Existence and solution methods for equilibria

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them

Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics

International audienceThis book concerns the numerical simulation of dynamical systems whose trajectories may not be differentiable everywhere. They are named nonsmooth dynamical systems. They make an important class of systems, firstly because of the many applications in which nonsmooth models are useful, secondly because they give rise to new problems in various fields of science. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution variational inequalities, each of these classes being itself split into several subclasses. With detailed examples of multibody systems with contact, impact and friction and electrical circuits with piecewise linear and ideal components, the book is is mainly intended for researchers in Mechanics and Electrical Engineering, but it will be attractive to researchers from other scientific communities like Systems and Control, Robotics, Physics of Granular Media, Civil Engineering, Virtual Reality, Haptic Systems, Computer Graphics, etc

Gap functions and error bounds for variational-hemivariational inequalities

In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results

The Gap Functions and Error Bounds of Solutions of a Class of Set-valued Mixed Variational Inequality and Related Algorithms

In this paper, we extend a class of generalizedized mixed variational inequality problem, and we study the gap functions, error bounds of solutions and related algorithms of a class of set-valued mixed variational inequalities. In order to solve these problems, we establish the corresponding generalized resolvent equations and prove the equivalence between these problems. Finally, we give three iterative algorithms and analyze the convergence of algorithms

Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v1 and v2. We consider a special type of boundary which can trap the particle for a random time. We found closed-form expressions for the stationary distribution of the position of the particle not only for the alternating Markov process but also for a broad class of semi-Markov processes