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Numerical Algebra, Control and Optimization (NACO)
 

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

Pages: 83 - 98, Volume 1, Issue 1, March 2011      doi:10.3934/naco.2011.1.83

 
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Ailing Zhang - Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan (email)
Shunsuke Hayashi - Department of Applied Mathematics and Physics,, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan (email)

Abstract: 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:  Fractional programming, Celis-Dennis-Tapia subproblems, nonconvex quadratic programming, bisection method, generalized Newton method.
Mathematics Subject Classification:  Primary: 90C32, 90C20; Secondary: 90C26.

Received: August 2010;      Revised: October 2010;      Available Online: February 2011.

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