Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints

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2011, 1(1): 83-98. doi: 10.3934/naco.2011.1.83

Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints

Ailing Zhang ^1, and Shunsuke Hayashi ^2,

1.	Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan
2.	Department of Applied Mathematics and Physics,, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan

Received August 2010 Revised October 2010 Published February 2011

Abstract
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In this paper, we focus on fractional programming problems that minimize the ratio of two indefinite quadratic functions subject to two quadratic constraints. Utilizing the relationship between fractional programming and parametric programming, we transform the original problem into a univariate nonlinear equation. To evaluate the function in the equation, we need to solve a problem of minimizing a nonconvex quadratic function subject to two quadratic constraints. This problem is commonly called a Celis-Dennis-Tapia (CDT) subproblem, which arises in some trust region algorithms for equality constrained optimization. In the outer iterations of the algorithm, we employ the bisection method and/or the generalized Newton method. In the inner iterations, we utilize Yuan's theorem to obtain the global optima of the CDT subproblems. We also show some numerical results to examine the efficiency of the algorithm. Particularly, we will observe that the generalized Newton method is more robust to the erroneous evaluation for the univariate functions than the bisection method.

Keywords: nonconvex quadratic programming, Fractional programming, generalized Newton method., Celis-Dennis-Tapia subproblems, bisection method.

Mathematics Subject Classification: Primary: 90C32, 90C20; Secondary: 90C2.

Citation: Ailing Zhang, Shunsuke Hayashi. Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 83-98. doi: 10.3934/naco.2011.1.83

References:

[1]	A. Beck, A. Ben-Tal and M. Teboulle, Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares,, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 425. doi: 10.1137/040616851.
[2]	A. Beck and Y. C. Eldar, Strong duality in nonconvex quadratic optimization with two quadratic constraints,, SIAM Journal on Optimization, 17 (2006), 844. doi: 10.1137/050644471.
[3]	H. P. Benson, Fractional programming with convex quadratic forms and functions,, European Journal of Operational Research, 173 (2006), 351. doi: 10.1016/j.ejor.2005.02.069.
[4]	G. R. Bitran and T. L. Magnanti, Duality and sensitivity analysis for fractional programs,, Operations Research, 24 (1976), 675. doi: 10.1287/opre.24.4.675.
[5]	M. R. Celis, J. E. Dennis and R. A. Tapia, A trust region strategy for nonlinear equality constrained optimization,, in, (1985), 71.
[6]	A. Charnes and W. W. Cooper, Programming with linear fractional functionals,, Naval Research Logistics Quarterly, 9 (1962), 181. doi: 10.1002/nav.3800090303.
[7]	X. Chen and Y. Yuan, On local solutions of the Celis-Dennis-Tapia subproblem,, SIAM Journal on Optimization, 10 (2000), 359. doi: 10.1137/S1052623498335018.
[8]	X. Chen and Y. Yuan, On maxima of dual function of the CDT subproblem,, Journal of Computational Mathematics, 19 (2001), 113.
[9]	X. Chen and Y. Yuan, Optimality conditions for CDT subproblem,, in, (1999), 111.
[10]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000). doi: 10.1137/1.9780898719857.
[11]	J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming,, Mathematical Programming, 52 (1991), 191. doi: 10.1007/BF01582887.
[12]	W. Dinkelbach, On nonlinear fractional programming,, Management Science, 13 (1967), 492. doi: 10.1287/mnsc.13.7.492.
[13]	J. Gotoh and H. Konno, Maximization of the ratio of two convex quadratic functions over a polytope,, Computational Optimization and Applications, 20 (2001), 43. doi: 10.1023/A:1011219422283.
[14]	T. Ibaraki, Parametric approaches to fractional programs,, Mathematical Programming, 26 (1983), 345. doi: 10.1007/BF02591871.
[15]	T. Ibaraki, H. Ishii, J. Iwase, T. Hasegawa and H. Mine, Algorithms for quadratic fractional programming problems,, Journal of Operational Research Society of Japan, 19 (1976), 174.
[16]	R. Jagannathan, On some properties of programming problems in parametric form pertaining to fractional programming,, Management Science, 12 (1966), 609. doi: 10.1287/mnsc.12.7.609.
[17]	G. Li and Y. Yuan, Compute a Celis-Dennis-Tapia step,, Journal of Computational Mathematics, 23 (2005), 463.
[18]	J. Peng and Y. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints,, SIAM Journal on Optimization, 7 (1997), 579. doi: 10.1137/S1052623494261520.
[19]	M. J. D. Powell and Y. Yuan, A trust region algorithm for equality constrained optimization,, Mathematical Programming, 49 (1991), 189. doi: 10.1007/BF01588787.
[20]	J. Von Neumann, Über ein es Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpuntsatzes,, in, (1937), 73.
[21]	Y. Ye and S. Zhang, New results on quadratic minimization,, SIAM Journal on Optimization, 14 (2003), 245. doi: 10.1137/S105262340139001X.
[22]	Y. Yuan, On a subproblem of trust region algorithms for constrained optimization,, Mathematical Programming, 47 (1990), 53. doi: 10.1007/BF01580852.
[23]	Y. Yuan, A dual algorithm for minimizing a quadratic function with two quadratic constraints,, Journal of Computational Mathematics, 9 (1991), 348.
[24]	Y. Zhang, Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization,, Mathematical Programming, 55 (1992), 109. doi: 10.1007/BF01581194.

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References:

[1]	A. Beck, A. Ben-Tal and M. Teboulle, Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares,, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 425. doi: 10.1137/040616851.
[2]	A. Beck and Y. C. Eldar, Strong duality in nonconvex quadratic optimization with two quadratic constraints,, SIAM Journal on Optimization, 17 (2006), 844. doi: 10.1137/050644471.
[3]	H. P. Benson, Fractional programming with convex quadratic forms and functions,, European Journal of Operational Research, 173 (2006), 351. doi: 10.1016/j.ejor.2005.02.069.
[4]	G. R. Bitran and T. L. Magnanti, Duality and sensitivity analysis for fractional programs,, Operations Research, 24 (1976), 675. doi: 10.1287/opre.24.4.675.
[5]	M. R. Celis, J. E. Dennis and R. A. Tapia, A trust region strategy for nonlinear equality constrained optimization,, in, (1985), 71.
[6]	A. Charnes and W. W. Cooper, Programming with linear fractional functionals,, Naval Research Logistics Quarterly, 9 (1962), 181. doi: 10.1002/nav.3800090303.
[7]	X. Chen and Y. Yuan, On local solutions of the Celis-Dennis-Tapia subproblem,, SIAM Journal on Optimization, 10 (2000), 359. doi: 10.1137/S1052623498335018.
[8]	X. Chen and Y. Yuan, On maxima of dual function of the CDT subproblem,, Journal of Computational Mathematics, 19 (2001), 113.
[9]	X. Chen and Y. Yuan, Optimality conditions for CDT subproblem,, in, (1999), 111.
[10]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,", SIAM, (2000). doi: 10.1137/1.9780898719857.
[11]	J. P. Crouzeix and J. A. Ferland, Algorithms for generalized fractional programming,, Mathematical Programming, 52 (1991), 191. doi: 10.1007/BF01582887.
[12]	W. Dinkelbach, On nonlinear fractional programming,, Management Science, 13 (1967), 492. doi: 10.1287/mnsc.13.7.492.
[13]	J. Gotoh and H. Konno, Maximization of the ratio of two convex quadratic functions over a polytope,, Computational Optimization and Applications, 20 (2001), 43. doi: 10.1023/A:1011219422283.
[14]	T. Ibaraki, Parametric approaches to fractional programs,, Mathematical Programming, 26 (1983), 345. doi: 10.1007/BF02591871.
[15]	T. Ibaraki, H. Ishii, J. Iwase, T. Hasegawa and H. Mine, Algorithms for quadratic fractional programming problems,, Journal of Operational Research Society of Japan, 19 (1976), 174.
[16]	R. Jagannathan, On some properties of programming problems in parametric form pertaining to fractional programming,, Management Science, 12 (1966), 609. doi: 10.1287/mnsc.12.7.609.
[17]	G. Li and Y. Yuan, Compute a Celis-Dennis-Tapia step,, Journal of Computational Mathematics, 23 (2005), 463.
[18]	J. Peng and Y. Yuan, Optimality conditions for the minimization of a quadratic with two quadratic constraints,, SIAM Journal on Optimization, 7 (1997), 579. doi: 10.1137/S1052623494261520.
[19]	M. J. D. Powell and Y. Yuan, A trust region algorithm for equality constrained optimization,, Mathematical Programming, 49 (1991), 189. doi: 10.1007/BF01588787.
[20]	J. Von Neumann, Über ein es Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpuntsatzes,, in, (1937), 73.
[21]	Y. Ye and S. Zhang, New results on quadratic minimization,, SIAM Journal on Optimization, 14 (2003), 245. doi: 10.1137/S105262340139001X.
[22]	Y. Yuan, On a subproblem of trust region algorithms for constrained optimization,, Mathematical Programming, 47 (1990), 53. doi: 10.1007/BF01580852.
[23]	Y. Yuan, A dual algorithm for minimizing a quadratic function with two quadratic constraints,, Journal of Computational Mathematics, 9 (1991), 348.
[24]	Y. Zhang, Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization,, Mathematical Programming, 55 (1992), 109. doi: 10.1007/BF01581194.

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