Multi-valued, singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.Comment: 39 pages, in press: J. Math. Pures Appl. (2013

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

Quasi Variational Inclusions Involving Three Operators

In this paper, we consider some new classes of the quasi-variational inclusions involving three monotone operators. Some interesting problems such as variational inclusions involving sum of two monotone operators, difference of two monotone operators, system of absolute value equations, hemivariational inequalities and variational inequalities are the special cases of quasi variational inequalities. It is shown that quasi-variational inclusions are equivalent to the implicit fixed point problems. Some new iterative methods for solving quasi-variational inclusions and related optimization problems are suggested by using resolvent methods, resolvent equations and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under monotonicity. Some special cases are discussed as applications of the main results

Nonsmooth Lagrangian mechanics and variational collision integrators

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated

Inertial Douglas-Rachford splitting for monotone inclusion problems

We propose an inertial Douglas-Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties. To this end we formulate first the inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating the set of fixed points of a nonexpansive operator, for which we also provide an exhaustive convergence analysis. By using a product space approach we employ these results to the solving of monotone inclusion problems involving linearly composed and parallel-sum type operators and provide in this way iterative schemes where each of the maximally monotone mappings is accessed separately via its resolvent. We consider also the special instance of solving a primal-dual pair of nonsmooth convex optimization problems and illustrate the theoretical results via some numerical experiments in clustering and location theory.Comment: arXiv admin note: text overlap with arXiv:1402.529

A Douglas-Rachford splitting for semi-decentralized equilibrium seeking in generalized aggregative games

We address the generalized aggregative equilibrium seeking problem for noncooperative agents playing average aggregative games with affine coupling constraints. First, we use operator theory to characterize the generalized aggregative equilibria of the game as the zeros of a monotone set-valued operator. Then, we massage the Douglas-Rachford splitting to solve the monotone inclusion problem and derive a single layer, semi-decentralized algorithm whose global convergence is guaranteed under mild assumptions. The potential of the proposed Douglas-Rachford algorithm is shown on a simplified resource allocation game, where we observe faster convergence with respect to forward-backward algorithms.Comment: arXiv admin note: text overlap with arXiv:1803.1044