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Ba Khiet Le

Publications and source records attributed to Ba Khiet Le.

17 recordsLinked to original sources
arXiv2026

Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions

In this paper, we establish, for the first time in the literature, the convergence of the practical versions of the Inexact Proximal Point Algorithm (IPPA) and the Inexact Tseng Algorithm (ITA) for computing approximate solutions to monotone inclusions in Hilbert spaces under the the presence of nonsummable errors. Our ap- proach relies on Tikhonov regularization, the contraction property of the associated monotone operators, and the recently developed R-continuity theory. The proposed techniques and results can be extended to analyze various important inexact algorithms in optimization-related problems with nonsummable errors.

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arXiv2026

Reducing the Lipschitz Constant in Tseng Algorithm and the Convergence Analysis

In this paper, we prove the weak and linear convergence of Tseng algorithm applied to the zero of sum of two operators problem where each operator is not necessarily monotone while existing researches in the literature require the monotonicity of both operators. Consequently it allows us to decompose more efficiently and can reduce the Lipschitz constant significantly, which is meaningful especially in big data. Numerical examples supporting the theoretical results are also provided.

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arXiv2026

Splitting Algorithms Using Pairs Monotonicity to Solve the Sum of Two Non-monotone Operators Problem

This paper is concerned with solving the problem of finding a zero of the sum of a single-valued operator and a set-valued operator, neither of which is assumed to be monotone, in a real Hilbert space. Under the framework of pair monotonicity, we propose a generalized forward--backward algorithm and a generalized Tseng algorithm based on transformed and warped resolvents associated with an auxiliary linear operator. We establish weak, strong and linear convergence of the proposed algorithms under suitable assumptions involving pair strong monotonicity, pair cocoercivity, and pair Lipschitz continuity. Finally, numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed algorithms.

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arXiv2026

Stationarity Floors and Vanishing Perturbations in Sharpness-Aware Minimization

We study a deterministic family of sharpness-aware minimization methods for smooth nonconvex functions. The perturbation is $$ y_k=x_k+ρ\, \frac{\nabla f(x_k)}{\norm{\nabla f(x_k)}^α}, \qquad 0\leqα\leq 1, $$ so that its effective radius is $ρ\norm{\nabla f(x_k)}^{1-α}$. For $0<α\leq1$, we give an explicit complexity bound above the stationarity level $(Lρ)^{1/α}$. A one-dimensional quadratic example reaches this level exactly, showing that the bound describes a real limitation of the constant-parameter rule. The unnormalized case $α=0$ is treated separately and requires $Lρ<1$. We then introduce a clipped rule which agrees with the constant-$ρ$ rule away from stationary points and becomes proportional to the gradient near them. The clipped method has $\norm{\nabla f(x_k)}\to0$ and the usual $O(T^{-1/2})$ stationarity bound. Numerical tests on a quadratic function, the Rosenbrock function, and a five-dimensional nonconvex function illustrate the stationarity floor of the unclipped rule and the effect of clipping.

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arXiv2026

Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version

In this paper, we introduce the notion of adaptive strong monotonicity and establish a linear convergence of the recently proposed Generalized Proximal Point Algorithm (GPPA) for computing approximate solutions to nonmonotone inclusion problems in Hilbert spaces. We also investigate the Inexact Generalized Proximal Point Algorithm (IGPPA) in the presence of nondiminishing errors.

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arXiv2026

Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.

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arXiv2025

Compact R-Continuity with Applications to Solving Inclusions and Convergence of Algorithms

This paper investigates the notion of compact R-continuity and its specifications for set-valued mappings between Banach spaces. We reveal several important properties of compact R-continuity in general settings and show that in finite dimensions, this notion is supported by the classical Lojasiewicz inequality for analytic functions. An application of compact R-continuity and the obtained results is given to convergence analysis for a broad class of descent algorithms in nonsmooth optimization. We also show that this notion is instrumental for the design and justification of a novel R-class of algorithms to solve inclusion problems.

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arXiv2025

New General Fixed-Point Approach to Compute the Resolvent of Composite Operators

In this paper, we propose a new general and stable fixed-point approach to compute the resolvents of the composition of a set-valued maximal monotone operator with a linear bounded mapping. Weak, strong and linear convergence of the proposed algorithms are obtained. Advantages of our method over the existing approaches are also thoroughly analyzed.

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arXiv2025

Solving Non-Monotone Inclusions Using Monotonicity of Pairs of Operators

In this paper, under the monotonicity of pairs of operators, we propose some Generalized Proximal Point Algorithms to solve non-monotone inclusions using warped resolvents and transformed resolvents. The weak, strong, and linear convergence of the proposed algorithms are established under very mild conditions.

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arXiv2024

Explicit Convergence Rate of The Proximal Point Algorithm under R-Continuity

The paper provides a thorough comparison between R-continuity and other fundamental tools in optimization such as metric regularity, metric subregularity and calmness. We show that R-continuity has some advantages in the convergence rate analysis of algorithms solving optimization problems. We also present some properties of R-continuity and study the explicit convergence rate of the Proximal Point Algorithm (PPA) under the R-continuity.

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arXiv2024

Sliding Mode Observers for Set-valued Lur'e Systems with Uncertainties Beyond Observational Range

In this paper, we introduce a new sliding mode observer for Lur'e set-valued dynamical systems, particularly addressing challenges posed by uncertainties not within the standard range of observation. Traditionally, most of Luenberger-like observers and sliding mode observer have been designed only for uncertainties in the range of observation. Central to our approach is the treatment of the uncertainty term which we decompose into two components: the first part in the observation subspace and the second part in its complemented subspace. We establish that when the second part converges to zero, an exact sliding mode observer for the system can be obtained. In scenarios where this convergence does not occur, our methodology allows for the estimation of errors between the actual state and the observer state. This leads to a practical interval estimation technique, valuable in situations where part of the uncertainty lies outside the observable range. Finally, we show that our observer is also a T- observer as well as a strong H-infinity observer.

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arXiv2023

State-Dependent Sweeping Processes: Asymptotic Behavior and Algorithmic Approaches

In this paper, we investigate the asymptotic properties of a particular class of state-dependent sweeping processes. While extensive research has been conducted on the existence and uniqueness of solutions for sweeping processes, there is a scarcity of studies addressing their behavior in the limit of large time. Additionally, we introduce novel algorithms designed for the resolution of quasi-variational inequalities. As a result, we introduce a new derivative-free algorithm to find zeros of nonsmooth Lipschitz continuous mappings with a linear convergence rate. This algorithm can be effectively used in nonsmooth and nonconvex optimization problems that do not possess necessarily second-order differentiability conditions of the data.

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arXiv2023

Sliding Mode Observer for Set-valued Lur'e Systems and Chattering Removing

In this paper, we study a sliding mode observer for a class of set-valued Lur'e systems subject to uncertainties. We show that our approach has obvious advantages than the existing Luenberger-like observers. Furthermore, we provide an effective continuous approximation to eliminate the chattering effect in the sliding mode technique.

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arXiv2019

Lur'e dynamical systems with state-dependent set-valued feedback

Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur'e dynamical systems where the set-valued feedback depends on both time and state. This work is a generalization of \cite{abc} where the time-dependent set-valued feedback is considered to acquire only weak solutions. Obviously, strong solutions and implicit discretization scheme are nice properties, especially for numerical simulation. We also provide some conditions such that the solutions are exponentially attractive. The obtained results can be used to study the time-varying Lur'e systems with errors in data. Our result is new even the set-valued feedback depends only on the time.

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arXiv2019

Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

In this paper, we introduce for the first time a class of state-dependent maximal monotone differential inclusions. Then the existence and uniqueness of solutions are obtained by using an implicit discretization scheme and a kind of hypo-monotonicity assumption respectively. In addition, a characterization for nonsmooth Lyapunov pairs associated with such systems is provided. Our result can be applied to study state-dependent sweeping processes and Lur'e dynamical systems. It is new even the involved maximal monotone operators depend only on the time.

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arXiv2016

On Properties of Differential Inclusions with Prox-regular Sets

In this paper, some regularity properties of solutions of the following differential inclusion \begin{equation}\nonumber \left\{ \begin{array}{l} \dot{x}(t) \in f\big(x(t)\big) -N_{C}\big(x(t)\big)\; {\rm a.e.} \; t \in [0,+\infty), x(0) = x_0\in C, \end{array}\right. \end{equation} are analyzed where $f: H\to H$ is Lipschitz continuous and $C$ is closed, uniformly prox-regular subset of a Hilbet space $H$. Here $N_{C}(\cdot)$ denotes the proximal normal cone of $C$. This work can be considered as an improvement of [Hantoute-Mazade 2013] since these properties are established without the additional tangential condition at each point in $C$.

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