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Smoothing methods for nonsmooth, nonconvex minimization. (English) Zbl 1266.90145
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.

##### MSC:
 90C26 Nonconvex programming, global optimization 90C30 Nonlinear programming 49M37 Numerical methods based on nonlinear programming 65K10 Numerical optimization and variational techniques