Thesis (Ph. D.)--University of Washington, 1990. / Vita. Includes bibliographical references (leaves [97]-100).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1951. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 46-47).

Thesis--University of Florida. / Description based on print version record. Typescript. Vita. Bibliography: leaves 107-111.

Thesis (Ph. D.)--Oregon State University, 1968. / Typescript (photocopy). Includes bibliographical references (leaves 57-58). Also available on the World Wide Web.

In the literature, error bound conditions have been widely used for studying the linear convergence rates of various first-order algorithms and the majority of literature focuses on how to sufficiently ensure these error bound conditions, usually posing more assumptions on the model under discussion. In this thesis, we focus on the alternating direction method of multipliers (ADMM), and show that the known error bound conditions for studying ADMM's linear convergence, can indeed be further weakened if the error bound is studied over the specific iterative sequence generated by ADMM. A so-called partial error bound condition, which is tailored for the specific ADMM's iterative scheme and weaker than known error bound conditions in the literature, is thus proposed to derive the linear convergence of ADMM. We further show that this partial error bound condition theoretically justifies the difference if the two primal variables are updated in different orders in implementing ADMM, which had been empirically observed in the literature yet no theory is known so far.

Let X be a Hausdorff locally convex space, U be a base for closed absolutely convex O-neighborhoods in X , K C X be nonempty. For each U є U , we denote by P[subscript u] the gauge of U. Then T : K ↦ K is said to be nonexpansive w.r.t. U if and only if for each U є U, P[subscript u](T(x) - T(y)) ≤ P[subscript u](x - y) for all x, y є K; T: K ↦ K is said to be strictly contractive w.r.t. U if and. only if for each U є U, there is a constant λ[subscript u] with 0 ≤ λ[subscript u] < 1 such that P[subscript u](T(x) - T(y)) ≤λ[subscript u]P[subscript u](x - y) for all x, y є K . The concept of nonexpansive (respectively strictly contractive) mappings is originally defined on a metric space. The above definitions are generalizations if the topology on X is induced by a translation invariant metric, and in particular if X is a normed space. An analogue of the Banach contraction mapping principle is proved and some examples together with an implicit function theorem are shown as applications. Moreover several fixed point theorems for various kinds of nonexpansive mappings are obtained. The convergence of nets of nonexpansive mappings and that of fixed points are also studied. Finally on sets with 'complete normal structure', a common fixed point theorem is obtained for an arbitrary family of 'commuting' nonexpansive mappings while on sets with 'normal structure', a common fixed point, theorem for an arbitrary family of (not necessarily commuting) 'weakly periodic' nonexpansive mappings is obtained. / Science, Faculty of / Mathematics, Department of / Graduate

Two topologies (Z,L) and (Z,L1) for the family of the non-degenerate convex curves in the projective plane are considered, where (Z,L) is the topology from the Lane's neighborhood system and (Z,L1) is the topology from the parabolic neighborhood system. It is shown that the definition of convexity in the affine plane can be extended to the projective plane so that the Blaschke selection theorem remains true for the projective convex sets. With the help of this theorem, the topological space (Z,L) is compactified by adding Lane's compactifying elements. Furthermore, it is shown that (Z,L) is metrizable but (Z,L1) is not metrizable. The Lane's topology (X,L), as a subspace of (Z,L) for the non-degenerate conics, is both metrizable and separable. A subspace (X,τ) of (Z,L1) is studied which is metrizable but not separable. / Science, Faculty of / Mathematics, Department of / Graduate

Let f be a real valued function with the domain dom(f) in some vector space X and let C be the collection of convex subsets of X. The following two questions are investigated; 1. Do there exist maximal convex restrictions g of f with dom(g) 2 C? 2. If f is convex with dom(f) 2 C, do there exist maximal convex extension g of f with dom(g) 2 C? We will show that the answer to both questions is positive under a certain condition on C. We also show that the extreme points of the epigraph of a real continuous strictly convex function are dense in the graph of such a function, and the set of such extreme points of an epigraph may be equal to the graph. Moreover we show that a set of extreme points of an epigraph may be equal to a graph of such a convex function under certain conditions. We also discuss conditions under which an epigraph of a real convex function on a Banach space X may, and may not, have extreme points, denting points and/or strongly exposed points. One of the interesting results in this discussion is that boundary points, extreme points, denting points and the graphs in an closed epigraph of a strictly convex function coincide. Moreover, we show that there is relationship between the extremal structure of an epigraph of a convex function and a point in a domain on which such a function attains its minimum.

No description available.

In this thesis, we investigate the results of gradient flows in Euclidean, metric, and Wasserstein space. Our primary objective is to provide a comprehensive and self-contained analysis of minimizing non-convex functions using the 2-Wasserstein gradient flow and some modified gradient flows. Firstly, we establish the equivalence between minimizing a continuous function in $\mathbb{R}^{d}$ and minimizing a relaxed functional $F[m]$ in the set of all probability measures in $\mathbb{R}^{d}$. Subsequently, we thoroughly examine minimizing the relaxed functional F[m] using the approach of gradient flows. To further enhance our understanding, we apply Swarm calculus and particle methods to numerically solve the gradient flow of $F[m]$.