IPGM: Inertial Proximal Gradient Method for Convolutional Dictionary Learning

Inspired by the recent success of the proximal gradient method (PGM) and recent efforts to develop an inertial algorithm, we propose an inertial PGM (IPGM) for convolutional dictionary learning (CDL) by jointly optimizing both an ℓ2-norm data fidelity term and a sparsity term that enforces an ℓ1 penalty. Contrary to other CDL methods, in the proposed approach, the dictionary and needles are updated with an inertial force by the PGM. We obtain a novel derivative formula for the needles and dictionary with respect to the data fidelity term. At the same time, a gradient descent step is designed to add an inertial term. The proximal operation uses the thresholding operation for needles and projects the dictionary to a unit-norm sphere. We prove the convergence property of the proposed IPGM algorithm in a backtracking case. Simulation results show that the proposed IPGM achieves better performance than the PGM and slice-based methods that possess the same structure and are optimized using the alternating-direction method of multipliers (ADMM).

An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

Improved inertial projection and contraction method for solving pseudomonotone variational inequality problems

The objective of this article is to solve pseudomonotone variational inequality problems in a real Hilbert space. We introduce an inertial algorithm with a new self-adaptive step size rule, which is based on the projection and contraction method. Only one step projection is used to design the proposed algorithm, and the strong convergence of the iterative sequence is obtained under some appropriate conditions. The main advantage of the algorithm is that the proof of convergence of the algorithm is implemented without the prior knowledge of the Lipschitz constant of cost operator. Numerical experiments are also put forward to support the analysis of the theorem and provide comparisons with related algorithms.

An Inertial Algorithm for Solving Hammerstein Equations

An inertial algorithm for solving Hammerstein equations is presented. This algorithm is obtained as a consequence of a new inertial algorithm proposed and studied for solving nonlinear equations involving operators that are m-accretive. Some strong convergence theorems are proved in real Banach spaces that are uniformly smooth. Furthermore, comparisons of the numerical performance of our algorithms with the numerical performance of some recent important algorithms are presented.

A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem

<p style='text-indent:20px;'>In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view.</p>

Modified inertial algorithm for solving mixed equilibrium problems in Hadamard spaces

<p style='text-indent:20px;'>The main purpose of this paper is to introduce the concept of modified inertial algorithm in Hadamard spaces. We emphasize that, as far as we know, this is the first time that this concept is being considered in this setting. Under some weak assumptions, we prove that the modified inertial algorithm converges strongly to a common solution of a finite family of mixed equilibrium problems and fixed point problem of a nonexpansive mapping. We also give a primary numerical illustration in the framework of Hadamard spaces, to show the efficiency of the modified inertial term in our proposed algorithm.</p>