arXiv subjects

Phan Tu Vuong

Publications and source records attributed to Phan Tu Vuong.

17 recordsLinked to original sources

arXiv2026

Anderson Accelerated Primal-Dual Hybrid Gradient for solving LP

We present the Anderson Accelerated Primal--Dual Hybrid Gradient (AA-PDHG), a fixed-point-based framework that integrates Anderson Acceleration into the PDHG method for solving linear programming (LP) problems. A central motivation is to investigate whether Anderson Acceleration, which systematically exploits multi-step historical information, can serve as a viable alternative to the restart strategy for PDHG. We establish the global convergence of AA-PDHG under a safeguard condition and propose a filtered variant (FAA-PDHG) that enforces the uniform boundedness of the coefficient matrix through angle and length filtering, thereby providing a rigorous convergence guarantee. Numerical experiments on LP instances derived from MIPLIB 2017 demonstrate that both AA-PDHG and FAA-PDHG deliver significant speedups over vanilla PDHG. On pre-solved MIPLIB instances, AA-PDHG is the fastest method on about 70% of the benchmark when neither method uses primal-weight updates, and remains competitive when both AA-PDHG and restart PDHG use their respective primal-weight update strategies, establishing Anderson Acceleration as a competitive alternative to the restart mechanism.

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arXiv2026

Faster than Fast-LTS: Robust Regression and Outlier Detection with DC Programming

When datasets contain outliers, robust regression is a well-established alternative to Ordinary Least Squares. A commonly employed robust estimator is Least Trimmed Squares (LTS), which computes the regression coefficients from a subset of observations. Determining the exact solution corresponds to a combinatorial problem with prohibitive computational costs, even for instances of moderate dimension. Thus, the most prevalent approach in practice remains a heuristic known as Fast-LTS. Although the heuristic often performs effectively, certain elements of the approach remain open to improvement. In particular, its core procedure provides robust results only when initialized with a large number of starting points. To address the heuristic's limitations, this paper reformulates the LTS problem as a concave minimization problem subject to a capped simplex constraint, and proposes the successive Boosted Difference of Convex Functions Algorithm (sBDCA) as a solution method. Theoretically, we establish via the Łojasiewicz property that sBDCA converges to a local solution with a linear rate in the fastest case. To ensure robustness from a single initialization in practice, we derive and integrate a problem-specific preconditioning matrix into the algorithmic setup. Building on this theoretical foundation, we conduct numerical studies on various synthetic and real-world datasets to demonstrate the effectiveness of sBDCA with preconditioning. Specifically, we show that our approach is up to 3.25 times faster than Fast-LTS and achieves up to 90% lower objective function values, particularly in high-dimensional settings. As all code is openly available, this paper further provides a practical guide to robust regression in Python.

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arXiv2026

The Popov's Algorithm with Optimal Bounded Stepsize for Generalized Monotone Variational Inequalities

For solving constrained (pseudo)-monotone variational inequality, we prove that the upper bound of stepsize $\frac{1}{2L}$ established for the Popov's algorithm and the forward-reflected-backward algorithm is tight. For unconstrained case, we can enlarge the upper bound to $\frac{1}{\sqrt{3}L}$ and show that this upper bound is also tight. The convergence analysis is carried out by using a new Lyapunov-type function.

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arXiv2026

Second order continuous and discrete dynamical systems for solving inverse quasi-variational inequalities

In this paper, we investigate the inverse quasi-variational inequality problem in finite-dimensional spaces. First, we introduce a second-order dynamical system whose trajectory converges exponentially to the solution of the inverse quasi-variational inequality, under the assumptions of Lipschitz continuity and strong monotonicity. Next, we discretize the proposed dynamical system to develop an algorithm, and prove that the iterations converge linearly to the unique solution of the inverse quasi-variational inequality. Finally, we present numerical experiments and applications to validate the theoretical results and compare the performance with existing methods.

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arXiv2025

Countering adversarial evasion in regression analysis

Adversarial machine learning challenges the assumption that the underlying distribution remains consistent throughout the training and implementation of a prediction model. In particular, adversarial evasion considers scenarios where adversaries adapt their data to influence particular outcomes from established prediction models, such scenarios arise in applications such as spam email filtering, malware detection and fake-image generation, where security methods must be actively updated to keep up with the ever-improving generation of malicious data. Game theoretic models have been shown to be effective at modelling these scenarios and hence training resilient predictors against such adversaries. Recent advancements in the use of pessimistic bilevel optimsiation which remove assumptions about the convexity and uniqueness of the adversary's optimal strategy have proved to be particularly effective at mitigating threats to classifiers due to its ability to capture the antagonistic nature of the adversary. However, this formulation has not yet been adapted to regression scenarios. This article serves to propose a pessimistic bilevel optimisation program for regression scenarios which makes no assumptions on the convexity or uniqueness of the adversary's solutions.

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arXiv2025

The Boosted Difference of Convex Functions Algorithm for Value-at-Risk Constrained Portfolio Optimization

A highly relevant problem of modern finance is the design of Value-at-Risk (VaR) optimal portfolios. Due to contemporary financial regulations, banks and other financial institutions are tied to use the risk measure to control their credit, market, and operational risks. Despite its practical relevance, the non-convexity induced by VaR constraints in portfolio optimization problems remains a major challenge. To address this complexity more effectively, this paper proposes the use of the Boosted Difference-of-Convex Functions Algorithm (BDCA) to approximately solve a Markowitz-style portfolio selection problem with a VaR constraint. As one of the key contributions, we derive a novel line search framework that allows the application of the algorithm to Difference-of-Convex functions (DC) programs where both components are non-smooth. Moreover, we prove that the BDCA linearly converges to a Karush-Kuhn-Tucker point for the problem at hand using the Kurdyka-Lojasiewicz property. We also outline that this result can be generalized to a broader class of piecewise-linear DC programs with linear equality and inequality constraints. In the practical part, extensive numerical experiments under consideration of best practices then demonstrate the robustness of the BDCA under challenging constraint settings and adverse initialization. In particular, the algorithm consistently identifies the highest number of feasible solutions even under the most challenging conditions, while other approaches from chance-constrained programming lead to a complete failure in these settings. Due to the open availability of all data sets and code, this paper further provides a practical guide for transparent and easily reproducible comparisons of VaR-constrained portfolio selection problems in Python.

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arXiv2025

Adversarial training with restricted data manipulation

Adversarial machine learning concerns situations in which learners face attacks from active adversaries. Such scenarios arise in applications such as spam email filtering, malware detection and fake image generation, where security methods must be actively updated to keep up with the everimproving generation of malicious data. Pessimistic Bilevel optimisation has been shown to be an effective method of training resilient classifiers against such adversaries. By modelling these scenarios as a game between the learner and the adversary, we anticipate how the adversary will modify their data and then train a resilient classifier accordingly. However, since existing pessimistic bilevel approaches feature an unrestricted adversary, the model is vulnerable to becoming overly pessimistic and unrealistic. When finding the optimal solution that defeats the classifier, it is possible that the adversary's data becomes nonsensical and loses its intended nature. Such an adversary will not properly reflect reality, and consequently, will lead to poor classifier performance when implemented on real-world data. By constructing a constrained pessimistic bilevel optimisation model, we restrict the adversary's movements and identify a solution that better reflects reality. We demonstrate through experiments that this model performs, on average, better than the existing approach.

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arXiv2024

Classification under strategic adversary manipulation using pessimistic bilevel optimisation

Adversarial machine learning concerns situations in which learners face attacks from active adversaries. Such scenarios arise in applications such as spam email filtering, malware detection and fake-image generation, where security methods must be actively updated to keep up with the ever improving generation of malicious data.We model these interactions between the learner and the adversary as a game and formulate the problem as a pessimistic bilevel optimisation problem with the learner taking the role of the leader. The adversary, modelled as a stochastic data generator, takes the role of the follower, generating data in response to the classifier. While existing models rely on the assumption that the adversary will choose the least costly solution leading to a convex lower-level problem with a unique solution, we present a novel model and solution method which do not make such assumptions. We compare these to the existing approach and see significant improvements in performance suggesting that relaxing these assumptions leads to a more realistic model.

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arXiv2024

A third order dynamical system for generalized monotone equation

We propose a third order dynamical system for solving a nonlinear equation in Hilbert spaces where the operator is cocoercive with respect to the solutions set. Under mild conditions on the parameters, we establish the existence and uniqueness of the generated trajectories as well as its asymptotic convergence to a solution of the equation. When the operator is strongly monotone with respect to the solutions set, we deliver an exponential convergence rate of $e^{-2t}$, which is significantly faster than the known results of second order dynamical systems. In particular, for convex optimization problems, the proposed dynamical system provides a fast convergence rate of {\bf $\mathcal{O}(\frac{1}{t^3})$} for the objective values. In addition, we discuss the applications of the proposed dynamical system to several splitting monotone inclusion problems.

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arXiv2023

The Boosted DC Algorithm for Clustering with Constraints

This paper aims to investigate the effectiveness of the recently proposed Boosted Difference of Convex functions Algorithm (BDCA) when applied to clustering with constraints and set clustering with constraints problems. This is the first paper to apply BDCA to a problem with nonlinear constraints. We present the mathematical basis for the BDCA and Difference of Convex functions Algorithm (DCA), along with a penalty method based on distance functions. We then develop algorithms for solving these problems and computationally implement them, with publicly available implementations. We compare old examples and provide new experiments to test the algorithms. We find that the BDCA method converges in fewer iterations than the corresponding DCA-based method. In addition, BDCA yields faster CPU running-times in all tested problems.

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arXiv2021

Stochastic Relaxed Inertial Forward-Backward-Forward splitting for Monotone Inclusions in Hilbert spaces

We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator $T$ and a single-valued monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw motivation from the seminal work by Attouch and Cabot on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that (RISFBF) produces a sequence that weakly converges to the solution set. Moreover, it is possible to estimate the rate at which the discrete velocity of the stochastic process vanishes. Under strong monotonicity, we demonstrate strong convergence, and give a detailed assessment of the iteration and oracle complexity of the scheme. When the mini-batch is raised at a geometric (polynomial) rate, the rate statement can be strengthened to a linear (suitable polynomial) rate while the oracle complexity of computing an $ε$-solution improves to $O(1/ε)$. Importantly, the latter claim allows for possibly biased oracles, a key theoretical advancement allowing for far broader applicability. By defining a restricted gap function based on the Fitzpatrick function, we prove that the expected gap of an averaged sequence diminishes at a sublinear rate of $O(1/k)$ while the oracle complexity of computing a suitably defined $ε$-solution is $O(1/ε^{1+a})$ where $a>1$. Numerical results on two-stage games and an overlapping group Lasso problem illustrate the advantages of our method compared to stochastic forward-backward-forward (SFBF) and SA schemes.

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arXiv2021

Computing Dynamic User Equilibrium on Large-Scale Networks Without Knowing Global Parameters

Dynamic user equilibrium (DUE) is a Nash-like solution concept describing an equilibrium in dynamic traffic systems over a fixed planning period. DUE is a challenging class of equilibrium problems, connecting network loading models and notions of system equilibrium in one concise mathematical framework. Recently, Friesz and Han introduced an integrated framework for DUE computation on large-scale networks, featuring a basic fixed-point algorithm for the effective computation of DUE. In the same work, they present an open-source MATLAB toolbox which allows researchers to test and validate new numerical solvers. This paper builds on this seminal contribution, and extends it in several important ways. At a conceptual level, we provide new strongly convergent algorithms designed to compute a DUE directly in the infinite-dimensional space of path flows. An important feature of our algorithms is that they give provable convergence guarantees without knowledge of global parameters. In fact, the algorithms we propose are adaptive, in the sense that they do not need a priori knowledge of global parameters of the delay operator, and which are provable convergent even for delay operators which are non-monotone. We implement our numerical schemes on standard test instances, and compare them with the numerical solution strategy employed by Friesz and Han.

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arXiv2020

The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces

Tseng's forward-backward-forward algorithm is a valuable alternative for Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich's method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward-backward-forward dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng's forward-backward-forward algorithm with relaxation parameters, which we prove to converge also when it is applied to pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. Numerical experiments are carried out for pseudo-monotone variational inequalities over polyhedral sets and fractional programming problems.

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arXiv2020

A Relaxed Inertial Forward-Backward-Forward Algorithm for Solving Monotone Inclusions with Application to GANs

We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system which approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time discretization, a convergence analysis in the general monotone case as well as when applied to the solving of pseudo-monotone variational inequalities. We illustrate the proposed method by applications to a bilinear saddle point problem, in the context of which we also emphasize the interplay between the inertial and the relaxation parameters, and to the training of Generative Adversarial Networks (GANs).

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arXiv2019

Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks

In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tseng's forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adaptive extension of the algorithm, freeing us from requiring knowledge of the global Lipschitz constant. We test the performance of the algorithm in the computationally challenging task to find dynamic user equilibria in traffic networks and verify that our scheme is at least competitive to state-of-the-art solvers, and in some case even improve upon them.

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arXiv2019

Forward-backward-forward methods with variance reduction for stochastic variational inequalities

We develop a new stochastic algorithm with variance reduction for solving pseudo-monotone stochastic variational inequalities. Our method builds on Tseng's forward-backward-forward (FBF) algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by pseudo-monotone, Lipschitz continuous operators. The main computational advantage of Tseng's algorithm is that it relies only on a single projection step and two independent queries of a stochastic oracle. Our algorithm incorporates a variance reduction mechanism and leads to almost sure (a.s.) convergence to an optimal solution. To the best of our knowledge, this is the first stochastic look-ahead algorithm achieving this by using only a single projection at each iteration..

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arXiv2018

An inertial extrapolation method for convex simple bilevel optimization

We consider a scalar objective minimization problem over the solution set of another optimization problem. This problem is known as simple bilevel optimization problem and has drawn a significant attention in the last few years. Our inner problem consists of minimizing the sum of smooth and nonsmooth functions while the outer one is the minimization of a smooth convex function. We propose and establish the convergence of a fixed-point iterative method with inertial extrapolation to solve the problem. Our numerical experiments show that the method proposed in this paper outperforms the currently best known algorithm to solve the class of problem considered.

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