Theory and Algorithms for Nonlinear Optimization and Variational Inequalities

We consider three separate topics in nonlinear optimization, one theoretical topic and two algorithmic topics. Each of these topics deals with a broad class of nonlinear optimization problems. We first introduce and analyze a generalized parametric variational inequality problem $PVI(E, T, C, \psi$, Z) in locally convex Hausdorff topological vector spaces. Based on Nikaido's coincidence theorem, we establish several general existence theorems, even without requiring convexity nor contractibility on T and C, but merely a certain acyclic property. The case where C is not compact is considered. We also analyze asymptotic convergence of the partial proximal point algorithm for solving the generalized nonlinear equation $0\in T(x),$ where $T : H\to H$ is a maximal monotone multifunction, and $H := H\sb1 \times H\sb2$ is a product of two real Hilbert spaces. Under the mild feasibility assumption $O\in int(coR(T)),$ we show that the partial proximal point algorithm has the same convergence properties as does Rockafellar's proximal point algorithm. Moreover, the partial convergence rates are shown to depend upon how rapidly $T\sp{-1}$ grows away from the solution set $T\sb{-1}(0).$ The partial rate of convergence is linear, superlinear, or quadratic, depending upon certain parameters. Finally, we describe a primal-dual path-following interior point algorithm for solving the pair of primal-dual problems (P) and (D):

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Chu, Liang-Ju

Contributors dc:contributor

• McLinden, L.,

• Mathematics

Identifier

(UMI)AAI9305491

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72533