arXiv subjects

Radu Ioan Bot

Publications and source records attributed to Radu Ioan Bot.

At least 19 recordsLinked to original sources

arXiv2026

The Iterates of Nesterov's Accelerated Algorithm Converge in The Critical Regimes

In this paper, we prove that the iterates of the accelerated Nesterov's algorithm in the critical regime do converge in the weak topology to a global minimizer of an $L$-smooth function in a real Hilbert space, hence answering positively a conjecture posed by H. Attouch and co-authors a decade ago. This result is the algorithmic case of a very recent result on the continuous-time system posted by E. Ryu on X, with assistance from ChatGPT.

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arXiv2026

Failure of Rockafellar's Sum Conjecture via an Explicit Non-(FPV) Operator

Building on a construction of Weifeng Yang, we present a simplified counterexample to Rockafellar's sum conjecture on $c_0(\N_0\times\N)$. The central observation is geometric: the domain of a maximally monotone operator can have a nonconvex norm closure. In our example, this gives a short proof that the operator is not of type (FPV), which in turn implies the failure of the sum conjecture. We define an explicit operator using geometric series and provide a detailed proof of its maximal monotonicity. The example was developed with assistance from GPT-6 Astra. The purpose is to simplify and explain the construction, not to claim an independent counterexample mechanism.

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arXiv2026

Regularized extragradient method for structured bilevel optimization in continuous and discrete time

In a real Hilbert space, we study a bilevel optimization problem that consists in minimizing an outer convex function over the zero set of a maximally monotone operator. In the smooth setting, where the outer objective is convex and Fréchet differentiable and the inner operator is single-valued, continuous and monotone, we associate with the problem a first-order dynamical system that can be viewed as a monotone flow applied to a dynamically regularized operator. Under suitable geometric conditions on the inner problem --- either a weak Attouch-Czarnecki-type integrability condition or the stronger assumption of sharpness --- we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the trajectories to optimal solutions of the bilevel problem. In the smooth+nonsmooth setting, we enrich the outer objective with a proper, convex, and lower semicontinuous function, while the inner operator is augmented by the subdifferential of a function with the same properties. We propose a regularized proximal-extragradient algorithm in which both the forward and backward steps are performed with respect to dynamically regularized operators and functions, respectively. Under geometric assumptions on the inner problem analogous to those in the smooth setting, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the iterates to optimal solutions of the bilevel problem.

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arXiv2026

Variational convexity: new characterizations, calculus rules, and applications

Introduced by R.T. Rockafellar in 2019, variational convexity is a generalized notion of convexity under which stationary points of nonconvex optimization problems can still be guaranteed to exhibit local optimality. In this paper, we provide characterizations of variationally convex functions through their proximal hulls and epigraphs, and investigate operations that preserve variational convexity, including nonlinear and linear composition, summation, and proximal averaging. We further apply these results to identify variational convexity in nonlinear programming problems with possibly nonsmooth objectives, continuously differentiable inequality constraints, and affine equality constraints. This leads to new conditions ensuring local minimizers, rather than merely stationary points, for such problems, extending beyond current state-of-the-art results that typically require twice continuously differentiable objectives and constraints.

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arXiv2025

Fast Reflected Forward-Backward algorithm: achieving fast convergence rates for convex optimization with linear cone constraints

In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our approach extends the class of reflected forward-backward methods by introducing a Nesterov momentum term and a correction term, resulting in enhanced convergence performance. The iterative sequence of the proposed algorithm is proven to converge weakly, and the Fast RFB algorithm demonstrates impressive convergence rates, achieving $o\left( \frac{1}{k} \right)$ as $k \to +\infty$ for both the discrete velocity and the tangent residual at the \emph{last-iterate}. When applied to minimax problems with a smooth coupling term and nonsmooth convex regularizers, the resulting algorithm demonstrates significantly improved convergence properties compared to the current state of the art in the literature. For convex optimization problems with linear cone constraints, our approach yields a fully splitting primal-dual algorithm that ensures not only the convergence of iterates to a primal-dual solution, but also a \emph{last-iterate} convergence rate of $o\left( \frac{1}{k} \right)$ as $k \to +\infty$ for the objective function value, feasibility measure, and complementarity condition. This represents the most competitive theoretical result currently known for algorithms addressing this class of optimization problems. Numerical experiments are performed to illustrate the convergence behavior of Fast RFB.

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arXiv2025

Long-Time Analysis of Stochastic Heavy Ball Dynamics for Convex Optimization and Monotone Equations

In a separable real Hilbert space, we study the problem of minimizing a convex function with Lipschitz continuous gradient in the presence of noisy evaluations. To this end, we associate a stochastic Heavy Ball system, incorporating a friction coefficient, with the optimization problem. We establish existence and uniqueness of trajectory solutions for this system. Under a square integrability condition for the diffusion term, we prove almost sure convergence of the trajectory process to an optimal solution, as well as almost sure convergence of its time derivative to zero. Moreover, we derive almost sure and expected convergence rates for the function values along the trajectory towards the infimal value. Finally, we show that the stochastic Heavy Ball system is equivalent to a Su-Boyd-Candès-type system for a suitable choice of the parameter function, and we provide corresponding convergence rate results for the latter. In the second part of this paper, we extend our analysis beyond the optimization framework and investigate a monotone equation induced by a monotone and Lipschitz continuous operator, whose evaluations are assumed to be corrupted by noise. As before, we consider a stochastic Heavy Ball system with a friction coefficient and a correction term, now augmented by an additional component that accounts for the time derivative of the operator. We establish analogous convergence results for both the trajectory process and its time derivative, and derive almost sure as well as expected convergence rates for the decay of the residual and the gap function along the trajectory. As a final result, we show that a particular instance of the stochastic Heavy Ball system for monotone equations is equivalent to a stochastic second-order dynamical system with a vanishing damping term. Remarkably, this system exhibits fast convergence rates for both the residual and gap functions.

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arXiv2025

Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization

In this paper, we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions towards a weak Pareto optimal point. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. Furthermore, we conduct numerical experiments to validate the theoretical results and investigate the qualitative behavior of the dynamical system. This work extends results from convex single objective optimization into the multiobjective setting. The results presented in this paper lay the groundwork for the development of fast gradient and proximal point methods in multiobjective optimization, offering strong convergence guarantees.

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arXiv2025

A full splitting algorithm for structured difference-of-convex programs

In this paper, we study a class of nonconvex and nonsmooth structured difference-of-convex (DC) programs, which contain in the convex part the sum of a nonsmooth linearly composed convex function and a differentiable function, and in the concave part another nonsmooth linearly composed convex function. Among the various areas in which such problems occur, we would like to mention in particular the recovery of sparse signals. We propose an adaptive double-proximal, full-splitting algorithm with a moving center approach in the final subproblem, which addresses the challenge of evaluating compositions by decoupling the linear operator from the nonsmooth component. We establish the subsequential convergence of the generated sequence of iterates to an approximate stationary point and prove its global convergence under the Kurdyka-Łojasiewicz property. We also discuss the tightness of the convergence results and provide insights into the rationale for seeking an approximate KKT point. This is illustrated by constructing a counterexample showing that the algorithm can diverge when seeking exact solutions. Finally, we present a practical version of the algorithm that incorporates a nonmonotone line search, which significantly improves the convergence performance.

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arXiv2025

Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals

In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. We demonstrate that when the monotone operator flow is augmented with a Tikhonov regularization term, the resulting trajectory converges strongly to the element of the set of zeros with minimal norm. In addition, rates of convergence in norm for the trajectory's velocity and the operator along the trajectory can be derived in terms of the regularization function. In some particular cases, these rates of convergence can outperform the ones of the coercive operator flows and can be as fast as $O(\frac{1}{t})$ as $t \rightarrow +\infty$. In this way, we emphasize a surprising acceleration feature of the Tikhonov regularization. Additionally, we explore these properties for monotone operator flows that incorporate time rescaling and an anchor point and show that they are closely linked to second-order dynamics with a vanishing damping term. The convergence and convergence rate results we achieve for these systems complement recent findings for the Fast Optimistic Gradient Descent Ascent (OGDA) dynamics. When the monotone operator is defined as the identity minus a nonexpansive operator, the monotone equations transform into a fixed point problem. In such cases, explicitly discretizing the system with Tikhonov regularization, enhanced by an anchor point, leads to the Halpern fixed point iteration. We identify two regimes for the regularization sequence which ensure that the generated sequence of iterates converges strongly to the fixed point nearest to the anchor point. Furthermore, we establish a general theoretical framework that provides convergence rates for the vanishing of the discrete velocity and the fixed point residual. For certain regularization sequences, we derive specific convergence rates that align with those observed in continuous time.

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arXiv2025

Recovering Nesterov accelerated dynamics from Heavy Ball dynamics via time rescaling

In a real Hilbert space, we consider two classical problems: the global minimization of a smooth and convex function $f$ (i.e., a convex optimization problem) and finding the zeros of a monotone and continuous operator $V$ (i.e., a monotone equation). Attached to the optimization problem, first we study the asymptotic properties of the trajectories generated by a second-order dynamical system which features a constant viscous friction coefficient and a positive, monotonically increasing function $b(\cdot)$ multiplying $\nabla f$. For a generated solution trajectory $y(t)$, we show small $o$ convergence rates dependent on $b(t)$ for $f(y(t)) - \min f$, and the weak convergence of $y(t)$ towards a global minimizer of $f$. In 2015, Su, Boyd and Candés introduced a second-order system which could be seen as the continuous-time counterpart of Nesterov's accelerated gradient. As the first key point of this paper, we show that for a special choice for $b(t)$, these two seemingly unrelated dynamical systems are connected: namely, they are time reparametrizations of each other. Every statement regarding the continuous-time accelerated gradient system may be recovered from its Heavy Ball counterpart. As the second key point of this paper, we observe that this connection extends beyond the optimization setting. Attached to the monotone equation involving the operator $V$, we again consider a Heavy Ball-like system which features an additional correction term which is the time derivative of the operator along the trajectory. We establish a time reparametrization equivalence with the Fast OGDA dynamics introduced by Bot, Csetnek and Nguyen in 2022, which can be seen as an analog of the continuous accelerated gradient dynamics, but for monotone operators. Again, every statement regarding the Fast OGDA system may be recovered from a Heavy Ball-like system.

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arXiv2025

Fast convex optimization via closed-loop time scaling of gradient dynamics

In a Hilbert setting, for convex differentiable optimization, we develop a general framework for adaptive accelerated gradient methods. They are based on damped inertial dynamics where the coefficients are designed in a closed-loop way. Specifically, the damping is a feedback control of the velocity, or of the gradient of the objective function. For this, we develop a closed-loop version of the time scaling and averaging technique introduced by the authors. We thus obtain autonomous inertial dynamics which involve vanishing viscous damping and implicit Hessian driven damping. By simply using the convergence rates for the continuous steepest descent and Jensen's inequality, without the need for further Lyapunov analysis, we show that the trajectories have several remarkable properties at once: they ensure fast convergence of values, fast convergence of the gradients towards zero, and they converge to optimal solutions. Our approach leads to parallel algorithmic results, that we study in the case of proximal algorithms. These are among the very first general results of this type obtained using autonomous dynamics. Since the proposed numerical methods are based on proximal techniques, the results can be extended to a broader class, specifically to the problem of minimizing a proper, lower semicontinuous, and convex function. Numerical experiments are conducted to demonstrate the efficiency of the proposed methods.

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arXiv2024

Fast second-order dynamics with slow vanishing damping approaching the zeros of a monotone and continuous operator

In this work, we approach the problem of finding the zeros of a continuous and monotone operator through a second-order dynamical system with a damping term of the form $1/t^{r}$, where $r\in [0, 1]$. The system features the time derivative of the operator evaluated along the trajectory, which is a Hessian-driven type damping term when the governing operator comes from a potential. Also entering the system is a time rescaling parameter $β(t)$ which satisfies a certain growth condition. We derive $o\left(\frac{1}{t^{2r}β(t)}\right)$ convergence rates for the norm of the operator evaluated along the generated trajectories as well as for a gap function which serves as a measure of optimality for the associated variational inequality. The parameter $r$ enters the growth condition for $β(t)$: when $r < 1$, the damping $1/t^{r}$ approaches zero at a slower speed than Nesterov's $1/t$ damping; in this case, we are allowed to choose $β(t)$ to be an exponential function, thus having linear convergence rates for the involved quantities. We also show weak convergence of the trajectories towards zeros of the governing operator. Through a particular choice for the operator, we establish a connection with the problem of minimizing a smooth and convex function with linear constraints. The convergence rates we derived in the operator case are inherited by the objective function evaluated at the trajectories and for the feasibility gap. We also prove weak convergence of the trajectories towards primal-dual solutions of the problem. A discretization of the dynamical system yields an implicit algorithm that exhibits analogous convergence properties to its continuous counterpart. We complement our theoretical findings with two numerical experiments.

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arXiv2024

On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator

In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $M : \R^n \rightarrow \R^n$ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process $X(t)$ to a zero of $M$ and of $\|M(X(t))\|$ to $0$ as $t \rightarrow +\infty$. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for $\|M(X(t))\|^2$ and $\langle M(X(t), X(t)-x^*\rangle$, where $x^*$ is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties.

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arXiv2024

Fast Optimistic Gradient Descent Ascent (OGDA) method in continuous and discrete time

In the framework of real Hilbert spaces we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator $V$. The starting poin is a second order dynamical system that combines a vanishing damping term with the time derivative of $V$ along the trajectory. Our method exhibits fast convergence rates of order $o \left( \frac{1}{tβ(t)} \right)$ for $\|V(z(t))\|$, wher $β(\cdot)$ is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function. We also prove the weak convergence of the trajectory to a zero of $V$. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method, for which we prove that the generated sequence of iterates shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order $o \left( \frac{1}{kβ_k} \right)$ for $\|V(z^k)\|$ and the restricted gap function, where $(β_k)_{k \geq 0}$ is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm we show by additionally assuming that the operator $V$ is Lipschitz continuous convergence rates of order $o \left( \frac{1}{k} \right)$ for $\|V(z^k)\|$ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition we prove for both algorithms the convergence of the iterates to a zero of $V$. To our knowledge, our study exhibits the best known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods for monotone equations.

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arXiv2023

Fast Forward-Backward splitting for monotone inclusions with a convergence rate of the tangent residual of $o(1/k)$

We address the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive operator. Our approach introduces a modification to the forward-backward method by integrating an inertial/momentum term alongside a correction term. We demonstrate that the sequence of iterations thus generated converges weakly towards a solution for the monotone inclusion problem. Furthermore, our analysis reveals an outstanding attribute of our algorithm: it displays rates of convergence of the order $o(1/k)$ for the discrete velocity and the tangent residual approaching zero. These rates for tangent residuals can be extended to fixed-point residuals frequently discussed in the existing literature. Specifically, when applied to minimize a nonsmooth convex function subject to linear constraints, our method evolves into a primal-dual full splitting algorithm. Notably, alongside the convergence of iterates, this algorithm possesses a remarkable characteristic of nonergodic/last iterate $o(1/k)$ convergence rates for both the function value and the feasibility measure. Our algorithm showcases the most advanced convergence and convergence rate outcomes among primal-dual full splitting algorithms when minimizing nonsmooth convex functions with linear constraints.

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arXiv2023

Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o\left(\frac{1}{k}\right)$

The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.

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arXiv2023

A Fast Optimistic Method for Monotone Variational Inequalities

We study monotone variational inequalities that can arise as optimality conditions for constrained convex optimisation or convex-concave minimax problems and propose a novel algorithm that uses only one gradient/operator evaluation and one projection onto the constraint set per iteration. The algorithm, which we call fOGDA-VI, achieves a $o \left( \frac{1}{k} \right)$ rate of convergence in terms of the restricted gap function as well as the natural residual for the last iterate. Moreover, we provide a convergence guarantee for the sequence of iterates to a solution of the variational inequality. These are the best theoretical convergence results for numerical methods for (only) monotone variational inequalities reported in the literature. To empirically validate our algorithm we investigate a two-player matrix game with mixed strategies of the two players. Concluding, we show promising results regarding the application of fOGDA-VI to the training of generative adversarial nets.

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arXiv2023

Accelerated Griffin-Lim algorithm: A fast and provably converging numerical method for phase retrieval

The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. In this paper, we present a fast inertial/momentum based algorithm for the phase retrieval problem and we prove a convergence guarantee for the new algorithm and for the Fast Griffin-Lim algorithm, whose convergence remained unproven in the past decade. In the final chapter, we compare the algorithm for the Short Time Fourier transform phase retrieval with the Griffin-Lim algorithm and FGLA and to other iterative algorithms typically used for this type of problem.

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