arXiv subjects

Szilárd Csaba László

Publications and source records attributed to Szilárd Csaba László.

13 recordsLinked to original sources

arXiv2025

Solving nonconvex optimization problems via a second order dynamical system with unbounded damping

In this paper we study a second order dynamical system with variable coefficients in connection to the minimization problem of a smooth nonconvex function. The convergence of the trajectories generated by the dynamical system to a critical point of the objective function is assured, provided a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. We also provide convergence rates for the trajectories generated by the dynamical system, formulated in terms of the Łojasiewicz exponent, and we show that the unbounded damping considered in our dynamical system significantly improves the convergence rates known so far in the literature, that is, instead of linear rates we obtain superlinear rates.

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arXiv2024

Strong convergence and fast rates for systems with Tikhonov regularization

We introduce and investigate the asymptotic behaviour of the trajectories of a second order dynamical system with Tikhonov regularization for solving a monotone equation with single valued, monotone and continuous operator acting on a real Hilbert space. We consider a vanishing damping which is correlated with the Tikhonov parameter and which is in line with recent developments in the literature on this topic. A correction term which involves the time derivative of the operator along the trajectory is also involved in the system and makes the link with Newton and Levenberg-Marquardt type methods. We obtain strong convergence of the trajectory to the minimal norm solution and fast convergence rates for the velocity and a quantity involving the operator along the trajectory. The rates are very closed to the known fast convergence results for systems without Tikhonov regularization, the novelty with respect to this is that we also obtain strong convergence of the trajectory to the minimal norm solution. As an application we introduce a primal-dual dynamical system for solving linearly constrained convex optimization problems, where the strong convergence of the trajectories is highlighted together with fast convergence rates for the feasibility measure and function values.

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arXiv2024

A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution

We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection to the minimization problem of the sum of a convex lower semi-continuous function $f$ and a smooth convex function $g$. For the appropriate setting of the parameters we provide strong convergence of the generated sequence $(x_k)$ to the minimum norm minimizer of our objective function $f+g$. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another settings of the parameters the optimal rate of order $\mathcal{O}(k^{-2})$ for the potential energy $(f+g)(x_k)-\min(f+g)$ can be obtained.

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arXiv2024

On the convergence of an inertial proximal algorithm with a Tikhonov regularization term

This paper deals with an inertial proximal algorithm that contains a Tikhonov regularization term, in connection to the minimization problem of a convex lower semicontinuous function $f$. We show that for appropriate Tikhonov regularization parameters the value of the objective function in the sequences generated by our algorithm converges fast (with arbitrary rate) to the global minimum of the objective function and the generated sequences converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the discrete velocities towards zero and some sum estimates. Nevertheless, our main goal is to obtain strong convergence results and also pointwise and sum estimates for the same constellation of the parameters involved. Our analysis reveals that the extrapolation coefficient and the Tikhonov regularization coefficient are strongly correlated and there is a critical setting of the parameters that separates the cases when strong respective weak convergence results can be obtained.

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arXiv2024

Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization

This paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $\argmin$ set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.

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arXiv2023

A Nesterov type algorithm with double Tikhonov regularization: fast convergence of the function values and strong convergence to the minimal norm solution

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function $f.$ We show that the generated sequences converge strongly to the minimal norm element from $\argmin f$. We also show that from a practical point of view the Tikhonov regularization does not affect Nesterov's optimal convergence rate of order $\mathcal{O}(n^{-2})$ for the potential energies $f(x_n)-\min f$ and $f(y_n)-\min f$, where $(x_n),\,(y_n)$ are the sequences generated by our algorithm. Further, we obtain fast convergence to zero of the discrete velocity, but also some estimates concerning the value of the gradient of the objective function in the generated sequences.

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arXiv2021

Forward-backward algorithms with different inertial terms for structured non-convex minimization problems

We investigate two inertial forward-backward algorithms in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithms are formulated in the spirit of the famous FISTA method, however the setting is non-convex and we allow different inertial terms. Moreover, the inertial parameters in our algorithms can take negative values too. We also treat the case when the non-smooth function is convex and we show that in this case a better step size can be allowed. We prove some abstract convergence results which applied to our numerical schemes allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we obtain a general result that applied to our numerical schemes ensures convergence rates for the generated sequences and for the objective function values formulated in terms of the KL exponent of a regularization of the objective function. Finally, we apply our results to image restoration.

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arXiv2020

Tikhonov regularization of a second order dynamical system with Hessian driven damping

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

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arXiv2020

Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization

We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allows us to show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent. Finally, we obtain sublinear convergence rates for the objective function values in the iterates in the case when the objective function is convex.

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arXiv2019

An extension of the second order dynamical system that models Nesterov's convex gradient method

In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Frèchet differentiable objective function. We show that inertial algorithms, such as Nesterov's algorithm, can be obtained via the natural explicit discretization from our dynamical system. Our dynamical system can be viewed as a perturbed version of the heavy ball method with vanishing damping, however the perturbation is made in the argument of the gradient of the objective function. This perturbation seems to have a smoothing effect for the energy error and eliminates the oscillations obtained for this error in the case of the heavy ball method with vanishing damping, as some numerical experiments show. We prove that the value of the objective function in a generated trajectory converges in order O(1/t^2) to the global minimum of the objective function. Moreover, we obtain that a trajectory generated by the dynamical system converges to a minimum point of the objective function.

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arXiv2018

Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization

We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we provide convergence rates for the generated sequences and the function values formulated in terms of the Łojasiewicz exponent.

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arXiv2017

Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.

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arXiv2017

Second order dynamical systems with penalty terms associated to monotone inclusions

In this paper we investigate in a Hilbert space setting a second order dynamical system of the form $$\ddot{x}(t)+\g(t)\dot{x}(t)+x(t)-J_{λ(t) A}\big(x(t)-λ(t) D(x(t))-λ(t)β(t)B(x(t))\big)=0,$$ where $A:{\mathcal H}\toto{\mathcal H}$ is a maximal monotone operator, $J_{λ(t) A}:{\mathcal H}\To{\mathcal H}$ is the resolvent operator of $λ(t)A$ and $D,B: {\mathcal H}\rightarrow{\mathcal H}$ are cocoercive operators, and $λ,β:[0,+\infty)\rightarrow (0,+\infty)$, and $γ:[0,+\infty)\rightarrow (0,+\infty)$ are step size, penalization and, respectively, damping functions, all depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy-Lipschitz-Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator $A+D+{N}_C,$ where $C=\zer B$ and $N_C$ denotes the normal cone operator of $C$. To this end we use Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show strong convergence for trajectories to the unique zero of $A+D+{N}_C$, provided that $A$ is a strongly monotone operator.

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