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.

翻译：本文针对一类约束集上的分式优化问题建立了全局最优解，其关键特征如下：1）目标函数的分子与分母在约束集上均为凸、半代数、Lipschitz连续且具有Lipschitz连续梯度的可微函数；2）约束集为闭凸半代数集。与Dinkelbach方法相比，本文的创新点在于：1）Dinkelbach方法每次迭代需求解一个凹最大化问题，获得解较为困难，而本文方法每次迭代仅需执行一次邻近梯度运算；2）Dinkelbach方法要求分子至少存在一个非负点才能推进算法，本文方法则无此限制，因此适用于更广泛的情形；3）Dinkelbach方法要求约束集为闭且有界，而本文方法仅需闭性而不必有界性。因此，本文方法可用于更多实际模型，如数学金融中的夏普比率（SR）或信息比率优化。数值实验表明，本文方法在两个简单实例中能够获得真实解。对于真实金融数据，其在SR最大化问题上优于现有多种方法。