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

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

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

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.
Citation: Yong Xia, Yu-Jun Gong, Sheng-Nan Han. A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions. Numerical Algebra, Control & 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.  doi: 10.1023/A:1008364005245.  Google Scholar

[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.  doi: 10.1007/s11590-006-0018-1.  Google Scholar

[3]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust-Region Methods,, MPS/SIAM Series on Optimization. SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[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.   Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming,, version 1. 21, (2010).   Google Scholar

[6]

Y. Hsia, Complexity and Nonlinear Semidefinite Programming Reformulation of l1-constrained Nonconvex Quadratic Optimization,, Optimization Letters, 8 (2014), 1433.  doi: 10.1007/s11590-013-0670-1.  Google Scholar

[7]

S. Khot and A. Naor, Grothendieck-type inequalities in combinatorial optimization,, Communications on Pure and Applied Mathematics, 65 (2012), 992.  doi: 10.1002/cpa.21398.  Google Scholar

[8]

G. Kindler, A. Naor and G. Schechtman, The UGC hardness threshold of the Grothendieck problem,, Math. Oper. Res., 35 (2010), 267.  doi: 10.1287/moor.1090.0425.  Google Scholar

[9]

L. Lovasz and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization,, SIAM. J. Optimization, 1 (1991), 166.  doi: 10.1137/0801013.  Google Scholar

[10]

R. Luss and M. Teboulle, Convex Approximations to Sparse PCA via Lagrangian Duality,, Operations Research Letters, 39 (2011), 57.  doi: 10.1016/j.orl.2010.11.005.  Google Scholar

[11]

J. M. Martínez, Local minimizers of quadratic functions on Euclidean balls and spheres,, SIAM. J. Optimization. 4 (1994), 4 (1994), 159.  doi: 10.1137/0804009.  Google Scholar

[12]

Y. Nesterov, Global Quadratic Optimization via Conic Relaxation,, in Handbook of Semidefinite Programming, (2000), 363.   Google Scholar

[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.  doi: 10.1051/ro:2006023.  Google Scholar

[14]

J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimation over symmetric cones,, Optimization Methods and Software, 11-12 (1999), 11.  doi: 10.1080/10556789908805766.  Google Scholar

[15]

Y. Xia, New results on semidefinite bounds for l1-constrained nonconvex quadratic optimization,, RAIRO-Operations Research, 47 (2013), 285.  doi: 10.1051/ro/2013039.  Google Scholar

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.  doi: 10.1023/A:1008364005245.  Google Scholar

[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.  doi: 10.1007/s11590-006-0018-1.  Google Scholar

[3]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust-Region Methods,, MPS/SIAM Series on Optimization. SIAM, (2000).  doi: 10.1137/1.9780898719857.  Google Scholar

[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.   Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming,, version 1. 21, (2010).   Google Scholar

[6]

Y. Hsia, Complexity and Nonlinear Semidefinite Programming Reformulation of l1-constrained Nonconvex Quadratic Optimization,, Optimization Letters, 8 (2014), 1433.  doi: 10.1007/s11590-013-0670-1.  Google Scholar

[7]

S. Khot and A. Naor, Grothendieck-type inequalities in combinatorial optimization,, Communications on Pure and Applied Mathematics, 65 (2012), 992.  doi: 10.1002/cpa.21398.  Google Scholar

[8]

G. Kindler, A. Naor and G. Schechtman, The UGC hardness threshold of the Grothendieck problem,, Math. Oper. Res., 35 (2010), 267.  doi: 10.1287/moor.1090.0425.  Google Scholar

[9]

L. Lovasz and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization,, SIAM. J. Optimization, 1 (1991), 166.  doi: 10.1137/0801013.  Google Scholar

[10]

R. Luss and M. Teboulle, Convex Approximations to Sparse PCA via Lagrangian Duality,, Operations Research Letters, 39 (2011), 57.  doi: 10.1016/j.orl.2010.11.005.  Google Scholar

[11]

J. M. Martínez, Local minimizers of quadratic functions on Euclidean balls and spheres,, SIAM. J. Optimization. 4 (1994), 4 (1994), 159.  doi: 10.1137/0804009.  Google Scholar

[12]

Y. Nesterov, Global Quadratic Optimization via Conic Relaxation,, in Handbook of Semidefinite Programming, (2000), 363.   Google Scholar

[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.  doi: 10.1051/ro:2006023.  Google Scholar

[14]

J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimation over symmetric cones,, Optimization Methods and Software, 11-12 (1999), 11.  doi: 10.1080/10556789908805766.  Google Scholar

[15]

Y. Xia, New results on semidefinite bounds for l1-constrained nonconvex quadratic optimization,, RAIRO-Operations Research, 47 (2013), 285.  doi: 10.1051/ro/2013039.  Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[2]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[3]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[4]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[5]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[6]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[7]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[8]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[9]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[10]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[11]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[12]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[13]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[14]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

 Impact Factor: 

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]