CEU Electronic Theses and Dissertations, 2013
| Author | Trinh, Hai Thi |
|---|---|
| Title | Proximal point algorithms |
| Summary | Many nonlinear operator equations (inclusions) are of the form $0\in A(x)$, where $A$ is a (possibly set-valued) monotone operator in a Hilbert space $H$. There are different iterative methods that are used to approximate the solutions of this equation. One of the most popular methods is the so-called proximal point algorithm (PPA), first introduced by B. Martinet (1970) and later developed extensively by R.T. Rockafellar and many other researchers. The researchers have investigated the convergence of this iterative process and in some cases gave the rate of convergence of this method. Many applications have been investigated as well. In particular, this algorithm is used to approximate the solutions of some variational inequalities or minimizers of convex cost functionals. In this thesis, we first provide a description of the particular case introduced by Martinet. Then, it is shown how Rockafellar derived the general PPA from Martinet's algorithm. Unfortunately, the sequence $\{ x_n \}$ converges only weakly, as shown by O. G\"{u}ler (1991). Many mathematicians (M. V. Solodov and B. F. Svaiter, S. Kamimura and W. Takahashi , H. K. Xu and others) have tried to modify the PPA in such a way that the new iterative methods generate strongly convergent sequences. On the other hand, the above summability condition on errors is too strong from a numerical point of view. This summability condition can be relaxed and there are already several results in this direction (for examples, see \cite{4}). Furthermore, there are extensions of the PPA to the case of two monotone operators $A$ and $B$. These extensions are generalizations of the old method of alternating projections introduced by J. von Neumann in early thirtieths. \\ The structure of the thesis is organized as follows. In Chapter 1, we recall some fundamental concepts, notations and results, which will be used in the following chapters. Chapter 2 presents a short introduction to the proximal point algorithms. Then, in Chapter 3, we include some important results regarding the boundedness and convergence of the sequences generated by the PPA. In Chapter 4, we discuss some generalizations of the regularization method. In particular, the modified two parameter method as well as the modified four parameter method will be discussed. Finally, Chapter 5 is concerned with the method of alternating resolvents, i.e., the proximal point algorithm involving two monotone operators. |
| Supervisor | Morosanu, Gheorghe |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2013/trinh_hai.pdf |
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