Smoothing methods for nonsmooth, nonconvex minimization

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.

The smoothing Fletcher-Reeves conjugate gradient method for solving finite minimax problems

“…As in Ref. 7, we introduce the definition of the smoothing function and the gradient consistency property.…”

“…7). The methods for solving the nonsmooth unconstrained optimization problem are more complex than the methods for solving the smooth optimization problem.…”

“…Reeves conjugate gradient method with smoothing techniques [2][3][4]7 . Also, the simple use of smoothing techniques can help prevent ill-conditioning.…”

Métodos de pontos interiores/exteriores e o problema multiobjetivo de despacho econômico e ambiental

“…Neste trabalho, uma estratégia de suavizaçãoé empregada a esta função, possibilitando a obtenção de uma função aproximada e diferenciável. De acordo com o método da suavização hiperbólica, [2], a função suavizanteé definida por:…”

Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems

“…What is more, there are many other smoothing functions with the gradient consistency property which are not generated by the integral-convolution with bounded supports. The reader is referred to [12,14,16,17] for more details.…”