Article ID: | iaor200970814 |
Country: | Netherlands |
Volume: | 45 |
Issue: | 3 |
Start Page Number: | 337 |
End Page Number: | 353 |
Publication Date: | Nov 2009 |
Journal: | Journal of Global Optimization |
Authors: | Fang Shu-Cherng, Xing Wenxun, Gao DavidY, Sheu Ruey-Lin |
Keywords: | duality |
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.