The last twenty years has seen considerable and fruitful research in the field of nonsmooth boundary value problems (BVP’s) for partial differential equations. The objective is to understand the behavior and properties of solutions to either variable coefficient equations with minimal regularity assumptions on the coefficients or to linear constant coefficient equations in domains with nonsmoot...

We survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic PDEs with nonsmooth coeecients. We highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid eeciency for nonsmooth coeecient PDEs on structured or unstructured grids. In particular, we shall describe various approache...

We consider stochastic zeroth-order optimization over Riemannian submanifolds embedded in Euclidean space, where the task is to solve problems with only noisy objective function evaluations. Toward this, our main contribution propose estimators of gradient and Hessian from evaluations, based on a version Gaussian smoothing technique. The proposed overcome difficulty nonlinearity manifold constr...

control problems with nonsmoothdata

We consider the bilevel problem of minimizing a nonsmooth convex function over the set of minimizers of another nonsmooth convex function. Standard convex constrained optimization is a particular case in this framework, corresponding to taking the lower level function as a penalty of the feasible set. We develop an explicit bundle-type algorithm for solving the bilevel problem, where each itera...

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and gener...

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...

The simulation of multibody systems with rigid contacts entails the solution of nonsmooth equations of motion. The dynamics is nonsmooth because of the discontinuous nature of noninterpenetration, collision, and adhesion constraints. We propose a solver that is able to handle the simulation of multibody systems of vast complexity, with more than 100,000 colliding rigid bodies. The huge number o...

Concave-Convex Procedure (CCCP) has been widely used to solve nonconvex d.c.(difference of convex function) programs occur in learning problems, such as sparse support vector machine (SVM), transductive SVM, sparse principal componenent analysis (PCA), etc. Although the global convergence behavior of CCCP has been well studied, the convergence rate of CCCP is still an open problem. Most of d.c....

Many discrete-time dynamic models of current interest are based on functions that, while generally continuous, are nonsmooth; examples include speci c multimodels, hinging hyperplane models, and hybrid systems. We consider two models for which we can vary the smoothness and examine its in uence on qualitative behavior. In the smooth regime, both models exhibit asymptotic stability for suÆcientl...