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1. Strong Metric (Sub)regularity of Karush–Kuhn–Tucker Mappings for Piecewise Linear-Quadratic Convex-Composite Optimization and the Quadratic Convergence of Newton’s Method

   https://doi.org/10.1287/moor.2019.1027  

   Burke, James V.; Engle, Abraham (August 2020, Mathematics of Operations Research)

   This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: where one minimizes an objective that can be written as the composition of a convex function with one that is continuiously differentiable. We focus on the case in which the convex function is a potentially infinite-valued piecewise linear-quadratic function. Such problems include nonlinear programming, mini-max optimization, and estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problems into a generalized equation. We establish conditions for strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. This allows us to extend classical convergence of Newton and quasi-Newton methods to the broader class of nonfinite valued piecewise linear-quadratic convex-composite optimization problems. In particular, we establish local quadratic convergence of the Newton method under conditions that parallel those in nonlinear programming. 

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2. The geometry of monotone operator splitting methods

   https://doi.org/10.1017/S0962492923000065  

   Combettes, Patrick L (July 2024, Acta Numerica)

   We propose a geometric framework to describe and analyse 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. 

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3. Multivariate Monotone Inclusions in Saddle Form

   https://doi.org/10.1287/moor.2021.1161  

   Bùi, Minh N.; Combettes, Patrick L. (May 2022, Mathematics of Operations Research)

   We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is 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 model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented. 

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4. Parametrized Families of Resolvent Compositions

   https://doi.org/10.1007/s11228-025-00745-7  

   Cornejo, Diego J (March 2025, Set-Valued and Variational Analysis)

   Abstract This paper presents an in-depth analysis of a parametrized version of the resolvent composition, an operation that combines a set-valued operator and a linear operator. We provide new properties and examples, and show that resolvent compositions can be interpreted as parallel compositions of perturbed operators. Additionally, we establish new monotonicity results, even in cases when the initial operator is not monotone. Finally, we derive asymptotic results regarding operator convergence, specifically focusing on graph-convergence and the$$\rho $$ $\rho$-Hausdorff distance. 

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5. Hilbert direct integrals of monotone operators

   https://doi.org/10.4153/S0008414X2400049X  

   Bùi, Minh N; Combettes, Patrick L (August 2025, Canadian Journal of Mathematics)

   Abstract Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address by introducing the Hilbert direct integral of a family of monotone operators. The properties of this construct are studied, and conditions under which the direct integral inherits the properties of the factor operators are provided. The question of determining whether the Hilbert direct integral of a family of subdifferentials of convex functions is itself a subdifferential leads us to introducing the Hilbert direct integral of a family of functions. We establish explicit expressions for evaluating the Legendre conjugate, subdifferential, recession function, Moreau envelope, and proximity operator of such integrals. Next, we propose a duality framework for monotone inclusion problems involving integrals of linearly composed monotone operators and show its pertinence toward the development of numerical solution methods. Applications to inclusion and variational problems are discussed. 

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