Global optimization for a class of fractional programming problems

This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric su...

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Bibliographic Details
Published inJournal of global optimization Vol. 45; no. 3; pp. 337 - 353
Main Authors Fang, Shu-Cherng, Gao, David Y., Sheu, Ruey-Lin, Xing, Wenxun
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.11.2009
Springer Nature B.V
Subjects
Computer Science
Eigenvalues
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Ratios
Real Functions
Canonical duality
Global optimization
Sum-of-ratios
Quadratic fractional programming
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Summary:This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the “problem-defining” matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-008-9378-7