Convergence of a Class of Nonmonotone Descent Methods for Kurdyka–Łojasiewicz Optimization Problems
研究了一类非单调下降方法在最小化Kurdyka–Łojasiewicz函数时的收敛性，证明了序列收敛到临界点，并给出了线性或次线性收敛速率条件，对非凸非光滑问题的算法设计有参考价值。
This note is concerned with a class of nonmonotone descent methods for minimizing a proper lower semicontinuous Kurdyka–Łojasiewicz (KL) function , which generates a sequence satisfying a nonmonotone decrease condition and a relative error tolerance. Under suitable assumptions, we prove that the whole sequence converges to a limiting critical point of , and, when is a KL function of exponent , the convergence admits a linear rate if and a sublinear rate associated to if . These assumptions are shown to be sufficient and necessary if in addition is weakly convex on a neighborhood of the set of critical points. Our results not only extend the convergence results of monotone descent methods for KL optimization problems but also resolve the convergence problem on the iterate sequence generated by a class of nonmonotone line search algorithms for nonconvex and nonsmooth problems.