Complete solutions to nonconvex fractional programming problems

TL;DR

This paper introduces a canonical dual approach for solving a class of non-convex fractional programming problems with elliptic constraints, providing conditions for global optimality and duality gap absence.

Contribution

It develops a novel canonical dual framework that transforms complex non-convex fractional problems into tractable concave maximization problems with no duality gap.

Findings

Canonical dual problems are two-dimensional concave maximizations.

No duality gap exists under certain conditions.

Provides optimality and existence conditions for global minimizers.

Abstract

This paper presents a canonical dual approach to the problem of minimizing the sum of a quadratic function and the ratio of nonconvex function and 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 certain conditions, we show that the canonical dual of each subproblem becomes a two-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.