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Nam Van Tran

Publications and source records attributed to Nam Van Tran.

11 recordsLinked to original sources
arXiv2026

Third-Order Dynamical Systems for Generalized Inverse Mixed Variational Inequality Problems

In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the system under suitable continuity assumptions, and prove their exponential convergence to the unique solution under strong monotonicity and Lipschitz continuity conditions. Furthermore, we derive an explicit discretization of the proposed dynamical system, leading to a forward -backward algorithm with double inertial effects. We then establish the linear convergence of the generated iterates to the unique solution.

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arXiv2026

A novel neural network with predefined-time stability for solving generalized monotone inclusion problems with applications

We propose a novel dynamical framework for solving inclusion problems of the form \(0 \in F(x) + G(x)\) in Hilbert spaces, where \(F\) is a maximal set-valued operator and \(G\) is a single-valued mapping. The analysis is conducted under a generalized monotonicity assumption, which relaxes the classical monotonicity conditions commonly imposed in the literature and thereby extends the applicability of the proposed approach. Under mild conditions on the system parameters, we establish both fixed-time and predefined-time stability of the resulting dynamical system. The fixed-time stability guarantees a uniform upper bound on the settling time that is independent of the initial condition, whereas the predefined-time stability framework allows the system parameters to be selected \emph{a priori} in order to ensure convergence within a user-specified time horizon. Moreover, we investigate an explicit forward Euler discretization of the continuous-time dynamics, leading to a novel forward--backward iterative algorithm. A rigorous convergence analysis of the resulting discrete scheme is provided. Finally, the effectiveness and versatility of the proposed method are illustrated through several classes of problems, including constrained optimization problems, mixed variational inequalities, and variational inequalities, together with numerical experiments that corroborate the theoretical results.

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arXiv2026

A Second-Order Dynamical System for Solving Generalized Inverse Mixed Variational Inequality problems

In this paper, we study a class of generalized inverse mixed variational inequality problems (GIMVIPs). We propose a novel projection-based second-order time-varying dynamical system for solving GIMVIPs. Under the assumptions that the underlying operators are strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solution trajectories and prove their global exponential convergence to the unique solution of the GIMVIP. Furthermore, a discrete-time realization of the continuous dynamical system is developed, resulting in an inertial projection algorithm. We show that the proposed algorithm achieves linear convergence under suitable choices of parameters. Finally, numerical experiments are presented to illustrate the effectiveness and convergence behavior of the proposed method in solving GIMVIPs.

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arXiv2026

Novel Dynamical Systems with Finite-Time and Predefined-Time Stability for Generalized Inverse Mixed Variational Inequality Problems

This paper investigates a class of generalized inverse mixed variational inequality problems (GIMVIPs), which consist in finding a vector $\overline{w}\in \R^d$ such that \[ F(\bar w)\in Ω\quad \text{and} \quad \langle h(\bar w), v-F(\bar w) \rangle + g(v)-g(F(\bar w)) \ge 0, \quad \forall v\in Ω, \] where \(h,F:\R^d\to\R^d\) are single-valued operators, \(g:Ω\to\R\cup\{+\infty\}\) is a proper function, and \(Ω\) is a closed convex set. Two novel continuous-time dynamical systems are proposed to study the finite-time and predefined-time stability of solutions to GIMVIPs in finite-dimensional Hilbert spaces. Under suitable assumptions on the involved operators and model parameters, Lyapunov-based techniques are employed to establish finite-time and predefined-time convergence of the generated trajectories. Although both dynamical systems exhibit accelerated convergence, the settling time of the finite-time stable system depends on the initial condition, whereas the predefined-time stable system admits a uniform upper bound on the convergence time that is independent of the initial state and can be explicitly prescribed through user-selected parameters. Moreover, by applying a forward Euler discretization to the continuous-time dynamics, a proximal point-type iterative algorithm is derived, and its fixed-time convergence property is rigorously analyzed. Numerical experiments are provided to illustrate the effectiveness and advantages of the proposed methods.

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arXiv2025

A forward-reflected-anchored-backward splitting algorithm with double inertial effects for solving non-monotone inclusion problems

In this paper, we study inclusion problems where the involved operators may not be monotone in the classical sense. Specifically, we assume the operators to be generalized monotone, a weaker notion than classical monotonicity. This allows us to extend the applicability of our results to a broader class of operators. We apply the two-step inertial forward-reflected-anchored-backward splitting algorithm proposed in \cite{CHIN} to these non-monotone inclusion problems. We establish the strong convergence of the sequence generated by the algorithm and demonstrate its applicability to other optimization problems, including Constrained Optimization Problems, Mixed Variational Inequalities, and Variational Inequalities.

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arXiv2025

Globally Finite time and Globally Fixed-time stable Dynamical Systems for solving Inverse Quasi-variational inequality problems

In this paper, we propose two projection dynamical systems for solving inverse quasi-variational inequality problems in finite-dimensional Hilbert spaces-one ensuring finite-time stability and the other guaranteeing fixed-time stability. We first establish the connection between these dynamical systems and the solutions of inverse quasi-variational problems. Then, under mild conditions on the operators and parameters, we analyze the global finite-time and global fixed-time stability of the proposed systems. Both approaches offer accelerated convergence, however, while the settling time of a finite-time stable dynamical system depends on initial conditions, the fixed-time stable system achieves convergence within a predefined time, independent of initial conditions. To demonstrate their effectiveness, we provide numerical experiments, including an application to the traffic assignment problem.

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arXiv2022

Gaussian elimination for flexible systems of linear inclusions

Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using Gaussian elimination, a flexible system can be put into a row-echelon form with increasing error terms at the right-hand side. Then parameters are assigned to the error terms and the resulting system is solved by common methods of linear algebra. The solution set may have indeterminacy not only in terms of linear spaces, but also of modules. We determine maximal robustness for flexible systems.

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arXiv2021

On non-linear optimization with a perturbed objective function

A Lagrange multiplier theorem is derived for the case of an imprecise objective function and a precise constraint. The proof uses methods of analysis which deal in a direct, algebraic way with imprecisions. They include imprecise differentiation, and an approximate Fermat Lemma and Implicit Function Theorem. The tools are the external numbers of Nonstandard Analysis, which are models of Sorites imprecisions.

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arXiv2020

The explicit formula for Gauss-Jordan elimination and error analysis

The explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure for the solution of systems of linear equations is applied to error analysis. Stability conditions in terms of relative uncertainty and size of determinants are given such that the Gauss-Jordan procedure leads to a solution respecting the original imprecisions in the right-hand member. The solution is the same as given by Cramer's Rule. Imprecisions are modelled by scalar neutrices, which are convex groups of (nonstandard) real numbers. The resulting calculation rules extend informal error calculus, and permit to keep track of the errors at every stage.

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arXiv2020

On the explicit formula for Gauss-Jordan elimination

The elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure have the form of quotients of minors. Instead of the proof using identities of determinants of \cite{Li}, a direct proof by induction is given.

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