Globally Optimal Solutions to a Class of Fractional Optimization Problems Based on Proximal Gradient Algorithm

This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both being convex, semi-algebraic, Lipschitz continuous, and differentiable with Lipschitz continuous gradients over the constraint sets. The constrained sets associated with these problems are closed, convex, and semi-algebraic. We propose an efficient algorithm that is inspired by the proximal gradient method, and we provide a thorough convergence analysis. Our algorithm offers several benefits compared to existing methods. It requires only a single proximal gradient operation per iteration, thus avoiding the complicated inner-loop concave maximization usually required. Additionally, our method converges to a critical point without the typical need for a nonnegative numerator, and this critical point becomes a globally optimal solution with an appropriate condition. Our approach is adaptable to unbounded constraint sets as well. Therefore, our approach is viable for many more practical models. Numerical experiments show that our method not only reliably reaches ground-truth solutions in some model problems but also outperforms several existing methods in maximizing the Sharpe ratio with real-world financial data.