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广泛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，二者均为凸函数、半代数函数、Lipschitz连续，且在约束集上具有Lipschitz连续梯度。问题中的约束集是闭凸且半代数的。我们受近端梯度法启发，提出了一种高效算法，并进行了严谨的收敛性分析。与现有方法相比，该算法具有多项优势：每次迭代仅需执行一次近端梯度操作，避免了传统方法中复杂的内循环凹最大化步骤；算法收敛到临界点，且无需通常所需的非负分子假设；在适当条件下，该临界点即为全局最优解。此外，我们的方法同样适用于无界约束集。因此，该方法能够适用于更多实际模型。数值实验表明，该方法不仅在若干模型问题中可靠地收敛到真实解，而且在基于真实金融数据最大化夏普比率时，其表现优于多种现有方法。