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October  2017, 13(4): 1625-1640. doi: 10.3934/jimo.2017010

 School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China

* Corresponding author: Jinyan Fan

Received  April 2016 Revised  October 2016 Published  December 2016

Fund Project: The authors are partially supported by NSFC 11171217 and 11571234.

In this paper, we study subspace properties of the quadratically constrained quadratic program (QCQP). We prove that, if an appropriate subspace is chosen to satisfy subspace properties, then the solution of the QCQP lies in that subspace. So, we can solve the QCQP in that subspace rather than solve it in the original space. The computational cost could be reduced significantly if the dimension of the subspace is much smaller. We also show how to construct such subspaces and investigate their dimensions.

Citation: Xin Zhao, Jinyan Fan. On subspace properties of the quadratically constrained quadratic program. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1625-1640. doi: 10.3934/jimo.2017010
##### References:
 [1] F. A. Al-Khayyal, Generalized bilinear programming: Part Ⅰ. Models, applications and linear programming relaxation, European Journal of Operational Research, 60 (1992), 306-314.  doi: 10.1016/0377-2217(92)90082-K. [2] K. M. 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. [3] G. N. Grapiglia, J. Y. Yuan and Y. Yuan, A Subspace version of the Powell-Yuan trust-region algorithm for equality constrained optimization, Journal of the Operations Research Society of China, 1 (2013), 425-451.  doi: 10.1007/s40305-013-0029-4. [4] M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra and its Applications, 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0. [5] E. Phan-huy-Hao, Quadratically constrained quadratic programming: Some applications and a method for solution, Zeitschrift für Operations Research, 26 (1982), 105-119. [6] U. Raber, Nonconvex all-quadratic global optimization problems: Solution methods, application and related topics, 1999. [7] H. D. Sherali and W. P. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-4388-3. [8] N. Z. Shor, Dual estimates in multiextremal problems, Journal of Global Optimization, 2 (1992), 411-418.  doi: 10.1007/BF00122430. [9] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003. [10] Z. H. Wang and Y. X. Yuan, A subspace implementation of quasi-Newton trust region methods for unconstrained optimization, Numerische Mathematik, 104 (2006), 241-269.  doi: 10.1007/s00211-006-0021-6. [11] A. Weintraub and J. Vera, A cutting plane approach for chance constrained linear programs, Operations Research, 39 (1991), 776-785.  doi: 10.1287/opre.39.5.776. [12] Y. X. Yuan, Subspace techniques for nonlinear optimization, in Some Topics in Industrial and Applied Mathematics(eds. R. Jeltsch, D. Q. Li and I. H. Sloan), Higher Education Press, 8 (2007), 206-218 doi: 10.1142/9789812709356_0012. [13] Y. X. Yuan, Subspace methods for large scale nonlinear equations and nonlinear least squares, Optimization and Engineering, 10 (2009), 207-218.  doi: 10.1007/s11081-008-9064-0. [14] Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34.  doi: 10.3934/naco.2011.1.15. [15] X. Zhao and J. Y. Fan, Subspace choices for the Celis-Dennis-Tapia problem, Science China Mathematics, Accepted.

show all references

##### References:
 [1] F. A. Al-Khayyal, Generalized bilinear programming: Part Ⅰ. Models, applications and linear programming relaxation, European Journal of Operational Research, 60 (1992), 306-314.  doi: 10.1016/0377-2217(92)90082-K. [2] K. M. 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. [3] G. N. Grapiglia, J. Y. Yuan and Y. Yuan, A Subspace version of the Powell-Yuan trust-region algorithm for equality constrained optimization, Journal of the Operations Research Society of China, 1 (2013), 425-451.  doi: 10.1007/s40305-013-0029-4. [4] M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra and its Applications, 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0. [5] E. Phan-huy-Hao, Quadratically constrained quadratic programming: Some applications and a method for solution, Zeitschrift für Operations Research, 26 (1982), 105-119. [6] U. Raber, Nonconvex all-quadratic global optimization problems: Solution methods, application and related topics, 1999. [7] H. D. Sherali and W. P. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-4388-3. [8] N. Z. Shor, Dual estimates in multiextremal problems, Journal of Global Optimization, 2 (1992), 411-418.  doi: 10.1007/BF00122430. [9] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003. [10] Z. H. Wang and Y. X. Yuan, A subspace implementation of quasi-Newton trust region methods for unconstrained optimization, Numerische Mathematik, 104 (2006), 241-269.  doi: 10.1007/s00211-006-0021-6. [11] A. Weintraub and J. Vera, A cutting plane approach for chance constrained linear programs, Operations Research, 39 (1991), 776-785.  doi: 10.1287/opre.39.5.776. [12] Y. X. Yuan, Subspace techniques for nonlinear optimization, in Some Topics in Industrial and Applied Mathematics(eds. R. Jeltsch, D. Q. Li and I. H. Sloan), Higher Education Press, 8 (2007), 206-218 doi: 10.1142/9789812709356_0012. [13] Y. X. Yuan, Subspace methods for large scale nonlinear equations and nonlinear least squares, Optimization and Engineering, 10 (2009), 207-218.  doi: 10.1007/s11081-008-9064-0. [14] Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34.  doi: 10.3934/naco.2011.1.15. [15] X. Zhao and J. Y. Fan, Subspace choices for the Celis-Dennis-Tapia problem, Science China Mathematics, Accepted.
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