Smoothing methods for nonsmooth, nonconvex minimization

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 restoratio...

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Vydané v:Mathematical programming Ročník 134; číslo 1; s. 71 - 99
Hlavný autor: Chen, Xiaojun
Médium: Journal Article Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: Berlin/Heidelberg Springer-Verlag 01.08.2012
Springer
Springer Nature B.V
Predmet:
Applied mathematics
Applied sciences
Approximation
Calculus of variations and optimal control
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Eigenvalues
Equilibrium
Exact sciences and technology
Feature selection
Full Length Paper
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization algorithms
Sciences and techniques of general use
Stochastic models
Studies
Theoretical
Nonsmooth
Eigenvalue optimization
Smoothing methods
Regularized minimization problems
90C30
Nonconvex minimization
65K10
90C26
49M37
Stochastic variational inequality problems
Lipschitz condition
Non convex programming
Convex set
Image processing
Program optimization
Subdifferential
Iterative method
Non linear programming
Modeling
Image reconstruction
Nonsmooth analysis
Variational inequality
Variational calculus
Lipschitz function
Convex function
Eigenvalue problem
Signal reconstruction
Numerical convergence
Minimization
Image restoration
Smoothing method
Stationary condition
Smoothing
Optimal control
Problem solving
Stochastic control
Objective function
Non convex analysis
ISSN:0025-5610, 1436-4646
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Shrnutí: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0569-0