Sufficiency Conditions and a Duality Theory for Mathematical Programming Problems in Arbitrary Linear Spaces

摘要

The paper is devoted to an investigation of mathematical programming problems in arbitrary linear vector spaces. Two cases are considered: problems with a scalar-valued criterion function, and minimax problems. The constraints of the problem are assumed to be of three types: (a) the point must belong to a given (arbitrary) convex set in the underlying linear space, (b) a finite-dimensional equality constraint must be satisfied, (c) a generalized (possibly infinite-dimensional) inequality constraint, defined in terms of a convex body in a linear topological space, must be satisfied. Assuming that the equality constraints are affine, the the 'inequality' contraints are, in a certain generalized sense, convex, and that the problem is 'well-posed', Kuhn-Tucker type conditions which are both necessary and sufficient for optimality are obtained. A duality theory for obtaining the 'multipliers' in the generalized Kuhn-Tucker conditions is presented. An application to optimal control theory is also presented. (Author)