arXiv subjects

Matthew K. Tam

Publications and source records attributed to Matthew K. Tam.

At least 19 recordsLinked to original sources

arXiv2026

Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting

In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed point, as frequently assumed in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations.

math.OC↗

arXiv2026

Primal-dual splitting for structured composite monotone inclusions with or without cocoercivity

In this paper, we propose a primal-dual splitting algorithm for a broad class of structured composite monotone inclusions that involve finitely many set-valued operators, compositions of set-valued operators with bounded linear operators, and single-valued operators possibly without cocoercivity. The proposed algorithm is not only a unification for several contemporary algorithms but also a blueprint to generate new algorithms with graph-based structures using a single transparent convergence analysis. Our approach reduces dimensionality compared with the standard product space technique, which typically reformulates the original problem as the sum of two maximally monotone operators in order to apply splitting methods. It accommodates different cocoercive or Lipschitz constants as well as different resolvent parameters, and yields a larger allowable stepsize range than recent methods. We demonstrate the practicality of the approach by a numerical experiment on cancer detection using the decentralized fused LASSO problem.

math.OC↗

arXiv2026

A general approach to distributed operator splitting

Splitting methods have emerged as powerful tools to address complex problems by decomposing them into smaller solvable components. In this work, we develop a general approach to forward-backward splitting methods for solving monotone inclusion problems involving both set-valued and single-valued operators, where the latter may lack cocoercivity. Our proposed approach, based on some coefficient matrices, not only encompasses several important existing algorithms but also extends to new ones, offering greater flexibility for different applications. Moreover, by appropriately selecting the coefficient matrices, the resulting algorithms can be implemented in a distributed and decentralized manner.

math.OC↗

arXiv2026

Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations

We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems.

math.OC↗

arXiv2026

A First-Order Algorithm for Decentralised Min-Max Problems

In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial combination of two algorithms: PG-EXTRA for decentralised minimisation problems and the forward reflected backward method for (non-distributed) min-max problems. In each iteration of our algorithm, each agent computes the gradient of the smooth component of its local objective function as well as the proximal operator of its nonsmooth component, following by a round of communication with its neighbours. Our analysis shows that the sequence generated by the method converges under standard assumptions with non-decaying stepsize.

math.OC↗

arXiv2025

The Golden Ratio Primal-Dual Algorithm with Two New Stepsize Rules for Convex-Concave Saddle Point Problems

In this paper, we present two stepsize strategies for the extended Golden Ratio primal-dual algorithm (E-GRPDA) designed to address structured convex optimization problems in finite-dimensional real Hilbert spaces. The first rule features a non-increasing primal stepsize that remains bounded below by a positive constant and is updated adaptively at each iteration, eliminating the need to compute the Lipschitz constant of the gradient of the function and the norm of the operator, without using backtracking. The second stepsize rule is adaptive, adjusting based on the local smoothness of the smooth component function and the norm of the operator involved. In other words, we present an adaptive version of the E-GRPDA algorithm. We prove that E-GRPDA achieves an ergodic sublinear convergence rate with both stepsize rules, based on the function-value residual and constraint violation rather than on the so-called primal-dual gap function. Additionally, we establish an R-linear convergence rate for E-GRPDA with the first stepsize rule, under standard assumptions and with appropriately chosen parameters. Through numerical experiments on various convex optimization problems, we demonstrate the effectiveness of our approaches and compare their performance to existing ones.

math.OC↗

arXiv2025

A distributed proximal splitting method with linesearch for locally Lipschitz gradients

In this paper, we propose a distributed first-order algorithm with backtracking linesearch for solving multi-agent minimisation problems, where each agent handles a local objective involving nonsmooth and smooth components. Unlike existing methods that require global Lipschitz continuity and predefined stepsizes, our algorithm adjusts stepsizes using distributed linesearch procedures, making it suitable for problems where global constants are unavailable or difficult to compute. The proposed algorithm is designed within an abstract linesearch framework for a primal-dual proximal-gradient method to solve min-max convex-concave problems, enabling the consensus constraint to be decoupled from the optimisation task. Our theoretical analysis allows for gradients of functions to be locally Lipschitz continuous, relaxing the prevalent assumption of globally Lipschitz continuous gradients.

math.OC↗

arXiv2024

Linear Convergence of Resolvent Splitting with Minimal Lifting and its Application to a Primal-Dual Algorithm

We consider resolvent splitting algorithms for finding a zero of the sum of finitely many maximally monotone operators. The standard approach to solving this type of problem involves reformulating as a two-operator problem in the product-space and applying the Douglas-Rachford algorithm. However, existing results for linear convergence cannot be applied in the product-space formulation due to a lack of appropriate Lipschitz continuity and strong monotonicity. In this work, we investigate a different approach that does not rely on the Douglas-Rachford algorithm or the product-space directly. We establish linear convergence of the "resolvent splitting with minimal lifting" algorithm due to Malitsky and Tam for monotone inclusions with finitely many operators. Our results are then used to derive linear convergence of a primal-dual algorithm for convex minimization problems involving infimal convolutions. The theoretical results are demonstrated on numerical experiments in image denoising.

math.OC↗

arXiv2024

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that: (i) may have multiple solutions, (ii) do not satisfy strict complementary conditions, and (iii) possess non-linear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a "sharpness" property of the constraint set; a new property we introduce here.

math.OC↗

arXiv2023

Frugal and Decentralised Resolvent Splittings Defined by Nonexpansive Operators

Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.

math.OC↗

arXiv2022

Bregman Golden Ratio Algorithms for Variational Inequalities

Variational inequalities provide a framework through which many optimisation problems can be solved, in particular, saddle-point problems. In this paper, we study modifications to the so-called Golden RAtio ALgorithm (GRAAL) for variational inequalities -- a method which uses a fully explicit adaptive step-size, and provides convergence results under local Lipschitz assumptions without requiring backtracking. We present and analyse two Bregman modifications to GRAAL: the first uses a fixed step-size and converges under global Lipschitz assumptions, and the second uses an adaptive step-size rule. Numerical performance of the former method is demonstrated on a bimatrix game arising in network communication, and of the latter on two problems, namely, power allocation in Gaussian communication channels and $N$-person Cournot completion games. In all of these applications, an appropriately chosen Bregman distance simplifies the projection steps computed as part of the algorithm.

math.OC↗

arXiv2022

Resolvent Splitting for Sums of Monotone Operators with Minimal Lifting

In this work, we study fixed point algorithms for finding a zero in the sum of $n\geq 2$ maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying fixed point operator is necessarily defined on a $d$-fold Cartesian product space with $d\geq n-1$. Further, we show that this bound is unimprovable by providing a family of examples for which $d=n-1$ is attained. This family includes the Douglas-Rachford algorithm as the special case when $n=2$. Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.

math.OC↗

arXiv2022

Distributed Forward-Backward Methods for Ring Networks

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

math.OC↗

arXiv2021

Strengthened Splitting Methods for Computing Resolvents

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the "strengthening" of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs.

math.OC↗

arXiv2021

Convergence Rates for Boundedly Regular Systems

In this work, we consider a continuous dynamical system associated with the fixed point set of a nonexpansive operator which was originally studied by Boţ & Csetnek (2015). Our main results establish convergence rates for the system's trajectories when the nonexpansive operator satisfies an additional regularity property. This setting is the natural continuous-time analogue to discrete-time results obtained in Bauschke, Noll & Phan (2015) and Borwein, Li & Tam (2017) by using the same regularity properties.

math.OC↗

arXiv2021

Method of Alternating Projection for the Absolute Value Equation

A novel approach for solving the general absolute value equation $Ax+B|x| = c$ where $A,B\in \mathbb{R}^{m\times n}$ and $c\in \mathbb{R}^m$ is presented. We reformulate the equation as a feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on $A$ and $B$. Furthermore, we prove linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with $m\neq n$, both theoretically and numerically.

math.OC↗

arXiv2020

Gearhart-Koshy Acceleration for Affine Subspaces

The method of cyclic projections finds nearest points in the intersection of finitely many affine subspaces. To accelerate convergence, Gearhart and Koshy proposed a modification which, in each iteration, performs an exact line search based on minimising the distance to the solution. When the subspaces are linear, the procedure can be made explicit using feasibility of the zero vector. This work studies an alternative approach which does not rely on this fact, thus providing an efficient implementation in the affine setting.

math.OC↗

arXiv2020

A Douglas-Rachford construction of non-separable continuous compactly supported multidimensional wavelets

After re-casting the $n$-dimensional wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas--Rachford algorithm is employed in the search for one- and two-dimensional wavelets. New one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets in the case where the dilation on the plane is the standard $D_af(t)=a^{-1}f(t/a)$ $(t\in{\mathbb R}^n, a>0)$.

math.CA↗