An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints

• Published in: Management science Vol. 17; no. 11; pp. 759 - 773
• Main Author: Soland, Richard M
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.07.1971; Institute of Management Sciences; Institute for Operations Research and the Management Sciences
• Series: Management Science
• Subjects: Algorithms; Industrial refining; Integer programming; Integers; Mathematical functions; Mathematical programming; Objective functions; Perceptron convergence procedure; Rectangles
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.17.11.759

We extend a previous algorithm in order to solve mathematical programming problems of the form: Find x = ( x 1 , ..., x n ) to minimize i 0 ( x i ) subject to x G , l x L and ij ( x i ) 0, j = 1, ..., m . Each ij is assumed to be lower semicontinuous, possibly nonconvex, and G is assumed to be closed. The algorithm is of the branch and bound type and solves a sequence of problems in each of which the objective function is convex. In case G is convex each problem in the sequence is a convex programming problem. The problems correspond to successive partitions of the set C = { x | l x L }. Two different rules for refining the partitions are considered; these lead to convergence of the algorithm under different requirements on the problem functions. An example is given, and computational considerations are discussed.