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A wedge trust region method with self-correcting geometry for derivative-free optimization
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 |
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. |
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