Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (topological) linear spaces. We follow basically the separation approach by Kasimbeyli (2010, SIAM J. Optim. 20) based on augmented dual cones and normlinear separation functions. Classical separation theorems for convex sets will be the key tool for proving our main nonlinear cone separation theorems. Also in the setting of a real reflexive Banach space, we are able to extend the cone separation result derived by Kasimbeyli

Properly optimal elements in vector optimization with variable ordering structures

In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions

Domination and Decomposition in Multiobjective Programming

During the last few decades, multiobjective programming has received much attention for both its numerous theoretical advances as well as its continued success in modeling and solving real-life decision problems in business and engineering. In extension of the traditionally adopted concept of Pareto optimality, this research investigates the more general notion of domination and establishes various theoretical results that lead to new optimization methods and support decision making. After a preparatory discussion of some preliminaries and a review of the relevant literature, several new findings are presented that characterize the nondominated set of a general vector optimization problem for which the underlying domination structure is defined in terms of different cones. Using concepts from linear algebra and convex analysis, a well known result relating nondominated points for polyhedral cones with Pareto solutions is generalized to nonpolyhedral cones that are induced by positively homogeneous functions, and to translated polyhedral cones that are used to describe a notion of approximate nondominance. Pareto-oriented scalarization methods are modified and several new solution approaches are proposed for these two classes of cones. In addition, necessary and sufficient conditions for nondominance with respect to a variable domination cone are developed, and some more specific results for the case of Bishop-Phelps cones are derived. Based on the above findings, a decomposition framework is proposed for the solution of multi-scenario and large-scale multiobjective programs and analyzed in terms of the efficiency relationships between the original and the decomposed subproblems. Using the concept of approximate nondominance, an interactive decision making procedure is formulated to coordinate tradeoffs between these subproblems and applied to selected problems from portfolio optimization and engineering design. Some introductory remarks and concluding comments together with ideas and research directions for possible future work complete this dissertation

On generalizations of the Arrow-Barankin-Blackwell Theorem in vector optimization.

Chan Ka Wo.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 114-118).Abstracts in English and Chinese.Introduction --- p.iiiConventions of This Thesis --- p.viPrerequisites --- p.xiiiChapter 1 --- Cones in Real Vector Spaces --- p.1Chapter 1.1 --- The Fundamentals of Cones --- p.2Chapter 1.2 --- Enlargements of a Cone --- p.22Chapter 1.3 --- Special Cones in Real Vector Spaces --- p.29Chapter 1.3.1 --- Positive Cones --- p.29Chapter 1.3.2 --- Bishop-Phelps Cones --- p.36Chapter 1.3.3 --- Quasi-Bishop-Phelps Cones --- p.42Chapter 1.3.4 --- Quasi*-Bishop-Phelps Cones --- p.45Chapter 1.3.5 --- Gallagher-Saleh D-cones --- p.47Chapter 2 --- Generalizations in Topological Vector Spaces --- p.52Chapter 2.1 --- Efficiency and Positive Proper Efficiency --- p.54Chapter 2.2 --- Type I Generalizations --- p.71Chapter 2.3 --- Type II Generalizations --- p.82Chapter 2.4 --- Type III Generalizations --- p.92Chapter 3 --- Generalizations in Dual Spaces --- p.97Chapter 3.1 --- Weak*-Support Points of a Set --- p.98Chapter 3.2 --- Generalizations in the Dual Space of a General Normed Space --- p.100Chapter 3.3 --- Generalizations in the Dual Space of a Banach Space --- p.104Epilogue: Glimpses Beyond --- p.112Bibliography --- p.11

Extended Second Welfare Theorem for Nonconvex Economies with Infinite Commodities and Public Goods

This paper is devoted to the study of nonconvex models of welfare economics with public goods and infinite-dimensional commodity spaces. Our main attention is paid to new extensions of the fundamental second welfare theorem to the models under consideration. Based on advanced tools of variational analysis and generalized differentiation, we establish appropriate approximate and exact versions of the extended second welfare theorem for Pareto, weak Pareto, and strong Pareto optimal allocations in both marginal price and decentralized price forms

Characterization of proper optimal elements with variable ordering structures

In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. As these vector optimization problems are not only of interest in applications but also mathematical challenging, in recent publications it was started to develop a comprehensive theory. In doing that also notions of proper efficiency where generalized to variable ordering structures. In this paper we study the relations between several types of proper optimality notions, among others based on local and global approximations of the considered sets. We give scalarization results based on new functionals defined by elements from the dual cones which allow characterizations also in the nonconvex case

Uniqueness for continuous superresolution by means of Choquet theory and geometric measure theory

The problem of superresolution is to recover an element of a vector space from data much smaller than the dimension of the space, using a prior assumption of sparsity. A famous example is compressive sensing, where the elements are images with a large finite resolution. On the other hand, we focus on a continuous form of superresolution. Given a measure $\mu$ on a continuous domain such as the two dimensional torus, can we recover $\mu$ from knowledge of only a finite number of its Fourier coefficients using a total variation minimization method? We will see that the answer depends on certain properties of $\mu$. Namely, a necessary condition is that $\mu$ be discrete.We use methods from geometric analysis to investigate the continuous superresolution problem. Tools from measure theory relate properties of the support of a measure, such as Hausdorff dimension, to properties of its Fourier transform. We also use measure theory to investigate the possibility of alternatives to total variation that may allow us to recover surface measures defined on space curves. There is a theorem of Choquet concerning representations of points in convex sets as sums of their extreme points. As it turns out, we can apply this to the superresolution problem because the extreme points of the set of measures with total variation $1$ are precisely the set of delta measures. We consider superresolution as a convex optimization problem, where the goal is to find representations of the initial data as sums of delta measures. Choquet theory provides tools to investigate the previously unresolved problem of uniqueness. We use this to give a novel sufficient condition for a measure to be uniquely superresolved, given data on a known finite set of frequencies

The horofunction boundary and Denjoy-Wolff type theorems

In this thesis we will study the horofunction boundary of metric spaces, in particular the Funk, reverse-Funk and Hilbert's metrics, and one of its applications, Denjoy-Wolff type theorems. In a Denjoy-Wolff type setting we will show that Beardon points are star points of the union of the ω-limit sets. We will also show that Beardon and Karlsson points are not unique in R2. In fact, we will show one can have a continuum of Karlsson points. We will establish two Denjoy-Wolff type theorem that confirm the Karlsson-Nussbaum conjecture for classes of non-expanding maps on Hilbert' metric spaces. For unital Euclidean Jordan algebras we will give a description of the intersection of closed horoballs with the boundary of the cone as the radius tends to minus infinity. We will expand on results by Walsh by establishing a general form for the Funk and reverse Funk horofunction boundaries of order-unit spaces. We will also give a full classification of the horofunctions of JH-algebras and the horofunctions and Busemann points of the spin factors for the Funk, reverse Funk and Hilbert metrics. Finally we will show that there exists a reverse-Funk non-Busemann horofunction for the cone of positive bounded self-adjoint operators on an infinite dimensional Hilbert space, the infinite dimensional spin factors and a space in which the pure states are weak* closed, answering a question raised by Walsh in [65]