An SDP approach for quadratic fractional problems with a two-sided quadratic constraint

We consider a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. On one hand, (P) can be solved under some technical conditions by the Dinkelbach iterative method [W. Dinkelbach, On nonlinear fractional programming, Manag. Sci. 13 (1967), pp. 492–498] which has dominated the development of the area for nearly half a century. On the other hand, some special case of (P), typically the one in Beck and Teboulle [A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid, Math. Program. Ser. A 118 (2009), pp. 13–35], could be directly solved via an exact semi-definite reformulation, rather than iteratively. In this paper, by a recent breakthrough of Xia et al. [S-Lemma with equality and its applications. Available at http://arxiv.org/abs/1403.2816] on the S-lemma with equality, we propose to analyse (P) with three cases and show that each of them admits an exact SDP relaxation. As a result, (P) can be completely solved in polynomial time without any condition. Finally, the paper is presented with many interesting examples to illustrate the idea of our approach and to visualize the structure of the problem.