Mathematical Programming with Increasing Constraint Functions

Abstract

The mathematical programming problem—find a non-negative n -vector x which maximizes f ( x ) subject to the constraints g i ( x ) ≥ O , i = 1,…, m —is investigated where f ( x ) is assumed to be concave or pseudo-concave and the g i ( x ) are increasing functions. It is shown that under certain conditions on g i ( x ), the Kuhn-Tucker-Lagrange conditions are necessary and sufficient for the optimality of x *. It is also shown that the g i ( x ) are a useful class of functions since, among other properties, they are closed under non-negative addition, under the addition of any scalar, and under multiplication of non-negative members of the class. Examples of the above programming problem with increasing constraint functions are found in many chance-constrained programming problems.