We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict deri...

In this paper, we study a general optimization model, which covers a large class of existing models for many applications in imaging sciences. To solve the resulting possibly nonconvex, nonsmooth and non-Lipschitz optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables, a...

This article is concerned with symmetry properties of the solutions of the reaction-diiusion equation u + f (u) = 0 in a bounded connected domain in R N (N = 2; 3; : : :). Of especial interest are nonlinear source terms f of the type f (u) = u p ? u q with 0 q < p 1. Two results are presented. The rst result concerns the solution of a free boundary problem, where the domain is unknown and u and...

We deal with backward stochastic differential equations (BSDE for short) driven by Teugel’s martingales and an independent Brownian motion. We study the existence, uniqueness and comparison of solutions for these equations under a Lipschitz as well as a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove that if the Lipschitz constant LN behaves as √ log(N) ...

It is shown that sufficiently close outer and inner parallel sets to a ddimensional Lipschitz manifold in R with boundary have locally positive reach and the normal cycle of the Lipschitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents, provided that the normal cycles of the parallel set have locally bounded mass. The Gauss-Bonnet formul...

In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds ...

Abstract In this paper, we consider a class of structured nonconvex nonsmooth optimization problems, in which the objective function is formed by sum possibly and differentiable with Lipschitz continuous gradient, subtracted weakly convex function. This general framework allows us to tackle problems involving loss functions specific constraints, it has many applications such as signal recovery,...

This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral optimization problems. More specifically, the objective function these problems is form $$f_{1}+f_{2}+h$$ , where $$f_{1}$$ $$f_{2}$$ are differentiable matrix functions with Lipschitz continuous gradients, $$h$$ proper closed convex functio...

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration and signal processing can be modelled in this form. First we propose the concept of the Karush-Kuhn-Tucker (KKT) stationary condition for the non-Lipschitz proble...

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration and signal processing can be modelled in this form. First we propose the concept of the Karush-Kuhn-Tucker (KKT) stationary condition for the non-Lipschitz proble...