Celis-Dennis-Tapia based approachto quadratic fractional programming problemswith two quadratic constraints

Ailing Zhang; Shunsuke Hayashi

doi:10.3934/naco.2011.1.83

Article Contents

2011, Volume 1, Issue 1: 83-98. Doi: 10.3934/naco.2011.1.83

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

Received: August 2010

Revised: October 2010

Published: January 2011

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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.

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Mathematics Subject Classification: Primary: 90C32, 90C20; Secondary: 90C26.

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