An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints

Richard M. Soland
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Richard M. Soland: Research Analysis Corporation, McLean, Virginia

Management Science, 1971, vol. 17, issue 11, 759-773

Abstract: We extend a previous algorithm in order to solve mathematical programming problems of the form: Find x = (x 1 , ..., x n ) to minimize \sum \varphi i0 (x i ) subject to x \in G, l \leqq x \leqq L and \sum \varphi ij (x i ) \leqq 0, j = 1, ..., m. Each \varphi 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 \leqq x \leqq 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.

Date: 1971
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