论文标题
一类分数优化问题的一阶算法
First-order algorithms for a class of fractional optimization problems
论文作者
论文摘要
我们在本文中考虑了一类单比分数最小化问题，其中目标的分子部分是非滑动非凸函数的总和和平滑的非凸函数，而分母部分是非平滑凸函数。在这项工作中，我们首先使用所涉及的三个函数的一阶操作员来得出其一阶必要的最佳条件。然后，我们开发了一阶算法，即，具有单调线搜索（PGSA_ML）和非单声道线搜索（PGSA_NL）的近端基因求学生算法（PGSA），带有单调线搜索（PGSA_ML）的PGSA和PGSA。结果表明，在轻度假设下，它们产生的序列的任何积累点都是问题的关键点。此外，我们通过进一步假设分子部分中非平滑函数的局部Lipschitz连续性，分母部分的平滑度和库尔迪卡·洛贾西奇（Kurdyka-Lojasiew）的属性。所提出的算法应用于与一对对称阳性半金属矩阵相关的稀疏广义特征值问题，并根据其一般收敛定理获得相应的收敛结果。我们执行一些初步的数值实验，以证明所提出算法的效率
We consider in this paper a class of single-ratio fractional minimization problems, in which the numerator part of the objective is the sum of a nonsmooth nonconvex function and a smooth nonconvex function while the denominator part is a nonsmooth convex function. In this work, we first derive its first-order necessary optimality condition, by using the first-order operators of the three functions involved. Then we develop first-order algorithms, namely, the proximity-gradient-subgradient algorithm (PGSA), PGSA with monotone line search (PGSA_ML) and PGSA with nonmonotone line search (PGSA_NL). It is shown that any accumulation point of the sequence generated by them is a critical point of the problem under mild assumptions. Moreover, we establish global convergence of the sequence generated by PGSA or PGSA_ML and analyze its convergence rate, by further assuming the local Lipschitz continuity of the nonsmooth function in the numerator part, the smoothness of the denominator part and the Kurdyka- Lojasiewicz property of the objective. The proposed algorithms are applied to the sparse generalized eigenvalue problem associated with a pair of symmetric positive semidefinite matrices and the corresponding convergence results are obtained according to their general convergence theorems. We perform some preliminary numerical experiments to demonstrate the efficiency of the proposed algorithms
