Global optimization for a class of fractional programming problems

  • Authors:

  • Shu-Cherng Fang;David Y. Gao;Ruey-Lin Sheu;Wenxun Xing

  • Affiliations:

  • Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA;Department of Mathematics, Virginia Tech, Blacksburgh, USA;Department of Mathematics, National Cheng Kung University, Tainan, Taiwan, ROC;Department of Mathematical Sciences, Tsinghua University, Beijing, China

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2009

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Abstract

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.