Advances in Douglas-Rachford method – Part II

FB-02: Advances in Douglas-Rachford method – Part II
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s): Cong Bang Vu, Dimitri Papadimitriou

Shadow Douglas-Rachford splitting
Matthew Tam
In this talk, I will introduce the shadow Douglas-Rachford method 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 discretisation of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretisation with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.

The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators
Aviv Gibali, Francisco Javier Aragón Artacho, Yair Censor
The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this talk we present a structural generalization that allows to use r-sets-DR operators in a cyclic fashion and also demonstrates great computational advantages over existing results.

Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions with applications
Yuchao Tang
In this paper, we propose a forward-partial inverse-half-forward splitting (FPIHFS) algorithm for finding a zero of the sum of a maximally monotone operator, a monotone Lipschitzian operator, a cocoercive operator, and a normal cone of a closed vector subspace. The FPIHFS algorithm is derived from a combination of the partial inverse method with the forward-backward-half-forward splitting algorithm. As applications, we apply it to solve the Projection on Minkowski sums of convex sets problem and the generalized Heron problem.

Multivariate Monotone Inclusions in Saddle Form
Patrick Combettes, Minh Bui
We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. We study a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, uses the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to all of them. Applications are presented.

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