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Title:  Theory and Algorithms for Nonlinear Optimization and Variational Inequalities 
Author(s):  Chu, LiangJu 
Doctoral Committee Chair(s):  McLinden, L., 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  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 primaldual pathfollowing interior point algorithm for solving the pair of primaldual problems (P) and (D): (P) sk20inf$\sb{x}\{f(x); Ax=b, x\ge 0\}$ (D) sk20sup$\sb{x,s}\{g(x,s); Ax=b, x\ge 0$ and $s=\nabla f(x)  A\sp{T}y\ge 0$ for some $y\in R\sp{m}\}.$ Here, f is a continuously differentiable convex function in $R\sp{n},$ and g is defined by $g(x, s) := f(x)  x\sp{T}s.$ Under a kind of strict feasibility assumption, we show that the algorithm under modification requires a total of $O(\sqrt{n\ell})$ number of iterations, with total arithmetic operations of order $O(n\sp3\ell),$ where $\ell$ is the initial input size. As an application to usual linear or convex quadratic programming, this algorithm solves the pair (P) and (D) in at most $O(\sqrt{nL})$ iterations, with total arithmetic operations of order $O(n\sp3 L),$ where L is the input size. Moreover, we show the duality gap sequence goes to zero linearly, superlinearly, or quadratically, depending upon a certain parameter $\theta.$ Also, we show that any limit point of the induced sequence ($x\sp{k},s\sp{k})$ is a maximal complementary solution of a certain monotone multifunction. 
Issue Date:  1992 
Type:  Text 
Description:  169 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1992. 
URI:  http://hdl.handle.net/2142/72533 
Other Identifier(s):  (UMI)AAI9305491 
Date Available in IDEALS:  20141217 
Date Deposited:  1992 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois