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Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints

Author

Listed:
  • Richard E. Wendell

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

  • Arthur P. Hurter

    (Northwestern University, Evanston, Illinois)

Abstract

We consider an important class of mathematical programs, in which the vector variable can be partitioned into two subvectors corresponding to independent constraint sets. Necessary and sufficient conditions for optimal solutions are developed, and two approaches for obtaining solutions are reviewed. We present an enumeration approach, reducing the problem to a finite number of subproblems, and show that duality makes the solution of many of the subproblems unnecessary. Next, we develop an alternating approach, wherein the problem is solved for one of the subvectors while the other is held constant, and then the subvector roles are reversed. This procedure is observed to converge to partial optimum solutions. A widely applicable subclass of problems includes a linear program in one of the subvectors. For this subclass a sufficient condition for local optimality is determined. The condition is easily testable and fails to hold, in many cases, only if a better solution is obtained. Also, this condition shows that partial optimum solutions are almost always local optima.

Suggested Citation

  • Richard E. Wendell & Arthur P. Hurter, 1976. "Minimization of a Non-Separable Objective Function Subject to Disjoint Constraints," Operations Research, INFORMS, vol. 24(4), pages 643-657, August.
    • Handle: RePEc:inm:oropre:v:24:y:1976:i:4:p:643-657
    DOI: 10.1287/opre.24.4.643
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    Cited by:

    1. Bloemhof-Ruwaard, Jacqueline M. & Hendrix, Eligius M. T., 1996. "Generalized bilinear programming: An application in farm management," European Journal of Operational Research, Elsevier, vol. 90(1), pages 102-114, April.
    2. Carina Moreira Costa & Dennis Kreber & Martin Schmidt, 2022. "An Alternating Method for Cardinality-Constrained Optimization: A Computational Study for the Best Subset Selection and Sparse Portfolio Problems," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2968-2988, November.
    3. Jan Pablo Burgard & Carina Moreira Costa & Martin Schmidt, 2024. "Robustification of the k-means clustering problem and tailored decomposition methods: when more conservative means more accurate," Annals of Operations Research, Springer, vol. 339(3), pages 1525-1568, August.
    4. Dimitris Bertsimas & Xuan Vinh Doan & Karthik Natarajan & Chung-Piaw Teo, 2010. "Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 580-602, August.
    5. Thomas Kleinert & Martin Schmidt, 2021. "Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 198-215, January.
    6. Blanco, Víctor & Fernández, Elena & Puerto, Justo, 2017. "Minimum Spanning Trees with neighborhoods: Mathematical programming formulations and solution methods," European Journal of Operational Research, Elsevier, vol. 262(3), pages 863-878.
    7. Kovacevic, Raimund M. & Pflug, Georg Ch., 2014. "Electricity swing option pricing by stochastic bilevel optimization: A survey and new approaches," European Journal of Operational Research, Elsevier, vol. 237(2), pages 389-403.
    8. Skripnikov, A. & Michailidis, G., 2019. "Regularized joint estimation of related vector autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 139(C), pages 164-177.
    9. Björn Geißler & Antonio Morsi & Lars Schewe & Martin Schmidt, 2018. "Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 309-323, May.

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