Smoothing methods for nonsmooth, nonconvex minimization (Q715249): Difference between revisions

The author considers the optimization problem \[ f(x)\to\min,\quad\text{subject to }x\in X, \] where \(X\) is a closed convex subset of \(\mathbb{R}^n\) and \(f:\mathbb{R}^n\to \mathbb{R}\) is continuous and almost everywhere differentiable in \(X\). Special attention is devoted to the case of a nonsmooth and nonconvex objective function, which is not in general locally Lipschitzian. For solving such optimization problems, smoothing methods are proposed using smoothing functions. Properties of the smoothing functions and conditions ensuring the convergence of the proposed methods to a stationary point of the original problem are presented. The last section of the paper is devoted to the numerical implementation of the proposed procedures.