Boţ Radu Ioan, Csetnek Ernö Robert
Faculty of Mathematics, University of Vienna, Vienna, Austria.
Faculty of Mathematics and Computer Sciences, Babeş-Bolyai University, Cluj-Napoca, Romania.
Optimization. 2017 Aug 3;66(8):1383-1396. doi: 10.1080/02331934.2017.1294592. Epub 2017 Feb 24.
In this paper, we propose two proximal-gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either concave or convex. In the iterative schemes, we perform a proximal step with respect to the nonsmooth numerator and a gradient step with respect to the smooth denominator. The algorithm in case of a concave denominator has the particularity that it generates sequences which approach both the (global) optimal solutions set and the optimal objective value of the underlying fractional programming problem. In case of a convex denominator the numerical scheme approaches the set of critical points of the objective function, provided the latter satisfies the Kurdyka-ᴌojasiewicz property.
在本文中，我们针对实希尔伯特空间中的分式规划问题提出了两种近端梯度算法，其中分子是一个恰当、凸且下半连续的函数，分母是一个光滑函数，既可以是凹函数也可以是凸函数。在迭代格式中，我们针对非光滑的分子执行一个近端步，针对光滑的分母执行一个梯度步。分母为凹函数情况下的算法具有这样的特殊性，即它生成的序列既趋近于基础分式规划问题的（全局）最优解集，也趋近于最优目标值。在分母为凸函数的情况下，数值格式趋近于目标函数的临界点集，前提是后者满足库尔迪卡 - 洛亚西维茨性质。
