Inertial Proximal Algorithm Research Articles

Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.

Asymptotics for Some Proximal-like Method Involving Inertia and Memory Aspects

Given a Hilbert space H and a closed convex function Φ: $H\rightarrow{\mathbb{R}} \cup \{+\infty\}$ , we consider the inertial proximal algorithm $$ x_{n+1}-x_n-\alpha_n(x_n-x_{n-1})+\beta_n\partial\Phi(x_{n+1})\ni 0, \qquad \qquad \qquad \qquad({\mathcal{A}}) $$ where (α n ) and (β n ) are nonnegative sequences. The notation $\partial \Phi$ stands for the subdifferential of Φ in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to $\inf\Phi$ as n→ + ∞. When Φ has a unique minimum, the question of the convergence of (x n ) is solved. In the case of multiple minima, it is proved that if $\left(\prod_{k=1}^n \alpha_k\right)\not \in l^1$ and if a suitable geometric condition on the set argmin Φ is fulfilled, then non stationary sequences of (A) cannot converge.

Regularization inertial proximal point algorithm for common solutions of a finite family of inverse-strongly monotone equations

The aim of the paper is to propose an iterative regularization method of proximal point type for finding a common solution for a finite family of inverse-strongly monotone equations in Hilbert spaces.

Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces

The purpose of this paper is to present a regularization variant of the inertial proximal point algorithm for finding a common element of the set of solutions for a variational inequality problem involving a hemicontinuous monotone mapping and for a finite family of -inverse strongly monotone mappings from a closed convex subset of a Hilbert space into .

Iterative approximation of solution of generalized mixed set-valued variational inequality problem

In this paper, we consider a generalizedmixed set-valued variational inequality problem which includes many important known variational inequality problems and equilibrium problem, and its related some auxiliary variational inequality problems. We prove the existence of solutions of the auxiliary variational inequality problems and suggest a two-step iterative algorithm and an inertial proximal iterative algorithm. Further, we discuss the convergence analysis of iterative algorithms. The theorems presented in this paper improve and generalize many known results for solving equilibrium problems, variational inequality and complementarity problems in the literature. Mathematics subject classification (2000): 47H04, 47H06, 47H10, 47J20, 47J25, 47J30, 49J40.

An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results

Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm: $${{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} {\lambda ^{2}_{n} }}} \right. \kern-\nulldelimiterspace} {\lambda ^{2}_{n} } + A{\left( {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} {\lambda _{n} }}} \right. \kern-\nulldelimiterspace} {\lambda _{n} }} \right)} + B{\left( {x_{{n + 2}} } \right)} \ni 0,\,\,\,\,\,\,\,\,\,\,\,{\left( {\user1{\mathcal{A}}} \right)}$$ where (λn) is a sequence of positive steps. Algorithm \({\left( {\user1{\mathcal{A}}} \right)}\) may be viewed as the discretized equation of a nonlinear oscillator subject to friction. We prove that, if 0 ∈ int (A(0)) (condition of dry friction), then the sequence (xn) generated by \({\left( {\user1{\mathcal{A}}} \right)}\) is strongly convergent and its limit x∞ satisfies 0 ∈ A(0) + B(x∞). We show that, under a general condition, the limit x∞ is achieved in a finite number of iterations. When this condition is not satisfied, we prove in a rather large setting that the convergence rate is at least geometrical.