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

Zhang, Yi, Zhang, Liwei and Wu, Yue
(2012)
The augmented Lagrangian method for a type of inverse problems over second-order cones.
*TOP* (doi:10.1007/s11750-011-0227-3).
(In Press).

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## Description/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 (SC1) 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/ √ 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.

Item Type: | Article |
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ISSNs: | 1134-5764 (print) 1863-8279 (electronic) |

Keywords: | inverse optimization, second-order cone quadratic programming, augmented lagrangian method, rate of convergence, damped semismooth newton method |

Subjects: | H Social Sciences > HD Industries. Land use. Labor > HD28 Management. Industrial Management Q Science > QA Mathematics > QA76 Computer software |

Divisions: | Faculty of Business and Law > Southampton Management School > Management Science |

ePrint ID: | 336440 |

Date Deposited: | 26 Mar 2012 15:32 |

Last Modified: | 27 Mar 2014 20:20 |

URI: | http://eprints.soton.ac.uk/id/eprint/336440 |

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