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.
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Fang, SC., Gao, D.Y., Sheu, RL. et al. Global optimization for a class of fractional programming problems. J Glob Optim 45, 337–353 (2009). https://doi.org/10.1007/s10898-008-9378-7
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DOI: https://doi.org/10.1007/s10898-008-9378-7