On the Convergence and O(1/N) Complexity of a Class of Nonlinear Proximal Point Algorithms for Monotonic Variational Inequalities

Jian Wu,Gaohang Yu

doi:10.19139/soic.v2i2.73

Abstract

This paper presents a class of proximal point algorithms using a nonlinear proximal term for monotonic variational inequality problems. This work extents proximal point algorithms using Bregman distance for minimization problems, and differs with J. Eckstein's approximate iterations in Bregman-function-based proximal algorithms (1998). We study the convergence of the proposed algorithms and obtain a $O(1/N)$ computing complexity/convergence rate of the algorithms. Further more, connections to some existed popular methods were given, which shows that our algorithm can include these methods within a general form.

Highlights

Variational inequality (VI) has received a lot of attention due to its various applications in operation research, economic equilibrium, engineering design, and other fields [5, 6].In this paper, we will study iterative algorithms for monotonic VI problems, which can be summarized in a form as follows: find a point u∗ in Ω such that⟨u − u∗, F (u∗)⟩ ≥ 0, ∀u ∈ Ω, (1)ISSN 2310-5070 ISSN 2311-004XCopyright ⃝c 2014 International Academic PressJIAN WU AND GAOHANG YU where F : Rd → Rd is a mapping from a Euclidian space Rd to itself, and Ω a convex subset in Rd.VI has an important application in optimization
We will study iterative algorithms for monotonic VI problems, which can be summarized in a form as follows: find a point u∗ in Ω such that
A classical algorithm widely known as proximal point algorithm (PPA) is first proposed in [8] and developed in [9]

Summary

Introduction

Variational inequality (VI) has received a lot of attention due to its various applications in operation research, economic equilibrium, engineering design, and other fields [5, 6]. We will study iterative algorithms for monotonic VI problems, which can be summarized in a form as follows: find a point u∗ in Ω such that. This class of VIs has a nice property that its solution set is nonempty; secondly, it responds to the convex program in the optimization field. There is extensive literature on numerical algorithms for VIs [1, 2, 4, 7, 10] Among these solvers, a classical algorithm widely known as proximal point algorithm (PPA) (or proximal minimization algorithm specially for minimization problems [3]) is first proposed in [8] and developed in [9].

Motivation and Proposed Algorithm

Connection to Other Methods

Entropy Case

Projection Method: a Special Case

Conclusion