Mathematical Programming with Increasing Constraint Functions

• Published in: Management science Vol. 15; no. 7; pp. 416 - 425
• Main Author: Pierskalla, William P
• Format: Journal Article
• Language: English
• Published: Hanover, MD., etc INFORMS 01.03.1969; Institute of Management Sciences; Institute for Operations Research and the Management Sciences
• Series: Management Science
• Subjects: Addition; Decreasing functions; Increasing functions; Mathematical functions; Mathematical programming; Mathematical vectors; Necessary conditions for optimality; Random variables; Scalar functions; Sufficient conditions
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.15.7.416

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.