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Minh N. Bùi

Publications and source records attributed to Minh N. Bùi.

16 recordsLinked to original sources
arXiv2026

Variational Analysis in Spectral Decomposition Systems

This work is concerned with the variational analysis of functions defined on Euclidean spaces whose values depend solely on certain invariants (``spectrum'') of their arguments, a class we term ``spectral functions.'' Building on our previous work \cite{PartI} on the convex analysis of such functions, we work in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows the derivation of new results in more general settings such as normal decomposition systems. The main results of this work provide constructive formulae for computing the regular, limiting, and Clarke subdifferentials of a spectral function in terms of the corresponding objects of the associated invariant function. Finally, we obtain a generalization of Lidski\uı's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

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arXiv2026

A Block-Activated Decomposition Algorithm for Multi-Stage Stochastic Variational Inequalities

We develop a block-activated decomposition algorithm for multi-stage stochastic variational inequalities with nonanticipativity constraints. The algorithm features two computational novelties: (i) At each iteration, it activates only a user-chosen subset of scenarios. (ii) For each activated scenario, it employs the resolvent of the cost operator and the projector onto the constraint set separately. These reduce computational load and enhance tractability, in contrast with existing approaches, which often require evaluating the resolvent of the sum of the cost operator and the normal cone operator of the constraint set. As an application, we demonstrate that in risk-averse stochastic programming with conditional value-at-risk objective functions, our method requires only projecting onto constraint sets, together with solving univariate equations involving the proximity operators of the cost functions, thereby avoiding solving high-dimensional constrained subproblems as required by existing methods.

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arXiv2026

Convex Analysis in Spectral Decomposition Systems

This work is concerned with the convex analysis of functions defined on (not necessarily finite-dimensional) Hilbert spaces whose values depend solely on a certain ``spectrum'' of the arguments, a class we term ``spectral functions.'' We propose a notion of a spectral decomposition system which brings together a wide array of settings underlying important applications such as Fourier-phase-invariant functions, mixed-norm regularization, and functions of eigenvalues or (signed) singular values of matrices. We are particularly motivated by algorithmic requirements for evaluating convex analytical objects. Thus, a central contribution is a novel reduced minimization principle that enables the constructive reduction of minimization problems involving spectral functions to those of the simpler associated invariant functions. This result is then leveraged to explicitly evaluate the conjugates, subgradients, and set-valued Bregman proximity operators of spectral functions.

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arXiv2025

On a Lemma by Brézis and Haraux

We propose several applications of an often overlooked part of the 1976 paper by Brézis and Haraux, in which the Brézis--Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.

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arXiv2024

Interchange Rules for Integral Functions

We first present an abstract principle for the interchange of infimization and integration over spaces of mappings taking values in topological spaces. New conditions on the underlying space and the integrand are then introduced to convert this principle into concrete scenarios that are shown to capture those of various existing interchange rules. These results are leveraged to improve state-of-the-art interchange rules for evaluating Legendre conjugates, subdifferentials, recessions, Moreau envelopes, and proximity operators of integral functions by bringing the corresponding operations under the integral sign.

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arXiv2024

Integral Resolvent and Proximal Mixtures

Using the theory of Hilbert direct integrals, we introduce and study a monotonicity-preserving operation, termed the integral resolvent mixture. It combines arbitrary families of monotone operators acting on different spaces and linear operators. As a special case, we investigate the resolvent expectation, an operation which combines monotone operators in such a way that the resulting resolvent is the Lebesgue expectation of the individual resolvents. Along the same lines, we introduce an operation that mixes arbitrary families of convex functions defined on different spaces and linear operators to create a composite convex function. Such constructs have so far been limited to finite families of operators and functions. The subdifferential of the integral proximal mixture is shown to be the integral resolvent mixture of the individual subdifferentials. Applications to the relaxation of systems of composite monotone inclusions are presented.

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arXiv2024

Hilbert Direct Integrals of Monotone Operators

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 towards the development of numerical solution methods. Applications to inclusion and variational problems are discussed.

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arXiv2021

Block-Activated Algorithms for Multicomponent Fully Nonsmooth Minimization

Under consideration are multicomponent minimization problems involving a separable nonsmooth convex function penalizing the components individually, and nonsmooth convex coupling terms penalizing linear mixtures of the components. We investigate block-activated proximal algorithms for solving such problems, i.e., algorithms which, at each iteration, need to use only a block of the underlying functions, as opposed to all of them as in standard methods. For smooth coupling functions, several block-activated algorithms exist and they are well understood. By contrast, in the fully nonsmooth case, few block-activated methods are available and little effort has been devoted to assessing them. Our goal is to shed more light on the implementation, the features, and the behavior of these algorithms, compare their merits, and provide machine learning and image recovery experiments illustrating their performance.

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arXiv2021

Projective Splitting as a Warped Proximal Algorithm

We show that the asynchronous block-iterative primal-dual projective splitting framework introduced by P. L. Combettes and J. Eckstein in their 2018 Math. Program. paper can be viewed as an instantiation of the recently proposed warped proximal algorithm.

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arXiv2021

Multivariate Monotone Inclusions in Saddle Form

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. 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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arXiv2021

A Warped Resolvent Algorithm to Construct Nash Equilibria

We propose an asynchronous block-iterative decomposition algorithm to solve Nash equilibrium problems involving a mix of nonsmooth and smooth functions acting on linear mixtures of strategies. The methodology relies heavily on monotone operator theory and in particular on warped resolvents.

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arXiv2020

Bregman Forward-Backward Operator Splitting

We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.

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arXiv2020

Warped Proximal Iterations for Monotone Inclusions

Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.

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arXiv2019

The Douglas--Rachford Algorithm Converges Only Weakly

We show that the weak convergence of the Douglas--Rachford algorithm for finding a zero of the sum of two maximally monotone operators cannot be improved to strong convergence. Likewise, we show that strong convergence can fail for the method of partial inverses.

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