Globally optimal solutions to a class of fractional optimization problems based on proximity gradient algorithm

We establish globally optimal solutions to a class of fractional optimization problems on a class of constraint sets, whose key characteristics are as follows: 1) The numerator and the denominator of the objective function are both convex, semi-algebraic, Lipschitz continuous and differentiable with Lipschitz continuous gradients on the constraint set. 2) The constraint set is closed, convex and semi-algebraic. Compared with Dinkelbach's approach, our novelty falls into the following aspects: 1) Dinkelbach's has to solve a concave maximization problem in each iteration, which is nontrivial to obtain a solution, while ours only needs to conduct one proximity gradient operation in each iteration. 2) Dinkelbach's requires at least one nonnegative point for the numerator to proceed the algorithm, but ours does not, which is available to a much wider class of situations. 3) Dinkelbach's requires a closed and bounded constraint set, while ours only needs the closedness but not necessarily the boundedness. Therefore, our approach is viable for many more practical models, like optimizing the Sharpe ratio (SR) or the Information ratio in mathematical finance. Numerical experiments show that our approach achieves the ground-truth solutions in two simple examples. For real-world financial data, it outperforms several existing approaches for SR maximization.