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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 xXR n , g(x)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 η (x) at x 0 if the relations

f(x)-f(x 0 )[ x f(x o )] ' η(x),-g(x 0 )[ x g(x o )]η(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)[ x f(x)] ' η(x),and-g(x) x g(x)η(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

MSC:
90C30Nonlinear programming
49K05Free problems in one independent variable (optimality conditions)