In this paper, we investigate 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.

翻译：本文研究了一类在实际应用中出现的带约束分数优化问题。该类问题的目标函数由分子与分母之比构成，其中分子与分母均为凸函数、半代数函数、Lipschitz连续，且在约束集上具有Lipschitz连续的梯度。相关约束集是闭凸且半代数的。我们提出了一种受邻近梯度法启发的高效算法，并对其收敛性进行了深入分析。与现有方法相比，我们的算法具有多项优势：每次迭代仅需执行一次邻近梯度操作，避免了通常所需的复杂内环凹最大化过程；此外，该方法在无需假设分子非负的情况下即可收敛至临界点，该临界点在适当条件下即成为全局最优解。我们的方法同样适用于无界约束集。因此，本方法可应用于更多实际模型。数值实验表明，该方法不仅能在某些模型问题中稳定逼近真实解，而且在利用真实金融数据最大化夏普比率时优于若干现有方法。