Abstract. 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'skiȋ-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.

Abstract. We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Lojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. We illustrate the theoretical results by considering two numerical experiments: the first one concerns the ability of recovering the local optimal solutions of nonconvex optimization problems, while the second one refers to the restoration of a noisy blurred image.

We present two modified versions of the primal-dual splitting algorithm relying on forward-backward splitting proposed in [21] for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of O( 1 n ) and O(ω n ), for ω ∈ (0, 1), respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and support vector machines classification.

We begin by considering second order dynamical systems of the fromẍ(t)+γ(t)ẋ(t)+ λ(t)B(x(t)) = 0, where B : H → H is a cocoercive operator defined on a real Hilbert space H, λ : [0, +∞) → [0, +∞) is a relaxation function and γ : [0, +∞) → [0, +∞) a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator B. The framework allows to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when B is the gradient of a smooth convex function the value of the latter converges along the ergodic trajectory to its minimal value with a rate of O(1/t).

We introduce and investigate the convergence properties of an inertial forwardbackward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with intricate objective functions.

We study the existence and uniqueness of (locally) absolutely continuous trajectories of a dynamical system governed by a nonexpansive operator. The weak convergence of the orbits to a fixed point of the operator is investigated by relying on Lyapunov analysis. We show also an order of convergence of o 1 √ t for the fixed point residual of the trajectory of the dynamical system. We apply the results to dynamical systems associated with the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive one. Several dynamical systems from the literature turn out to be particular instances of this general approach.

We consider the primal problem of finding the zeros of the sum of a maximal monotone operator and the composition of another maximal monotone operator with a linear continuous operator. By formulating its Attouch-Théra-type dual inclusion problem, a primal-dual splitting algorithm which simultaneously solves the two problems in finite-dimensional spaces is presented. The proposed scheme uses at each iteration the resolvents of the maximal monotone operators involved in separate steps and aims to overcome the shortcoming of classical splitting algorithms when dealing with compositions of maximal monotone and linear continuous operators. The iterative algorithm is used for solving nondifferentiable convex optimization problems arising in image processing and in location theory.

In this work, we propose a new algorithm for finding a zero in the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas-Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.Keywords. monotone operator · operator splitting · Douglas-Rachford algorithm · dynamical systems MSC2010. 49M29, 90C25, 47H05, 47J20, 65K15