# Gauss回代交替方向法求解一类二次规划逆问题Gauss Back Substitution Alternating Direction Method for a Class of Inverse Quadratic Programming

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In this paper, we solve the inverse problem of quadratic programming whose objective function is the sum of matrix spectrum norm and vector inﬁnite norm. We transform the problem into a convex optimization problem with objective function separable and propose Gauss back substitution alternating direction method to solve it. We ﬁnd that one of its subproblems is still a convex optimization problem with objective function separable, but it is impossible to solve every variable accurately. So we use the inexact method to solve the subproblem. Finally, the numerical experiment of the problem in this paper is given. The data shows that the method in this paper can solve the inverse quadratic programming problem eﬃciently and quickly.

 [1] Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press, Princeton. [2] https://doi.org/10.1515/9781400873173 [3] John, D., Shai, S.S., Yoram, S., et al. (2008) E?cient Projections onto the ?1-Ball for Learning in High Dimensions. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland, July 2008, 272-279. [4] Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011) Distributed Optimization and Statistic al Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning, 3, 1-122.https://doi.org/10.1561/2200000016 [5] Eckstein, J. and Bertsekas, D.P. (1992) On the Douglas—Rachford Splitting Method and the Proximal Point Algorithm for Maximal Monotone Operators. Mathematical Programming, 55, 293-318. https://doi.org/10.1007/BF01581204 [6] Gabay, D. and Mercier, B. (1976) A Dual Algorithm for the Solution of Nonlinear Variational Problems via Finite Element Approximation. Computers Mathematics with Applications, 2, 17-40. https://doi.org/10.1016/0898-1221(76)90003-1 [7] He, B., Tao, M. and Yuan, X. (2012) Alternating Direction Method with Gaussian Back Substitution for Separable Convex Programming. SIAM Journal on Optimization, 22, 313- 340. https://doi.org/10.1137/110822347 [8] He, B., Wang, S. and Yang, H. (2003) A Modi?ed Variable-Penalty Alternating Directions Method for Monotone Variational Inequalities. Journal of Computational Mathematics, 21, 495-504. [9] He, B.S., Yang, H. and Wang, S.L. (2000) Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities. Journal of Optimization Theory and Applications, 106, 337-356. https://doi.org/10.1023/A:1004603514434 [10] 李姣芬, 宋丹丹, 李涛, 等. 核范数和谱范数下广义 Sylvester 方程最小二乘问题的有效算法 [J]. [11] 计算数学, 2017, 39(2): 129-150. [12] Ng, M.K., Wang, F. and Yuan, X. (2011) Inexact Alternating Direction Methods for Image Recovery. SIAM Journal on Scienti?c Computing, 33, 1643-1668. https://doi.org/10.1137/100807697 [13] Birgin, E.G., Mario, J. and Raydan, M.M. (2000) Nonmonotone Spectral Projected Gradient Methods on Convex Sets. SIAM Journal on Optimization, 10, 1196-1211. https://doi.org/10.1137/S1052623497330963