Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming

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July 2013, 9(3): 703-721. doi: 10.3934/jimo.2013.9.703

Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming

Ye Tian ^1, , Shu-Cherng Fang ^2, , Zhibin Deng ^2, and Wenxun Xing ^3,

1.	School of Business Administration, Southwestern University of Finance and Economics, Chengdu, 611130, China
2.	Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, United States, United States
3.	Department of Mathematical Sciences, Tsinghua University, Beijing

Received October 2012 Revised February 2013 Published April 2013

Abstract
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In this paper, we provide a computable representation of the cone of nonnegative quadratic forms over a general nontrivial second-order cone using linear matrix inequalities (LMI). By constructing a sequence of such computable cones over a union of second-order cones, an efficient algorithm is designed to find an approximate solution to a completely positive programming problem using semidefinite programming techniques. In order to accelerate the convergence of the approximation sequence, an adaptive scheme is adopted, and ``reformulation-linearization technique'' (RLT) constraints are added to further improve its efficiency.

Keywords: Cone of nonnegative quadratic forms general nontrivial second-order cone completely positive programming approximation adaptive scheme..

Mathematics Subject Classification: Primary: 90C26, 90C59, 90C22; Secondary: 30E1.

Citation: Ye Tian, Shu-Cherng Fang, Zhibin Deng, Wenxun Xing. Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. Journal of Industrial and Management Optimization, 2013, 9 (3) : 703-721. doi: 10.3934/jimo.2013.9.703

References:

[1]	K. Anstreicher, Semidefinite programming versus the Reformulation-Linearization Technique for nonconvex quadratically constrained quadratic programming, Journal of Global Optimization, 43 (2009), 471-484. doi: 10.1007/s10898-008-9372-0.
[2]	A. Ben-Tal and A. Nemirovski, "Lectures on Modern Convex Optimization Analysis, Algorithms and Engineering Applications," SIAM, Philadelphis, 2001. doi: 10.1137/1.9780898718829.
[3]	I. Bomze and E. de Klerk, Solving standard quadratic optimization problem via linear, semidefinite and copositive programming, Journal of Global Optimization, 24 (2002), 163-185. doi: 10.1023/A:1020209017701.
[4]	I. Bomze and G. Eichfelder, Copositivity Detection by difference-of-convex decomposition and $\omega$-subdivision, to appear in Mathematical Programming. doi: 10.1007/s10107-012-0543-x.
[5]	I. Bomze, F. Jarre and F. Rendl, Quadratic factorization heuristics for copositive programming, Mathematical Programming Computation, 3 (2011), 37-57. doi: 10.1007/s12532-011-0022-z.
[6]	M. Brockington and J. Culberson, "Camouflaging Independent Sets in Quasi-random Graphs," Clique Coloring and Satisfiability: Second DIMACS Implementation Challenge, 26, Amer Mathematical Society, 1994.
[7]	S. Bundfuss and M. Dür, An adaptive linear approximation algorithm for copositive programs, SIAM Journal on Optimization, 20 (2009), 30-53. doi: 10.1137/070711815.
[8]	S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.
[9]	S. Burer and K. Anstreicher, Second-order cone constraints for extended trust-region subproblems, submitted to SIAM Journal on Optimization, (2011). doi: 10.1137/110826862.
[10]	Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, accepted by European Journal of Operational Research, (2013). doi: 10.1016/j.ejor.2013.02.031.
[11]	, DIMACS Implementation Challenges. ftp://dimacs.rutgers.edu/pub/challenge/graph/benchmarks/clique
[12]	M. Dür, "Copositive Programming: A Survey," Recent Advances in Optimization and Its Application in Engineering, Springer, New York, 2012.
[13]	N. Govozdenovic and M. Laurent, The operator $\Psi$ for the chromatic number of a graph, SIAM Journal on Optimization, 19 (2008), 572-591. doi: 10.1137/050648237.
[14]	M. Grant and S. Boyd, "CVX: matlab Software for Disciplined Programming," version 1.2, 2010. http://cvxr.com/cvx
[15]	P. Hansen, B. Jaumard, M. Ruiz and J. Xiong, Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics, 40 (1993), 373-392. doi: 10.1002/1520-6750(199304)40:3<373::AID-NAV3220400307>3.0.CO;2-A.
[16]	K. Ikramov, Linear-time algorithm for verifying the copositivity of an acyclic matrix, Computational Mathematics and Mathmetical Physics, 42 (2002), 1701-1703.
[17]	E. de Klerk and D. Pasechnik, Approximation of the stability number of a graph via copositive programming, Journal of Global Optimization, 12 (2002), 875-892. doi: 10.1137/S1052623401383248.
[18]	C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM Journal on Optimization, 21 (2010), 1475-1490. doi: 10.1137/100793955.
[19]	C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions, Submitted to Mathematical Programming, (2011).
[20]	T. Motzkin and E. Straus, Maxima for graphs and a new proof of a theorem of Turan, Canadian Journal of Mathematics, 17 (1965), 533-540. doi: 10.4153/CJM-1965-053-6.
[21]	K. Murty and S. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39 (1987), 117-129. doi: 10.1007/BF02592948.
[22]	J. Povh and F. Rendl, Copositive and semidefinite relaxations of the quadratic assignment problem, Discrete Optimization, 6 (2009), 231-241. doi: 10.1016/j.disopt.2009.01.002.
[23]	J. Preisig, Copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints, SIAM Journal on Control and Optimization, 34 (1996), 1135-1150. doi: 10.1137/S0363012993251894.
[24]	R. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1996.
[25]	N. Sahinidis and M. Tawarmalani, "BARON 9.0.4: Global Optimization of Mixed-Integer Nonlinear Programs," 2010. http://archimedes.cheme.cmu.edu/baron/baron.html
[26]	L. Sanchis, Test case construction for the vertex cover problem, in "Computational Support for Discrete Mathematics," DIMACS Series in Discrete Mathematics and Theoretical American Mathematical Society, 15 (1992), Providence, Rhodle Island, (1992).
[27]	J. Sturm, SeDuMi 1.02, a matlab tool box for optimization over symmetric cones, Optimization Methods and Software, 11&12 (1999), 625-653. doi: 10.1080/10556789908805766.
[28]	J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.
[29]	Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.
[30]	J. Žilinskas, Copositive programming by simplicial partition, Informatica, 22 (2011), 601-614.

show all references

References:

[1]	K. Anstreicher, Semidefinite programming versus the Reformulation-Linearization Technique for nonconvex quadratically constrained quadratic programming, Journal of Global Optimization, 43 (2009), 471-484. doi: 10.1007/s10898-008-9372-0.
[2]	A. Ben-Tal and A. Nemirovski, "Lectures on Modern Convex Optimization Analysis, Algorithms and Engineering Applications," SIAM, Philadelphis, 2001. doi: 10.1137/1.9780898718829.
[3]	I. Bomze and E. de Klerk, Solving standard quadratic optimization problem via linear, semidefinite and copositive programming, Journal of Global Optimization, 24 (2002), 163-185. doi: 10.1023/A:1020209017701.
[4]	I. Bomze and G. Eichfelder, Copositivity Detection by difference-of-convex decomposition and $\omega$-subdivision, to appear in Mathematical Programming. doi: 10.1007/s10107-012-0543-x.
[5]	I. Bomze, F. Jarre and F. Rendl, Quadratic factorization heuristics for copositive programming, Mathematical Programming Computation, 3 (2011), 37-57. doi: 10.1007/s12532-011-0022-z.
[6]	M. Brockington and J. Culberson, "Camouflaging Independent Sets in Quasi-random Graphs," Clique Coloring and Satisfiability: Second DIMACS Implementation Challenge, 26, Amer Mathematical Society, 1994.
[7]	S. Bundfuss and M. Dür, An adaptive linear approximation algorithm for copositive programs, SIAM Journal on Optimization, 20 (2009), 30-53. doi: 10.1137/070711815.
[8]	S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.
[9]	S. Burer and K. Anstreicher, Second-order cone constraints for extended trust-region subproblems, submitted to SIAM Journal on Optimization, (2011). doi: 10.1137/110826862.
[10]	Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, accepted by European Journal of Operational Research, (2013). doi: 10.1016/j.ejor.2013.02.031.
[11]	, DIMACS Implementation Challenges. ftp://dimacs.rutgers.edu/pub/challenge/graph/benchmarks/clique
[12]	M. Dür, "Copositive Programming: A Survey," Recent Advances in Optimization and Its Application in Engineering, Springer, New York, 2012.
[13]	N. Govozdenovic and M. Laurent, The operator $\Psi$ for the chromatic number of a graph, SIAM Journal on Optimization, 19 (2008), 572-591. doi: 10.1137/050648237.
[14]	M. Grant and S. Boyd, "CVX: matlab Software for Disciplined Programming," version 1.2, 2010. http://cvxr.com/cvx
[15]	P. Hansen, B. Jaumard, M. Ruiz and J. Xiong, Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics, 40 (1993), 373-392. doi: 10.1002/1520-6750(199304)40:3<373::AID-NAV3220400307>3.0.CO;2-A.
[16]	K. Ikramov, Linear-time algorithm for verifying the copositivity of an acyclic matrix, Computational Mathematics and Mathmetical Physics, 42 (2002), 1701-1703.
[17]	E. de Klerk and D. Pasechnik, Approximation of the stability number of a graph via copositive programming, Journal of Global Optimization, 12 (2002), 875-892. doi: 10.1137/S1052623401383248.
[18]	C. Lu, S.-C. Fang, Q. Jin, Z. Wang and W. Xing, KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems, SIAM Journal on Optimization, 21 (2010), 1475-1490. doi: 10.1137/100793955.
[19]	C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions, Submitted to Mathematical Programming, (2011).
[20]	T. Motzkin and E. Straus, Maxima for graphs and a new proof of a theorem of Turan, Canadian Journal of Mathematics, 17 (1965), 533-540. doi: 10.4153/CJM-1965-053-6.
[21]	K. Murty and S. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39 (1987), 117-129. doi: 10.1007/BF02592948.
[22]	J. Povh and F. Rendl, Copositive and semidefinite relaxations of the quadratic assignment problem, Discrete Optimization, 6 (2009), 231-241. doi: 10.1016/j.disopt.2009.01.002.
[23]	J. Preisig, Copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints, SIAM Journal on Control and Optimization, 34 (1996), 1135-1150. doi: 10.1137/S0363012993251894.
[24]	R. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1996.
[25]	N. Sahinidis and M. Tawarmalani, "BARON 9.0.4: Global Optimization of Mixed-Integer Nonlinear Programs," 2010. http://archimedes.cheme.cmu.edu/baron/baron.html
[26]	L. Sanchis, Test case construction for the vertex cover problem, in "Computational Support for Discrete Mathematics," DIMACS Series in Discrete Mathematics and Theoretical American Mathematical Society, 15 (1992), Providence, Rhodle Island, (1992).
[27]	J. Sturm, SeDuMi 1.02, a matlab tool box for optimization over symmetric cones, Optimization Methods and Software, 11&12 (1999), 625-653. doi: 10.1080/10556789908805766.
[28]	J. Sturm and S. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.
[29]	Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.
[30]	J. Žilinskas, Copositive programming by simplicial partition, Informatica, 22 (2011), 601-614.

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