American Institute of Mathematical Sciences

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April  2009, 5(2): 319-339. doi: 10.3934/jimo.2009.5.319

On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems

Xiantao Xiao 1, , Liwei Zhang 1, and  Jianzhong Zhang 2,

1. 

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, LiaoNing, China, China
2. 

BJNU-HKBU United International College, Zhuhai, China

Received  February 2008 Revised  July 2008 Published  April 2009

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 semismoothly differentiable ($\mbox{SC}^1$) 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 $1/t$, and the rate of multiplier iterates is proportional to $1/\sqrt{t}$, 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.
Keywords: quadratic programming, Inverse optimization, Newton method., the augmented Lagrangian method, rate of convergence, the cone of positive semi-definite matrices.
Mathematics Subject Classification: Primary: 90C20; Secondary: 90C4.
Citation: Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319
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