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The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

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  • Yi Zhang
  • Liwei Zhang
  • Yue Wu

Abstract

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with cone constraints, and its dual, which has fewer variables than the original one, is a semismoothly differentiable (SC 1 ) convex programming problem with both a linear inequality constraint and a linear second-order cone constraint. We demonstrate the global convergence of the augmented Lagrangian method with an exact solution to the subproblem and prove that the convergence rate of primal iterates, generated by the augmented Lagrangian method, is proportional to 1/r, and the rate of multiplier iterates is proportional to $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. Furthermore, a semismooth Newton method with Armijo line search is constructed to solve the subproblems in the augmented Lagrangian approach. Finally, numerical results are reported to show the effectiveness of the augmented Lagrangian method with both an exact solution and an inexact solution to the subproblem for solving the inverse second-order cone quadratic programming problem. Copyright Sociedad de Estadística e Investigación Operativa 2014

Suggested Citation

  • Yi Zhang & Liwei Zhang & Yue Wu, 2014. "The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 45-79, April.
    • Handle: RePEc:spr:topjnl:v:22:y:2014:i:1:p:45-79
    DOI: 10.1007/s11750-011-0227-3
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    References listed on IDEAS

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    1. Y. J. Liu & L. W. Zhang, 2008. "Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones," Journal of Optimization Theory and Applications, Springer, vol. 139(3), pages 557-575, December.
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    5. Joydeep Dutta, 2005. "Generalized derivatives and nonsmooth optimization, a finite dimensional tour," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 185-279, December.
    6. Alexander Shapiro & Jie Sun, 2004. "Some Properties of the Augmented Lagrangian in Cone Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 479-491, August.
    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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