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A Perturbation approach for an inverse quadratic programming problem

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  • Jianzhong Zhang
  • Liwei Zhang
  • Xiantao Xiao

Abstract

We consider an inverse quadratic programming (QP) problem in which the parameters in both the objective function and the constraint set of a given QP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem with a positive semidefinite cone constraint. With the help of duality theory, we reformulate this problem as a linear complementarity constrained semismoothly differentiable (SC 1 ) optimization problem with fewer variables than the original one. We propose a perturbation approach to solve the reformulated problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. As the objective function of the problem is a SC 1 function involving the projection operator onto the cone of positively semi-definite symmetric matrices, the analysis requires an implicit function theorem for semismooth functions as well as properties of the projection operator in the symmetric-matrix space. Since an approximate proximal point is required in the inexact Newton method, we also give a Newton method to obtain it. Finally we report our numerical results showing that the proposed approach is quite effective. Copyright Springer-Verlag 2010

Suggested Citation

  • Jianzhong Zhang & Liwei Zhang & Xiantao Xiao, 2010. "A Perturbation approach for an inverse quadratic programming problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 379-404, December.
    • Handle: RePEc:spr:mathme:v:72:y:2010:i:3:p:379-404
    DOI: 10.1007/s00186-010-0323-4
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    References listed on IDEAS

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    1. Jianzhong Zhang & Zhongfan Ma, 1999. "Solution Structure of Some Inverse Combinatorial Optimization Problems," Journal of Combinatorial Optimization, Springer, vol. 3(1), pages 127-139, July.
    2. Zhang, Jianzhong & Liu, Zhenhong & Ma, Zhongfan, 2000. "Some reverse location problems," European Journal of Operational Research, Elsevier, vol. 124(1), pages 77-88, July.
    3. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    4. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    5. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
    6. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
    7. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
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    2. Abd Allah A. Mousa & Yousria Abo-Elnaga, 2020. "Stability of Solutions for Parametric Inverse Nonlinear Cost Transportation Problem," Mathematics, MDPI, vol. 8(11), pages 1-21, November.

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