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

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones
The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones
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/√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.
inverse optimization, second-order cone quadratic programming, augmented lagrangian method, rate of convergence, damped semismooth newton method
1134-5764
45-79
Zhang, Yi
34089853-d472-4945-be54-a895e8852662
Zhang, Liwei
10fce21c-16d9-4096-b07a-cf2cab1591c0
Wu, Yue
e279101b-b392-45c4-b894-187e2ded6a5c
Zhang, Yi
34089853-d472-4945-be54-a895e8852662
Zhang, Liwei
10fce21c-16d9-4096-b07a-cf2cab1591c0
Wu, Yue
e279101b-b392-45c4-b894-187e2ded6a5c

Zhang, Yi, Zhang, Liwei and Wu, Yue (2011) The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones. TOP, 22 (1), 45-79. (doi:10.1007/s11750-011-0227-3).

Record type: Article

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/√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.

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More information

Accepted/In Press date: 22 August 2011
e-pub ahead of print date: 23 September 2011
Published date: 23 September 2011
Keywords: inverse optimization, second-order cone quadratic programming, augmented lagrangian method, rate of convergence, damped semismooth newton method
Organisations: Centre of Excellence for International Banking, Finance & Accounting

Identifiers

Local EPrints ID: 336440
URI: http://eprints.soton.ac.uk/id/eprint/336440
ISSN: 1134-5764
PURE UUID: cba7758f-5e14-4cdb-94a5-b0a4e31c2c2c
ORCID for Yue Wu: ORCID iD orcid.org/0000-0002-1881-6003

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Date deposited: 26 Mar 2012 15:32
Last modified: 15 Mar 2024 03:20

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Contributors

Author: Yi Zhang
Author: Liwei Zhang
Author: Yue Wu ORCID iD

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