Mathematical Programming

, Volume 134, Issue 1, pp 71–99

Smoothing methods for nonsmooth, nonconvex minimization

Full Length Paper Series B

Abstract

We consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex, not differentiable and perhaps not even locally Lipschitz at the solutions. Such optimization problems arise from wide applications including image restoration, signal reconstruction, variable selection, optimal control, stochastic equilibrium and spherical approximations. In this paper, we focus on smoothing methods for solving such optimization problems, which use the structure of the minimization problems and composition of smoothing functions for the plus function (x)+. Many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods. We present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function. Moreover, we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem.

Keywords

Nonsmooth Nonconvex minimization Smoothing methods Regularized minimization problems Eigenvalue optimization Stochastic variational inequality problems 

Mathematics Subject Classification

90C30 49M37 65K10 90C26 

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© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong, China

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