Constraint Qualifications in Maximization Problems

Abstract

Many problems arising in logistics and in the application of mathematics to industrial planning are in the form of constrained maximizations with nonlinear maxirnands or constraint functions or both. Thus a depot facing random demands for several items may wish to place orders for each in such a way as to maximize the expected number of demands which are fulfilled; the total of orders placed is limited by a budget constraint. In this case, the maximand is certainly nonlinear. The constraint would also be nonlinear if, for example, the marginal cost of storage of the goods were increasing. Practical methods for solving such problems in nonlinear programming almost invariably depends on some use of Lagrange multipliers, either by direct solution of the resulting system of equations or by a gradient method of successive approximations (see [5], Part II). This article discusses a part of the sufficient conditions for the validity of the multiplier method.