American Institute of Mathematical Sciences

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July  2014, 10(3): 945-963. doi: 10.3934/jimo.2014.10.945

Quadratic optimization over one first-order cone

Xiaoling Guo 1, , Zhibin Deng 2, , Shu-Cherng Fang 2, and  Wenxun Xing 1,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
2. 

Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 27695-7906, United States, United States

Received  February 2013 Revised  June 2013 Published  November 2013

This paper studies the first-order cone constrained homogeneous quadratic programming problem. For efficient computation, the problem is reformulated as a linear conic programming problem. A union of second-order cones are designed to cover the first-order cone such that a sequence of linear conic programming problems can be constructed to approximate the conic reformulation. Since the cone of nonnegative quadratic forms over a union of second-order cones has a linear matrix inequalities representation, each linear conic programming problem in the sequence is polynomial-time solvable by applying semidefinite programming techniques. The convergence of the sequence is guaranteed when the union of second-order cones gets close enough to the first-order cone. In order to further improve the efficiency, an adaptive scheme is adopted. Numerical experiments are provided to illustrate the efficiency of the proposed approach.
Keywords: first-order cone, linear conic programming, adaptive scheme., cone of nonnegative quadratic forms, Quadratic programming.
Mathematics Subject Classification: Primary: 90C20, 90C26; Secondary: 15B48, 65Y2.
Citation: Xiaoling Guo, Zhibin Deng, Shu-Cherng Fang, Wenxun Xing. Quadratic optimization over one first-order cone. Journal of Industrial & Management Optimization, 2014, 10 (3) : 945-963. doi: 10.3934/jimo.2014.10.945
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F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5.  Google Scholar

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E. D. Andersen, C. Roos and T. Terlaky, Notes on duality in second order and $p$-order cone optimization, Optimization, 51 (2002), 627-643. doi: 10.1080/0233193021000030751.  Google Scholar

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P. Belotti, J. C. Góez, I. Pólik, T. K. Ralphs and T. Terlaky, On families of quadratic surfaces having fixed intersections with two hyperplanes, Discrete Applied Mathematics, 161 (2013), 2778-2793. Available from: http://www.optimization-online.org/DB_FILE/2012/08/3563.pdf. doi: 10.1016/j.dam.2013.05.017.  Google Scholar

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S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  Google Scholar

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S. Burer, On the copositive representation of binary and continous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.  Google Scholar

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S. Burer and K. M. Anstreicher, Second-order-cone constraints for extended trust-region subproblems, SIAM Journal on Optimization, 23 (2013), 432-451. doi: 10.1137/110826862.  Google Scholar

[9]

Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28. doi: 10.1016/j.ejor.2013.02.031.  Google Scholar

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G. Eichfelder and J. Povh, On reformulations of nonconvex quadratic programs over convex cones by set-semidefinite constraints, preprint, 2010. Available from: http://www.optimization-online.org/DB_FILE/2010/12/2843.pdf. Google Scholar

[11]

Q. Jin, Quadratically Constrained Quadratic Programming Problems and Extensions, Ph.D thesis, North Carolina State University, 2011.  Google Scholar

[12]

Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of Operations Research Society of China, 1 (2013), 107-134. doi: 10.1007/s40305-013-0009-8.  Google Scholar

[13]

C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions, working paper, 2011. Google Scholar

[14]

M. W. Margaret, Advances in cone-based preference modeling for decision making with multiple criteria, Decision Making in Manufacturing and Services, 1 (2007), 153-173.  Google Scholar

[15]

Y. Nesterov and A. Nemirovsky, Interior-point Polynomial Methods in Convex Programming, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[16]

R. T. Rockafellar, Convex Analysis, 2nd edition, Princeton University Press, Princeton, NJ, 1972. Google Scholar

[17]

J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods, Optimization Methods and Software, 11/12 (1999), 625-653. doi: 10.1080/10556789908805766.  Google Scholar

[18]

J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.  Google Scholar

[19]

Y. Tian, S.-C. Fang, Z. Deng and W. 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, 9 (2013), 703-721. doi: 10.3934/jimo.2013.9.703.  Google Scholar

[20]

S. A. Vavasis, Nonlinear Optimization: Complexity Issues, International Series of Monographs on Computer Science, 8, The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[21]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.  Google Scholar

show all references

References:
[1]

F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

E. D. Andersen, C. Roos and T. Terlaky, Notes on duality in second order and $p$-order cone optimization, Optimization, 51 (2002), 627-643. doi: 10.1080/0233193021000030751.  Google Scholar

[3]

P. Belotti, J. C. Góez, I. Pólik, T. K. Ralphs and T. Terlaky, On families of quadratic surfaces having fixed intersections with two hyperplanes, Discrete Applied Mathematics, 161 (2013), 2778-2793. Available from: http://www.optimization-online.org/DB_FILE/2012/08/3563.pdf. doi: 10.1016/j.dam.2013.05.017.  Google Scholar

[4]

A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2001. doi: 10.1137/1.9780898718829.  Google Scholar

[5]

E. Bishop and R. R. Phelps, The support functionals of a convex set, in Proceedings of Symposia in Pure Mathematics, Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, 27-35.  Google Scholar

[6]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  Google Scholar

[7]

S. Burer, On the copositive representation of binary and continous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.  Google Scholar

[8]

S. Burer and K. M. Anstreicher, Second-order-cone constraints for extended trust-region subproblems, SIAM Journal on Optimization, 23 (2013), 432-451. doi: 10.1137/110826862.  Google Scholar

[9]

Z. Deng, S.-C. Fang, Q. Jin and W. Xing, Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme, European Journal of Operational Research, 229 (2013), 21-28. doi: 10.1016/j.ejor.2013.02.031.  Google Scholar

[10]

G. Eichfelder and J. Povh, On reformulations of nonconvex quadratic programs over convex cones by set-semidefinite constraints, preprint, 2010. Available from: http://www.optimization-online.org/DB_FILE/2010/12/2843.pdf. Google Scholar

[11]

Q. Jin, Quadratically Constrained Quadratic Programming Problems and Extensions, Ph.D thesis, North Carolina State University, 2011.  Google Scholar

[12]

Q. Jin, Y. Tian, Z. Deng, S.-C. Fang and W. Xing, Exact computable representation of some second-order cone constrained quadratic programming problems, Journal of Operations Research Society of China, 1 (2013), 107-134. doi: 10.1007/s40305-013-0009-8.  Google Scholar

[13]

C. Lu, Q. Jin, S.-C. Fang, Z. Wang and W. Xing, An LMI based adaptive approximation scheme to cones of nonnegative quadratic functions, working paper, 2011. Google Scholar

[14]

M. W. Margaret, Advances in cone-based preference modeling for decision making with multiple criteria, Decision Making in Manufacturing and Services, 1 (2007), 153-173.  Google Scholar

[15]

Y. Nesterov and A. Nemirovsky, Interior-point Polynomial Methods in Convex Programming, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[16]

R. T. Rockafellar, Convex Analysis, 2nd edition, Princeton University Press, Princeton, NJ, 1972. Google Scholar

[17]

J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Interior point methods, Optimization Methods and Software, 11/12 (1999), 625-653. doi: 10.1080/10556789908805766.  Google Scholar

[18]

J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485.  Google Scholar

[19]

Y. Tian, S.-C. Fang, Z. Deng and W. 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, 9 (2013), 703-721. doi: 10.3934/jimo.2013.9.703.  Google Scholar

[20]

S. A. Vavasis, Nonlinear Optimization: Complexity Issues, International Series of Monographs on Computer Science, 8, The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[21]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267. doi: 10.1137/S105262340139001X.  Google Scholar

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