一类分式优化问题的带非单调线搜索的近端梯度次梯度算法研究

Research on the ProximalGradient-Subgradient Algorithm with Nonmonotonic Line Search for a Class of Fractional Optimization Problems

摘要: 本文主要研究一类分式优化问题，其中分子是凸非光滑连续函数与非凸光滑函数的和，分母为凸 非光滑函数。 首先给出了问题的一阶最优性条件，然后给出了求解分式优化问题的新算法，即带 非单调线搜索的近端梯度次梯度算法(简称NL-PGSA)。此外，基于Kurdyka-L- ojasiewicz性质， 可以保证算法生成的整个序列的全局收敛性，最后，对l₁/l₂稀疏信号恢复问题进行了数值实验，验 证了该算法的有效性。

Abstract: This paper mainly considers a class of fractional optimization problems where the numerator is a sum of a convex nonsmooth continuous function and a nonconvex smooth function, while the denominator is a convex nonsmooth function. We first give the first-order optimality condition of the problem, and then a new algorithm, called proximal gradient-subgradient algorithm with nonmonotonic line search (NL-PGSA), is proposed for solving the fractional optimization problems. Moreover, the global convergence of the entire sequence generated by NL-PGSA algorithm has been proven based on the Kurdyka-L- ojasiewicz property. Finally, some numerical experiments on the l₁/l₂ sparse signal recovery problems are conducted to demonstrate the efficiency of the proposed algorithm.