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1. 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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2. 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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3. Deep Neural Network Structures Solving Variational Inequalities

   https://doi.org/10.1007/s11228-019-00526-z  

   Combettes, Patrick L.; Pesquet, Jean-Christophe (January 2020, Set-Valued and Variational Analysis)

   Motivated by structures that appear in deep neural networks, we investigate nonlinear com- posite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic properties. It is shown that the limit of the resulting process solves a variational inequality which, in general, does not derive from a minimization problem. The analysis relies on tools from monotone operator theory and sheds some light on a class of neural networks structures with so far elusive asymptotic properties. 

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4. Geometries of topological groups

   https://doi.org/10.1090/bull/1807  

   Rosendal, Christian (October 2023, Bulletin of the American Mathematical Society)

   The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory. 

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5. Relationships between Global and Local Monotonicity of Operator

   Khanh, PD; Khoa, VVH; Martinez-Legaz, JE; Mordukhovich, BS (January 2025, Journal of Convex Analysis)

   The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between fnite-dimensional and infnite-dimensional spaces. We frst show that for single-valued operators with convex domains in locally convex topological spaces, their continuity ensures that their global monotonicity agrees with the local one around any point of the graph. This also holds for set-valued mappings defned on the real line under a certain connectedness condition. The situation is diferent for set-valued operators in multidimensional spaces as demonstrated by an example of locally monotone operator on the plane that is not globally monotone. Finally, we invoke coderivative criteria from variational analysis to characterize both global and local maximal monotonicity of set-valued operators in Hilbert spaces to verify the equivalence between these monotonicity properties under the closedgraph and global hypomonotonicity assumptions. 

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