report M 683 (1984) · Zbl 0622.49005
[4] Kaul, R. N.; Kaur, S.: Optimality criteria in nonlinear programming involving non-convex functions. J. math. Anal. appl. 105, 104-112 (1985) · Zbl 0553.90086