A Further Related Work on Nonsmooth Nonconvex Optimization

To appreciate the difficulty and the broad scope of the research agenda in nonsmooth nonconvex optimization, we start by describing the existing relevant literature. First, the existing work is mostly devoted to establishing the asymptotic convergence properties of various optimization algorithms, including gradient sampling (GS) methods [16-18, 57, 19], bundle methods [56, 40] and subgradient methods [8, 65, 30, 28, 12]. More specifically, Burke et al. [16] provided a systematic investigation of approximating the Clarke subdifferential through random sampling and proposed a gradient bundle method [17]--the precursor of GS methods--for optimizing a nonconvex, nonsmooth and non-Lipschitz function. Later, Burke et al. [18] and Kiwiel [57] proposed the GS methods by incorporating key modifications into the algorithmic scheme in Burke et al. [17] and proved that every cluster point of the iterates generated by GS methods is a Clarke stationary point. For an overview of GS methods, we refer to Burke et al. [19].