Proximal Point Algorithm Research Articles

Inertial proximal point algorithm for sum of two monotone vector fields in Hadamard manifold

Inertial proximal point algorithm for sum of two monotone vector fields in Hadamard manifold

A Bregman proximal point algorithm without convexity in Banach spaces

ABSTRACT In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

Variance Reduction Techniques for Stochastic Proximal Point Algorithms

AbstractIn the context of finite sums minimization, variance reduction techniques are widely used to improve the performance of state-of-the-art stochastic gradient methods. Their practical impact is clear, as well as their theoretical properties. Stochastic proximal point algorithms have been studied as an alternative to stochastic gradient algorithms since they are more stable with respect to the choice of the step size. However, their variance-reduced versions are not as well studied as the gradient ones. In this work, we propose the first unified study of variance reduction techniques for stochastic proximal point algorithms. We introduce a generic stochastic proximal-based algorithm that can be specified to give the proximal version of SVRG, SAGA, and some of their variants. For this algorithm, in the smooth setting, we provide several convergence rates for the iterates and the objective function values, which are faster than those of the vanilla stochastic proximal point algorithm. More specifically, for convex functions, we prove a sublinear convergence rate of O(1/k). In addition, under the Polyak-łojasiewicz condition, we obtain linear convergence rates. Finally, our numerical experiments demonstrate the advantages of the proximal variance reduction methods over their gradient counterparts in terms of the stability with respect to the choice of the step size in most cases, especially for difficult problems.

A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming

A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming

Proximal Point Algorithms with Inertial Extrapolation for Quasi-convex Pseudo-monotone Equilibrium Problems

Proximal Point Algorithms with Inertial Extrapolation for Quasi-convex Pseudo-monotone Equilibrium Problems

Asymptotic proximal point algorithm for wavefront sensorless adaptive optics.

Solving the distorted wavefront in wavefront sensorless adaptive optics (WFSL-AO) relies on excellent optimizers. Many local or global optimization algorithms have been applied to WFSL-AO; however, there is still a challenge to balance the effect and speed of correcting aberrations. To overcome this, a novel global optimization algorithm named asymptotic proximal point (APP) method is introduced into WFSL-AO in this Letter. We compare this algorithm with the various existing optimization algorithms in convergence speed and correction capability by performing numerical simulations. The results show that the APP method beats all competitors with a better correction effect and faster speed.

Strong convergence of a viscosity proximal point algorithm with new error sequences

In the paper, we consider a viscosity proximal point algorithm with over-relaxed factor for solving inclusion problem in the setting of Hilbert space. Moreover, we establish its strong convergences under two different error criteria, which generalize some existing conclusions.

New fast proximal point algorithms for monotone inclusion problems with applications to image recovery

The proximal point algorithm has many applications for convex optimization with several versions of the proximal point algorithm including generalized proximal point algorithms and accelerated proximal point algorithms that have been studied in the literature. In this paper, we propose accelerated versions of generalized proximal point algorithms to find a zero of a maximal monotone operator in Hilbert spaces. We give both weak and linear convergence results of our proposed algorithms under standard conditions. Numerical applications of our results to image recovery are given and numerical implementations show that our algorithms are effective and superior to other related accelerated proximal point algorithms in the literature.

A Semismooth Newton Stochastic Proximal Point Algorithm with Variance Reduction

A Semismooth Newton Stochastic Proximal Point Algorithm with Variance Reduction

Proximal point type algorithms with relaxed and inertial effects beyond convexity

We show that the recent relaxed-inertial proximal point algorithm due to Attouch and Cabot remains convergent when the function to be minimized is not convex, being only endowed with certain generalized convexity properties. Numerical experiments showcase the improvements brought by the relaxation and inertia features to the standard proximal point method in this setting, too.

New Modified Proximal Point Algorithm for Solving Minimization and Common Fixed Point Problem over CAT(κ) Spaces

In this paper, we present a newly proximal point algorithm for solving minimization and common fixed point problems in CAT(1) spaces. Under some mild conditions, we prove strong and ∆-convergence theorems. Additionally, a convex minimization application and a common fixed point problems in CAT(κ) spaces with the bounded κ ∈ (0, ∞) are provided. Our findings complement and advance the pertinent recent findings in the literature

Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications

Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications

Modified proximal point algorithms for variational inclusion problems on Hadamard manifolds

Modified proximal point algorithms for variational inclusion problems on Hadamard manifolds

On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications

In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of flat Hadamard spaces. To the best of our knowledge, this result presents the first hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and fixed point problems.

Generalized proximal point algorithms with correction terms and extrapolation

Generalized proximal point algorithms with correction terms and extrapolation

Strongly Convergent Inertial Proximal Point Algorithm Without On-line Rule

Strongly Convergent Inertial Proximal Point Algorithm Without On-line Rule

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Weak convergence of inertial proximal point algorithm for a family of nonexpansive mappings in Hilbert spaces

In this paper, we modified proximal point algorithm with some convex combination technique to approximate a minimizer, equilibrium point and a common fixed point of a family of nonexpansive mappings in Hilbert spaces. We establish a weak convergence theorem under some mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method.

Proximal Point Algorithm for Nonexpansive Mappings in Hadamard Spaces Based on SRJ Iteration Process

In this paper, We provide a new modified proximal point approach utilizing fixed point iterates of nonexpansive mappings in Hadamard space and show that the sequence created by our iterative process converges to a minimizer of a convex function and a fixed point of mappings. Finally, we present a numerical illustration for supporting our main result. Our results obtained in this paper improve, extend and unify results of Khan-Abbas [23], Cholamjiak et al. [10] and Dashputre et al. [11].

On the diagonal proximal point algorithms

We study the asymptotic behaviour of the diagonal proximal point algorithms governed by a sequence of maximally monotone operators. With a summability condition on the Brézis–Haraux functions of the sequence of operators, we obtain several ergodic and weak convergence results in the monotone and subgradient cases. This approach is of relevance to and motivated by Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790]. Some strong convergence results are presented as well. Our results in this paper include discrete versions of the results in Attouch et al. [Asymptotic behaviour of nonautonomous monotone and subgradient evolution equations. Trans Amer Math Soc. 2018;370:755–790], and also unify and improve many existing results in Attouch et al. [Prox-penalization and splitting methods for constrained variational problems. SIAM J Optim. 2011;21:149–173], Cabot [Proximal point algorithm controlled by a slowly vanishing term: Applications to hierarchical minimization. SIAM J Optim. 2005;15:555–572], Lehdili and Moudafi [Combining the proximal algorithm and Tikhonov regularization. Optimization. 1996;37:239–252], Xu [A regularization method for the proximal point algorithm. J Glob Optim. 2006;36:115–125].