Inertial Proximal Algorithm Research Articles

A modified inertial proximal minimization algorithm for structured nonconvex and nonsmooth problem

We introduce an enhanced inertial proximal minimization algorithm tailored for a category of structured nonconvex and nonsmooth optimization problems. The objective function in question is an aggregation of a smooth function with an associated linear operator, a nonsmooth function dependent on an independent variable, and a mixed function involving two variables. Throughout the iterative procedure, parameters are selected employing a straightforward approach, and weak inertial terms are incorporated into two subproblems within the update sequence. Under a set of lenient conditions, we demonstrate that the sequence engendered by our algorithm is bounded. Furthermore, we establish the global and strong convergence of the algorithmic sequence, contingent upon the assumption that the principal function adheres to the Kurdyka–Łojasiewicz (KL) property. Ultimately, the numerical outcomes corroborate the algorithm’s feasibility and efficacy.

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

On the strong convergence of an inertial proximal algorithm with a time scale, Hessian-driven damping, and a Tikhonov regularization term

On the strong convergence of an inertial proximal algorithm with a time scale, Hessian-driven damping, and a Tikhonov regularization term

New inertial proximal algorithms for solving multivalued variational inequalities

New inertial proximal algorithms for solving multivalued variational inequalities

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.

Inertial proximal point algorithm for the split common solution problem of monotone operator equations

Inertial proximal point algorithm for the split common solution problem of monotone operator equations

Double Inertial Proximal Gradient Algorithms for Convex Optimization Problems and Applications

Double Inertial Proximal Gradient Algorithms for Convex Optimization Problems and Applications

Weak convergence of inertial proximal algorithms with self adaptive stepsize for solving multivalued variational inequalities

In this work, we introduce an inertial proximal algorithm for solving multivalued variational inequality problems in a real Hilbert space. By using self-adaptive and inertial techniques via proximal operators, we establish the weak convergence of the iteration sequences generated by these algorithms when the multivalued cost mappings associated with the problems are monotone and Lipschitz continuous. Moreover, we present the nonasymptotic O ( 1 k ) convergence rate of the proposed algorithms. We also provide some numerical examples to illustrate the accuracy and efficiency of our algorithms by comparing with other recent algorithms in the literature.

Convergence results of two-step inertial proximal point algorithm

This paper proposes a two-point inertial proximal point algorithm to find zero of maximal monotone operators in Hilbert spaces. We obtain weak convergence results and non-asymptotic O(1/n) convergence rate of our proposed algorithm in non-ergodic sense. Applications of our results to various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers are given. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.

Inertial proximal point algorithm for variational inclusion in Hadamard manifolds

In this paper, we consider the inertial proximal point algorithm for finding a zero point of variational inclusions on Hadamard manifolds. Under suitable conditions, it is proved that the sequence generated by the algorithm converges to an element of the set of zero points of variational inclusion problem. As applications, we utilize our results to study the minimization problem and saddle point problem in the setting of Hadamard manifolds.

A parallel Tseng\u2019s splitting method for solving common variational inclusion applied to signal recovery problems

In this work we propose an accelerated algorithm that combines various techniques, such as inertial proximal algorithms, Tseng’s splitting algorithm, and more, for solving the common variational inclusion problem in real Hilbert spaces. We establish a strong convergence theorem of the algorithm under standard and suitable assumptions and illustrate the applicability and advantages of the new scheme for signal recovering problem arising in compressed sensing.

Interactive Photo Editing on Smartphones via Intrinsic Decomposition

AbstractIntrinsic decomposition refers to the problem of estimating scene characteristics, such as albedo and shading, when one view or multiple views of a scene are provided. The inverse problem setting, where multiple unknowns are solved given a single known pixel‐value, is highly under‐constrained. When provided with correlating image and depth data, intrinsic scene decomposition can be facilitated using depth‐based priors, which nowadays is easy to acquire with high‐end smartphones by utilizing their depth sensors. In this work, we present a system for intrinsic decomposition of RGB‐D images on smartphones and the algorithmic as well as design choices therein. Unlike state‐of‐the‐art methods that assume only diffuse reflectance, we consider both diffuse and specular pixels. For this purpose, we present a novel specularity extraction algorithm based on a multi‐scale intensity decomposition and chroma inpainting. At this, the diffuse component is further decomposed into albedo and shading components. We use an inertial proximal algorithm for non‐convex optimization (iPiano) to ensure albedo sparsity. Our GPU‐based visual processing is implemented on iOS via the Metal API and enables interactive performance on an iPhone 11 Pro. Further, a qualitative evaluation shows that we are able to obtain high‐quality outputs. Furthermore, our proposed approach for specularity removal outperforms state‐of‐the‐art approaches for real‐world images, while our albedo and shading layer decomposition is faster than the prior work at a comparable output quality. Manifold applications such as recoloring, retexturing, relighting, appearance editing, and stylization are shown, each using the intrinsic layers obtained with our method and/or the corresponding depth data.

非凸优化中一种带非单调线搜索的惯性邻近算法

非凸优化中一种带非单调线搜索的惯性邻近算法

New inertial proximal gradient methods for unconstrained convex optimization problems

The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

Convergence Rate of Inertial Proximal Algorithms with General Extrapolation and Proximal Coefficients

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

Multi-step inertial proximal contraction algorithms for monotone variational inclusion problems

"In this article, we introduce the multi-step inertial proximal contraction algorithms (MiPCA) to approximate a zero of the sum of two monotone operators, with one of the two operators being monotone and Lipschitz continuous. The weak convergence of the MiPCA is shown under the summability condition formulated in terms of the iterative sequence in a Hilbert space setting. We also investigate the unconditional convergence of the one-step inertial proximal contraction algorithm. Finally, numerical experiments are given to illustrate the advantage of the multi-step inertial proximal contraction algorithms."

Superiorization methodology and perturbation resilience of inertial proximal gradient algorithm with application to signal recovery

In this paper, we construct a novel algorithm for solving non-smooth composite optimization problems. By using inertial technique, we propose a modified proximal gradient algorithm with outer perturbations, and under standard mild conditions, we obtain strong convergence results for finding a solution of composite optimization problem. Based on bounded perturbation resilience, we present our proposed algorithm with the superiorization method and apply it to image recovery problem. Finally, we provide the numerical experiments to show efficiency of the proposed algorithm and comparison with previously known algorithms in signal recovery.

RETRACTED ARTICLE: Convergence Rate of Inertial Proximal Algorithms with General Extrapolation and Proximal Coefficients

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.