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An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints

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  • Richard M. Soland

    (Research Analysis Corporation, McLean, Virginia)

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.

Suggested Citation

  • Richard M. Soland, 1971. "An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints," Management Science, INFORMS, vol. 17(11), pages 759-773, July.
    • Handle: RePEc:inm:ormnsc:v:17:y:1971:i:11:p:759-773
    DOI: 10.1287/mnsc.17.11.759
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    Cited by:

    1. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2021. "A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems," IIMA Working Papers WP 2021-03-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
    2. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2023. "A general purpose exact solution method for mixed integer concave minimization problems," European Journal of Operational Research, Elsevier, vol. 309(3), pages 977-992.
    3. Wooseung Jang & J. George Shanthikumar, 2002. "Stochastic allocation of inspection capacity to competitive processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(1), pages 78-94, February.
    4. Sinha, Ankur & Das, Arka & Anand, Guneshwar & Jayaswal, Sachin, 2021. "A General Purpose Exact Solution Method for Mixed Integer Concave Minimization Problems (revised as on 12/08/2021)," IIMA Working Papers WP 2021-03-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
    5. Emmanuel Ogbe & Xiang Li, 2019. "A joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programs," Journal of Global Optimization, Springer, vol. 75(3), pages 595-629, November.

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