1 Introduction

In this course, we will start by introducing the maximal monotone inclusion problem and the associated splitting methods. This problem is quite general, encompassing many other optimization problems as special cases, including variational inequality problems, convex feasibility problems, and many other problems. A widely used approach for addressing this problem is the proximal point method, which serves as an essential framework for interpreting numerous other algorithms. We will also explore additional splitting methods designed for the maximal monotone inclusion problem such as the Douglash-Rachord Method and Spingarn’s splitting method.

2 Schedule

Monotone Operator
1. Properties of the (maximal) monotone monotone operators
2. Minty Theorem
3. Enlargement of maximal monotone operator

Fixed point Iteration
1. Contraction and Banach Fixed Point Theorem
2. Averaged operator and Krasnosel’ski˘ı-Mann Iteration
3. Fejer convergence and quasi-Fejer convergence.

Proximal Point Method
1. Definition and property of resolvent
2. Convergence analysis with summable error
3. Proximal point of multipliers method

Extentions of proximal point method
1. A hybrid approximate extragradient–proximal point algorithm
2. Strong convergence of the Proximal point method.

Some other Splitting menthod
1. Douglash Rachford Method
2. Forward-backward splitting Method.
3. Spingarn’s splitting method

 

References
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