In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set and the objective function has a nonsmooth, nonconvex regularizer. Such regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding and non-Lipschitz Lp penalties as special cases. Using the theory of the generalized directional derivative and the t...

Let f be a continuous function on Rn, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradient is easily computed where it is defined. We p...

The titles published in this series are listed at the end of this volume. 1 1 2 4 4 4 5 5 6 6 7 9 Multifunctions and Derivatives Particular Locally Lipschitz Functions and Related Definitions Generalized Jacobians of Locally Lipschitz Functions Pseudo-Smoothness and D°f Piecewise Functions NCP Functions 1.4 Definitions of Regularity Definitions of Lipschitz Properties Regularity Definitions Fun...

We present an algorithm for the minimization of f : Rn → R, assumed to be locally Lipschitz and continuously differentiable in an open dense subset D of Rn. The objective f may be nonsmooth and/or nonconvex. The method is based on the gradient sampling algorithm (GS) of Burke, Lewis, and Overton [SIAM J. Optim., 15 (2005), pp. 751-779]. It differs, however, from previously proposed versions of ...

Fast-Lipschitz optimization has been recently proposed as a new framework with numerous computational advantages for both centralized and decentralized convex and nonconvex optimization problems. Such a framework generalizes the interference function optimization, which plays an essential role distributed radio power optimization over wireless networks. The characteristics of Fast-Lipschitz met...

The relationship between a minimization problem on the in nite horizon and local stabilizability is studied for a ne control systems. A su cient condition for a system to be locally stabilizable in the sense of Filippov solutions is given in the case that the value function associated to the minimization problem is locally Lipschitz continuous.

we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

Abstract. Nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images. Constrained optimization can improve the image restoration using a priori information. In this paper, we study regularized nonsmooth nonconvex minimization with box constraints for image restoration. We present a computable positive constant θ for using nonconvex nonsmooth re...

In this paper, we propose a new algorithm called ModelBI by blending the Bregman iterative regularization method and model function technique for solving class of nonconvex nonsmooth optimization problems. On one hand, use technique, which is essentially first-order approximation to objective function, go beyond traditional Lipschitz gradient continuity. other generate solutions fitting certain...