A new semidefinite relaxation for L_{1}-constrained quadratic optimization and extensions

Previous Article
Continuity and stability of two-stage stochastic programs with quadratic continuous recourse
NACO Home
This Issue
Next Article
A wedge trust region method with self-correcting geometry for derivative-free optimization

2015, 5(2): 185-195. doi: 10.3934/naco.2015.5.185

A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions

Yong Xia ^1, , Yu-Jun Gong ^2, and Sheng-Nan Han ^2,

1.	State Key Laboratory of Software Development Environment, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
2.	LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China, China

Received September 2014 Revised April 2015 Published June 2015

Abstract
Related Papers

In this paper, by improving the variable-splitting approach, we propose a new semidefinite programming (SDP) relaxation for the nonconvex quadratic optimization problem over the $\ell_1$ unit ball (QPL1). It dominates the state-of-the-art SDP-based bound for (QPL1). As extensions, we apply the new approach to the relaxation problem of the sparse principal component analysis and the nonconvex quadratic optimization problem over the $\ell_p$ ($1< p<2$) unit ball and then show the dominance of the new relaxation.

Keywords: $\ell_1$ unit ball, Semidefinite programming, Sparse principal component analysis., Quadratic optimization.

Mathematics Subject Classification: 90C20, 90C2.

Citation: Yong Xia, Yu-Jun Gong, Sheng-Nan Han. A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 185-195. doi: 10.3934/naco.2015.5.185

References:

[1]	I. M. Bomze, M. Dür, E. De Klerk, C. Roos, A. J. Quist and T. Terlaky, On copositive programming and standard quadratic optimization problems, Journal of Global Optimization, 18 (2000), 301-320. doi: 10.1023/A:1008364005245.
[2]	I. M. Bomze, F. Frommlet and M. Rubey, Improved SDP bounds for minimizing quadratic functions over the l1-ball, Optimization Letters, 1 (2007), 49-59. doi: 10.1007/s11590-006-0018-1.
[3]	A. R. Conn, N. I. M. Gould and P. L. Toint, Trust-Region Methods, MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA, 2000 doi: 10.1137/1.9780898719857.
[4]	A. d'Aspremont, L. El Ghaoui, M. I. Jordan and G. R. G. Lanckriet, A direct formulation for sparse PCA using semidefinite programming, SIAM Review, 48 (2007), 434-448.
[5]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21, 2010. Available: http://cvxr.com/cvx.
[6]	Y. Hsia, Complexity and Nonlinear Semidefinite Programming Reformulation of l1-constrained Nonconvex Quadratic Optimization, Optimization Letters, 8 (2014), 1433-1442. doi: 10.1007/s11590-013-0670-1.
[7]	S. Khot and A. Naor, Grothendieck-type inequalities in combinatorial optimization, Communications on Pure and Applied Mathematics, 65 (2012), 992-1035. doi: 10.1002/cpa.21398.
[8]	G. Kindler, A. Naor and G. Schechtman, The UGC hardness threshold of the Grothendieck problem, Math. Oper. Res., 35 (2010), 267-283. doi: 10.1287/moor.1090.0425.
[9]	L. Lovasz and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM. J. Optimization, 1 (1991), 166-190. doi: 10.1137/0801013.
[10]	R. Luss and M. Teboulle, Convex Approximations to Sparse PCA via Lagrangian Duality, Operations Research Letters, 39 (2011), 57-61. doi: 10.1016/j.orl.2010.11.005.
[11]	J. M. Martínez, Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM. J. Optimization. 4 (1994), 159-176. doi: 10.1137/0804009.
[12]	Y. Nesterov, Global Quadratic Optimization via Conic Relaxation, in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal and L. Vandenberghe, eds., Kluwer Academic Publishers, Boston, (2000), 363-387.
[13]	M.Ç. Pinar and M. Teboulle, On semidefinite bounds for maximization of a non-convex quadratic objective over the l1 unit ball, RAIRO-Operations Research, 40 (2006), 253-265. doi: 10.1051/ro:2006023.
[14]	J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimation over symmetric cones, Optimization Methods and Software, 11-12 (1999), 625-653. doi: 10.1080/10556789908805766.
[15]	Y. Xia, New results on semidefinite bounds for l1-constrained nonconvex quadratic optimization, RAIRO-Operations Research, 47 (2013), 285-297. doi: 10.1051/ro/2013039.

show all references

References:

[1]	I. M. Bomze, M. Dür, E. De Klerk, C. Roos, A. J. Quist and T. Terlaky, On copositive programming and standard quadratic optimization problems, Journal of Global Optimization, 18 (2000), 301-320. doi: 10.1023/A:1008364005245.
[2]	I. M. Bomze, F. Frommlet and M. Rubey, Improved SDP bounds for minimizing quadratic functions over the l1-ball, Optimization Letters, 1 (2007), 49-59. doi: 10.1007/s11590-006-0018-1.
[3]	A. R. Conn, N. I. M. Gould and P. L. Toint, Trust-Region Methods, MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA, 2000 doi: 10.1137/1.9780898719857.
[4]	A. d'Aspremont, L. El Ghaoui, M. I. Jordan and G. R. G. Lanckriet, A direct formulation for sparse PCA using semidefinite programming, SIAM Review, 48 (2007), 434-448.
[5]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1. 21, 2010. Available: http://cvxr.com/cvx.
[6]	Y. Hsia, Complexity and Nonlinear Semidefinite Programming Reformulation of l1-constrained Nonconvex Quadratic Optimization, Optimization Letters, 8 (2014), 1433-1442. doi: 10.1007/s11590-013-0670-1.
[7]	S. Khot and A. Naor, Grothendieck-type inequalities in combinatorial optimization, Communications on Pure and Applied Mathematics, 65 (2012), 992-1035. doi: 10.1002/cpa.21398.
[8]	G. Kindler, A. Naor and G. Schechtman, The UGC hardness threshold of the Grothendieck problem, Math. Oper. Res., 35 (2010), 267-283. doi: 10.1287/moor.1090.0425.
[9]	L. Lovasz and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM. J. Optimization, 1 (1991), 166-190. doi: 10.1137/0801013.
[10]	R. Luss and M. Teboulle, Convex Approximations to Sparse PCA via Lagrangian Duality, Operations Research Letters, 39 (2011), 57-61. doi: 10.1016/j.orl.2010.11.005.
[11]	J. M. Martínez, Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM. J. Optimization. 4 (1994), 159-176. doi: 10.1137/0804009.
[12]	Y. Nesterov, Global Quadratic Optimization via Conic Relaxation, in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal and L. Vandenberghe, eds., Kluwer Academic Publishers, Boston, (2000), 363-387.
[13]	M.Ç. Pinar and M. Teboulle, On semidefinite bounds for maximization of a non-convex quadratic objective over the l1 unit ball, RAIRO-Operations Research, 40 (2006), 253-265. doi: 10.1051/ro:2006023.
[14]	J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimation over symmetric cones, Optimization Methods and Software, 11-12 (1999), 625-653. doi: 10.1080/10556789908805766.
[15]	Y. Xia, New results on semidefinite bounds for l1-constrained nonconvex quadratic optimization, RAIRO-Operations Research, 47 (2013), 285-297. doi: 10.1051/ro/2013039.

[1]	Pengbo Geng, Wengu Chen. Unconstrained $ \ell_1 $-$ \ell_2 $ minimization for sparse recovery via mutual coherence. Mathematical Foundations of Computing, 2020, 3 (2) : 65-79. doi: 10.3934/mfc.2020006
[2]	Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357
[3]	Huining Qiu, Xiaoming Chen, Wanquan Liu, Guanglu Zhou, Yiju Wang, Jianhuang Lai. A fast $\ell_1$-solver and its applications to robust face recognition. Journal of Industrial and Management Optimization, 2012, 8 (1) : 163-178. doi: 10.3934/jimo.2012.8.163
[4]	Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305
[5]	Saeid Ansary Karbasy, Maziar Salahi. Quadratic optimization with two ball constraints. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 165-175. doi: 10.3934/naco.2019046
[6]	Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems and Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447
[7]	Azam Moradi, Jafar Razmi, Reza Babazadeh, Ali Sabbaghnia. An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (2) : 855-879. doi: 10.3934/jimo.2018074
[8]	Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2345-2366. doi: 10.3934/jimo.2020072
[9]	Shouhong Yang. Semidefinite programming via image space analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1187-1197. doi: 10.3934/jimo.2016.12.1187
[10]	Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems and Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067
[11]	Ying Lin, Rongrong Lin, Qi Ye. Sparse regularized learning in the reproducing kernel banach spaces with the $ \ell^1 $ norm. Mathematical Foundations of Computing, 2020, 3 (3) : 205-218. doi: 10.3934/mfc.2020020
[12]	Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial and Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
[13]	Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial and Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65
[14]	Zongming Guo, Fangshu Wan. Weighted fourth order elliptic problems in the unit ball. Electronic Research Archive, 2021, 29 (6) : 3775-3803. doi: 10.3934/era.2021061
[15]	Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075
[16]	Paul B. Hermanns, Nguyen Van Thoai. Global optimization algorithm for solving bilevel programming problems with quadratic lower levels. Journal of Industrial and Management Optimization, 2010, 6 (1) : 177-196. doi: 10.3934/jimo.2010.6.177
[17]	Shu-Cherng Fang, David Y. Gao, Ruey-Lin Sheu, Soon-Yi Wu. Canonical dual approach to solving 0-1 quadratic programming problems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 125-142. doi: 10.3934/jimo.2008.4.125
[18]	Cheng Lu, Zhenbo Wang, Wenxun Xing, Shu-Cherng Fang. Extended canonical duality and conic programming for solving 0-1 quadratic programming problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 779-793. doi: 10.3934/jimo.2010.6.779
[19]	Charles Curry, Stephen Marsland, Robert I McLachlan. Principal symmetric space analysis. Journal of Computational Dynamics, 2019, 6 (2) : 251-276. doi: 10.3934/jcd.2019013
[20]	Jiani Wang, Liwei Zhang. Statistical inference of semidefinite programming with multiple parameters. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1527-1538. doi: 10.3934/jimo.2019015

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Yong Xia Yu-Jun Gong Sheng-Nan Han
on Google Scholar
Yong Xia Yu-Jun Gong Sheng-Nan Han

[Back to Top]