Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms.

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

In a Hilbert space $\mathcal{H}$ H , we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $ A = ∇ f + B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as $t\to +\infty $ t → + ∞ of the generated trajectories towards the zeros of $\nabla f +B$ ∇ f + B . The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operatorAand maximal monotone operatorsBwithD(B)⊂H:xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, forn=1,2,…,for givenx1in a real Hilbert spaceH, where(αn),(γn), and(δn)are sequences in(0,1)withαn+γn+δn=1for alln≥1,(en)denotes the error sequence, andf:H→His a contraction. The algorithm is known to converge under the following assumptions onδnanden: (i)(δn)is bounded below away from 0 and above away from 1 and (ii)(en)is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i)(δn)is bounded below away from 0 and above away from 3/2 and (ii)(en)is square summable in norm; and we still obtain strong convergence results.

Algorithm for Solving a New System of Generalized Nonlinear Quasi-Variational-Like Inclusions in Hilbert Spaces

We introduce and study a new system of generalized nonlinear quasi-variational-like inclusions with H(·,·)-cocoercive operator in Hilbert spaces. We suggest and analyze a class of iterative algorithms for solving the system of generalized nonlinear quasi-variational-like inclusions. An existence theorem of solutions for the system of generalized nonlinear quasi-variational-like inclusions is proved under suitable assumptions which show that the approximate solutions obtained by proposed algorithms converge to the exact solutions.

On the Solving of Variational Inequalities of Stationary Problems of Two-Phase Flow in Porous Media

We consider a variational inequalities of the second kind with cocoercive operator and a non-differentiable proper convex functional. Such inequalities arise in the mathematical modeling of the problem of finding the boundaries of ultimately-stable pillars of residual viscous-plastic oil. To solve the variational inequalities we suggest the iterative process and its convergence investigated. The numerical results confirm the efficiency of the proposed method.

Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

We investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

Strong Convergence of Iterative Algorithm for a New System of GeneralizedH·,·-η-Cocoercive Operator Inclusions in Banach Spaces

We introduce and study a new system of generalizedH·,·-η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated withH·,·-η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.