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Technical Note—A Duality Theory for Convex Programming with Set-Inclusive Constraints

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  • A. L. Soyster

    (Temple University, Philadelphia, Pennsylvania)

Abstract

This paper extends the notion of convex programming with set-inclusive constraints as set forth by Soyster [ Opns. Res. 21, 1154–1157 (1973)] by replacing the objective vector c with a convex set C and formulating a dual problem. The primal problem to be considered is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\sup\Bigl\{\inf_{c\in C} c \cdot x \mid x_{1}K_{1}+x_{2}K_{2}+ \ldots x_{n}K_{n}\subseteq K(b), x_{i}\geq 0\Bigr\}$$\end{document} where the sets { K j } are convex activity sets, K ( b ) is a polyhedral resource set, C is a convex set of objective vectors, and the binary operation + refers to addition of sets. Any feasible solution to the dual problem provides an upper bound to (I) and, at optimality conditions, the value of (I) is equal to the value of the dual. Furthermore, the optimal solution of the dual problem can be used to reduce (I) to an ordinary linear programming problem.

Suggested Citation

  • A. L. Soyster, 1974. "Technical Note—A Duality Theory for Convex Programming with Set-Inclusive Constraints," Operations Research, INFORMS, vol. 22(4), pages 892-898, August.
  • Handle: RePEc:inm:oropre:v:22:y:1974:i:4:p:892-898
    DOI: 10.1287/opre.22.4.892
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    Cited by:

    1. Hsien-Chung Wu, 2011. "Duality Theory in Interval-Valued Linear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 298-316, August.
    2. A. K. Bhurjee & G. Panda, 2016. "Sufficient optimality conditions and duality theory for interval optimization problem," Annals of Operations Research, Springer, vol. 243(1), pages 335-348, August.
    3. H. C. Wu, 2010. "Duality Theory for Optimization Problems with Interval-Valued Objective Functions," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 615-628, March.
    4. H. C. Wu, 2008. "Wolfe Duality for Interval-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 497-509, September.
    5. Viktoryia Buhayenko & Dick den Hertog, 2017. "Adjustable Robust Optimisation approach to optimise discounts for multi-period supply chain coordination under demand uncertainty," International Journal of Production Research, Taylor & Francis Journals, vol. 55(22), pages 6801-6823, November.
    6. Bram L. Gorissen & Hans Blanc & Dick den Hertog & Aharon Ben-Tal, 2014. "Technical Note---Deriving Robust and Globalized Robust Solutions of Uncertain Linear Programs with General Convex Uncertainty Sets," Operations Research, INFORMS, vol. 62(3), pages 672-679, June.
    7. Gorissen, B.L. & Ben-Tal, A. & Blanc, J.P.C. & den Hertog, D., 2012. "A New Method for Deriving Robust and Globalized Robust Solutions of Uncertain Linear Conic Optimization Problems Having General Convex Uncertainty Sets," Other publications TiSEM e4c05682-e13c-4d1a-bc3f-a, Tilburg University, School of Economics and Management.

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