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Title: An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems

Abstract

We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is a linearly constrained semismoothly differentiable (SC{sup 1}) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange method, is proportional to 1/r, and the rate of multiplier iterates is proportional to 1/{radical}r, where r is the penalty parameter in the augmented Lagrangian. As the objective function of the dual problem is a SC{sup 1} function involving the projection operator onto the cone of symmetrically semi-definite matrices, the analysis requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and properties of the projection operator in the symmetric-matrix space. Furthermore, the semismooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, whichmore » is proven to have global convergence and local quadratic rate. Finally numerical results, implemented by the augmented Lagrangian method, are reported.« less

Authors:
 [1];  [2]
  1. Beijing Normal University-Hong Kong Baptist University, United International College (China)
  2. Dalian University of Technology, School of Mathematical Sciences (China)
Publication Date:
OSTI Identifier:
21480277
Resource Type:
Journal Article
Resource Relation:
Journal Name: Applied Mathematics and Optimization; Journal Volume: 61; Journal Issue: 1; Other Information: DOI: 10.1007/s00245-009-9075-z; Copyright (c) 2010 Springer Science+Business Media, LLC
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; CONVERGENCE; LAGRANGIAN FUNCTION; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; MATRICES; MINIMIZATION; NEWTON METHOD; PROGRAMMING; PROJECTION OPERATORS; SYMMETRY; CALCULATION METHODS; FUNCTIONS; ITERATIVE METHODS; MATHEMATICAL OPERATORS; OPTIMIZATION; SPACE

Citation Formats

Zhang Jianzhong, E-mail: jzzhang@uic.edu.h, and Zhang Liwei, E-mail: lwzhang@dlut.edu.c. An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems. United States: N. p., 2010. Web. doi:10.1007/S00245-009-9075-Z.
Zhang Jianzhong, E-mail: jzzhang@uic.edu.h, & Zhang Liwei, E-mail: lwzhang@dlut.edu.c. An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems. United States. doi:10.1007/S00245-009-9075-Z.
Zhang Jianzhong, E-mail: jzzhang@uic.edu.h, and Zhang Liwei, E-mail: lwzhang@dlut.edu.c. Mon . "An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems". United States. doi:10.1007/S00245-009-9075-Z.
@article{osti_21480277,
title = {An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems},
author = {Zhang Jianzhong, E-mail: jzzhang@uic.edu.h and Zhang Liwei, E-mail: lwzhang@dlut.edu.c},
abstractNote = {We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with a positive semidefinite cone constraint and its dual is a linearly constrained semismoothly differentiable (SC{sup 1}) convex programming problem with fewer variables than the original one. We demonstrate the global convergence of the augmented Lagrangian method for the dual problem and prove that the convergence rate of primal iterates, generated by the augmented Lagrange method, is proportional to 1/r, and the rate of multiplier iterates is proportional to 1/{radical}r, where r is the penalty parameter in the augmented Lagrangian. As the objective function of the dual problem is a SC{sup 1} function involving the projection operator onto the cone of symmetrically semi-definite matrices, the analysis requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and properties of the projection operator in the symmetric-matrix space. Furthermore, the semismooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, which is proven to have global convergence and local quadratic rate. Finally numerical results, implemented by the augmented Lagrangian method, are reported.},
doi = {10.1007/S00245-009-9075-Z},
journal = {Applied Mathematics and Optimization},
number = 1,
volume = 61,
place = {United States},
year = {Mon Feb 15 00:00:00 EST 2010},
month = {Mon Feb 15 00:00:00 EST 2010}
}
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