Sufficient optimality conditions for complex programming with quasi-concave constraints

Abstract

Consider the problem, to minimize f{x) subject to are differentiable functions. Mangasariak has given sufficient Lagrangian conditions for x _o to be an optimal point, when / is pseudo-convex and g _t is quasi-concave whenever g _i(x _Q) = 0. Various authors have recently extended these and similar results to the problem, to minimize Re f(z,[zbar]) subject to g(z,[zbar]) where are analytic functions, and S is a convex cone. Sufficiency theorems were obtained, assuming that / has pseudo-convex real part, and g is concave with respect to S. The latter was not weakened to multi concavity for the tight constraints only, since g(a,ⱥ) on the boundary of the cone S does not correspond to certain components of g(a,ⱥ) vanishing. We show here how the assumptions on gcan be weakened to quasi-concavity with respect to a- convex cone containing S. When S is polyhedral, this corresponds to quasi-concavity over tight constraints, as in the real case.