Mathematical Programming with Increasing Constraint Functions

The mathematical programming problem—find a non-negative n-vector x which maximizes f(x) subject to the constraints gⁱ(x) ≥ O, i = 1,…, m—is investigated where f(x) is assumed to be concave or pseudo-concave and the gⁱ(x) are increasing functions. It is shown that under certain conditions on gⁱ(x), the Kuhn-Tucker-Lagrange conditions are necessary and sufficient for the optimality of x*. It is also shown that the gⁱ(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.