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September  2020, 16(5): 2425-2437. doi: 10.3934/jimo.2019061

## Inverse quadratic programming problem with $l_1$ norm measure

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 College of Science, Liaoning Technical University, Fuxin 123000, China

* Corresponding author: Lidan Li

Received  October 2018 Revised  January 2019 Published  May 2019

Fund Project: The third author is supported by the National Natural Science Foundation of China under project grant No.11571059 and No.11731013

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 involving $l_1$ vector norm with a positive semidefinite cone constraint. By utilizing convex optimization theory, we rewrite its first order optimality condition as a generalized equation. Under extremely simple assumptions, we prove that any element of the generalized Jacobian of the equation at its solution is nonsingular. Based on this, we construct an inexact Newton method with Armijo line search to solve the equation and demonstrate its global convergence. Finally, we report the numerical results illustrating effectiveness of the Newton methods.

Citation: Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $l_1$ norm measure. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061
##### References:

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##### References:
Numerical results of Problem (5)
 n p iter I-norm-F T-norm-F time(s) 50 10 4 9.8e+003 9.9e-012 0.06 100 10 4 7.5e+004 2.8e-007 0.19 100 20 8 6.6e+004 1.0e-009 0.38 100 50 8 1.9e+005 8.6e-007 0.39 500 50 12 9.3e+006 1.1e-013 52.77 500 100 17 8.2e+006 4.0e-010 79.76 1000 100 25 7.7e+007 9.2e-009 916.90 1000 500 30 3.8e+007 6.0e-008 1191.39 2000 500 34 4.7e+008 3.5e-011 9300.16
 n p iter I-norm-F T-norm-F time(s) 50 10 4 9.8e+003 9.9e-012 0.06 100 10 4 7.5e+004 2.8e-007 0.19 100 20 8 6.6e+004 1.0e-009 0.38 100 50 8 1.9e+005 8.6e-007 0.39 500 50 12 9.3e+006 1.1e-013 52.77 500 100 17 8.2e+006 4.0e-010 79.76 1000 100 25 7.7e+007 9.2e-009 916.90 1000 500 30 3.8e+007 6.0e-008 1191.39 2000 500 34 4.7e+008 3.5e-011 9300.16
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