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An algorithm for global solution to bi-parametric linear complementarity constrained linear programs

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  • Yu-Ching Lee
  • Jong-Shi Pang
  • John Mitchell

Abstract

A linear program with linear complementarity constraints (LPCC) is among the simplest mathematical programs with complementarity constraints. Yet the global solution of the LPCC remains difficult to find and/or verify. In this work we study a specific type of the LPCC which we term a bi-parametric LPCC. Reformulating the bi-parametric LPCC as a non-convex quadratically constrained program, we develop a domain-partitioning algorithm that solves a series of the linear subproblems and/or convex quadratically constrained subprograms obtained by the relaxations of the complementarity constraint. The choice of an artificial constants-pair allows us to control the domain on which the partitioning is done. Numerical results of robustly solving 105 randomly generated bi-parametric LPCC instances of different structures associated with different numbers of complementarity constraints by the algorithm are presented. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Yu-Ching Lee & Jong-Shi Pang & John Mitchell, 2015. "An algorithm for global solution to bi-parametric linear complementarity constrained linear programs," Journal of Global Optimization, Springer, vol. 62(2), pages 263-297, June.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:2:p:263-297
    DOI: 10.1007/s10898-014-0228-5
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    References listed on IDEAS

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    1. Wayne F. Bialas & Mark H. Karwan, 1984. "Two-Level Linear Programming," Management Science, INFORMS, vol. 30(8), pages 1004-1020, August.
    2. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
    3. C. Audet & G. Savard & W. Zghal, 2007. "New Branch-and-Cut Algorithm for Bilevel Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 353-370, August.
    4. Alireza Ghaffari-Hadigheh & Habib Ghaffari-Hadigheh & Tamás Terlaky, 2008. "Bi-parametric optimal partition invariancy sensitivity analysis in linear optimization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 215-238, June.
    5. Van Voorhis, Tim & Al-Khayyal, Faiz A., 2003. "Difference of convex solution of quadratically constrained optimization problems," European Journal of Operational Research, Elsevier, vol. 148(2), pages 349-362, July.
    6. Paris, Quirino, 1983. "Multiple Optimal Solutions In Quadratic Programming Models," Western Journal of Agricultural Economics, Western Agricultural Economics Association, vol. 8(2), pages 1-14, December.
    7. Lijie Bai & John Mitchell & Jong-Shi Pang, 2013. "On convex quadratic programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 54(3), pages 517-554, April.
    8. Al-Khayyal, Faiz A., 1992. "Generalized bilinear programming: Part I. Models, applications and linear programming relaxation," European Journal of Operational Research, Elsevier, vol. 60(3), pages 306-314, August.
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