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

Forward-Half-Reflected-Partial inverse-Backward Splitting Algorithm for Solving Monotone Inclusions

In this article, we proposed a method for numerically solving monotone inclusions in real Hilbert spaces that involve the sum of a maximally monotone operator, a monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a vector subspace. Our algorithm splits and exploits the intrinsic properties of each operator involved in the inclusion. The proposed method is derived by combining partial inverse techniques and the {\it forward-half-reflected-backward} (FHRB) splitting method proposed by Malitsky and Tam (2020). Our method inherits the advantages of FHRB, equiring only one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator. Furthermore, we develop a method for solving primal-dual inclusions involving a mixture of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones. We apply our method to constrained composite convex optimization problems as a specific example. Finally, in order to compare our proposed method with existing methods in the literature, we provide numerical experiments on constrained total variation least-squares optimization problems. The numerical results are promising

The Geometry of Monotone Operator Splitting Methods

We propose a geometric framework to describe and analyze a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones

New strong convergence method for the sum of two maximal monotone operators

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis