Title of Paper:ON CONVERGENCE OF AUGMENTED LAGRANGIAN METHOD FOR INVERSE SEMI-DEFINITE QUADRATIC PROGRAMMING PROBLEMS

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Date of Publication:2009-05-01

Journal:JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION

Included Journals:SCIE、Scopus

Volume:5

Issue:2

Page Number:319-339

ISSN No.:1547-5816

Key Words:Inverse optimization; quadratic programming; the augmented Lagrangian method; the cone of positive semi-definite matrices; rate of convergence; Newton method

Abstract:We consider an inverse problem raised from the semi-definite quadratic programming (SDQP) problem. In the inverse problem, the parameters in the objective function of a given SDQP 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 semi-definite cone constraint and its dual is a linearly positive semi-definite cone constrained semi smoothly differentiable (SC1) 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 primaliterates, generated by the augmented Lagrange method, is proportionalto 1/t, and the rate of multiplier iterates is proportional to 1/root t, where t is the penalty parameter in the augmented Lagrangian. The numerical results are reported to show the effectiveness of the augmented Lagrangian method for solving the inverse semi-definite quadratic programming problem.