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pth Power Lagrangian Method for Integer Programming

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  • Duan Li
  • Douglas White

Abstract

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search. Copyright Kluwer Academic Publishers 2000

Suggested Citation

  • Duan Li & Douglas White, 2000. "pth Power Lagrangian Method for Integer Programming," Annals of Operations Research, Springer, vol. 98(1), pages 151-170, December.
    • Handle: RePEc:spr:annopr:v:98:y:2000:i:1:p:151-170:10.1023/a:1019252306512
    DOI: 10.1023/A:1019252306512
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    Cited by:

    1. Jingqun Li & R. Tharmarasa & Daly Brown & Thia Kirubarajan & Krishna R. Pattipati, 2019. "A novel convex dual approach to three-dimensional assignment problem: theoretical analysis," Computational Optimization and Applications, Springer, vol. 74(2), pages 481-516, November.
    2. Xiaoling Sun & Duan Li, 2000. "Asymptotic Strong Duality for Bounded Integer Programming: A Logarithmic-Exponential Dual Formulation," Mathematics of Operations Research, INFORMS, vol. 25(4), pages 625-644, November.
    3. Charles I. Nkeki, 2018. "Optimal investment risks management strategies of an economy in a financial crisis," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-24, March.

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