458 research outputs found
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
On the Newton method for solving fuzzy optimization problems
In this article we consider optimization problems where the objectives are fuzzy functions (fuzzy-valued functions). For this class of fuzzy optimization problems we discuss the Newton method to find a non-dominated solution. For this purpose, we use the generalized Hukuhara differentiability notion, which is the most general concept of existing differentiability for fuzzy functions. This work improves and correct the Newton Method
previously proposed in the literature.Fondo Nacional de Desarrollo Científico y Tecnológico (Chile)Ministerio de Ciencia y TecnologíaConselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil)Centro de Pesquisa em Matemática Aplicada à Indústria (Fundação de Amparo à Pesquisa do Estado de São Paulo
A Novel Euler's Elastica based Segmentation Approach for Noisy Images via using the Progressive Hedging Algorithm
Euler's Elastica based unsupervised segmentation models have strong
capability of completing the missing boundaries for existing objects in a clean
image, but they are not working well for noisy images. This paper aims to
establish a Euler's Elastica based approach that properly deals with random
noises to improve the segmentation performance for noisy images. We solve the
corresponding optimization problem via using the progressive hedging algorithm
(PHA) with a step length suggested by the alternating direction method of
multipliers (ADMM). Technically, all the simplified convex versions of the
subproblems derived from the major framework of PHA can be obtained by using
the curvature weighted approach and the convex relaxation method. Then an
alternating optimization strategy is applied with the merits of using some
powerful accelerating techniques including the fast Fourier transform (FFT) and
generalized soft threshold formulas. Extensive experiments have been conducted
on both synthetic and real images, which validated some significant gains of
the proposed segmentation models and demonstrated the advantages of the
developed algorithm
Integral Transformation, Operational Calculus and Their Applications
The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects
Recent developments in vector optimization
Lee, Hon Leung."August 2011."Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (leaves 98-101) and index.Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.6Chapter 2 --- Preliminaries --- p.11Chapter 2.1 --- Functional analysis --- p.11Chapter 2.2 --- Convex analysis --- p.14Chapter 2.3 --- Relative interiors --- p.19Chapter 2.4 --- Multifunctions --- p.21Chapter 2.5 --- Variational analysis --- p.22Chapter 3 --- A unified notion of optimality --- p.29Chapter 3.1 --- Basic notions of minimality --- p.29Chapter 3.2 --- A unified notion --- p.32Chapter 4 --- Separation theorems --- p.38Chapter 4.1 --- Zheng and Ng fuzzy separation theorem --- p.38Chapter 4.2 --- Extremal principles and other consequences --- p.43Chapter 5 --- Necessary conditions for the unified notion of optimality --- p.49Chapter 5.1 --- Local asymptotic closedness --- p.49Chapter 5.2 --- First order necessary conditions --- p.56Chapter 5.2.1 --- Introductory remark --- p.56Chapter 5.2.2 --- Without operator constraints --- p.59Chapter 5.2.3 --- With operator constraints --- p.66Chapter 5.3 --- Comparisons with known necessary conditions --- p.74Chapter 5.3.1 --- Finite-dimensional setting --- p.74Chapter 5.3.2 --- Zheng and Ng's work --- p.76Chapter 5.3.3 --- Dutta and Tammer's work --- p.80Chapter 5.3.4 --- Bao and Mordukhovich's previous work --- p.81Chapter 6 --- A weak notion: approximate efficiency --- p.84Chapter 6.1 --- Approximate minimality --- p.85Chapter 6.2 --- A scalarization result --- p.86Chapter 6.3 --- Variational approach --- p.94Bibliography --- p.98Index --- p.10
Different optimum notions for fuzzy functions and optimality conditions associated
Fuzzy numbers have been applied on decision and optimization problems
in uncertain or imprecise environments. In these problems, the necessity to define
optimal notions for decision-maker’s preferences as well as to prove necessary and
sufficient optimality conditions for these optima are essential steps in the resolution
process of the problem. The theoretical developments are illustrated and motivated
with several numerical examples.The research in this paper has been supported by MTM2015-66185 (MINECO/FEDER, UE) and
Fondecyt-Chile, Project 1151154
Generalized equilibrium in an economy without the survival assumption
It is well known that an equilibrium in the Arrow-Debreu model may
fail to exist if a very restrictive condition called the survival
assumption is not satisfied. We study two approaches that allow
for the relaxation of this condition. Danilov and Sotskov (1990),
and Florig (2001) developed a concept of a generalized equilibrium
based on a notion of hierarchic prices. Marakulin (1990) proposed
a concept of an equilibrium with non-standard prices. In this
paper, we establish the equivalence between non-standard and
hierarchic equilibria. Furthermore, we show that for any specified
system of dividends the set of such equilibria is generically
finite. We also provide a generic characterization of hierarchic
equilibria and give an easy proof of the core equivalence result
Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications
Tesis por compendioMathematical models have extensively been used in problems related to
engineering, computer sciences, economics, social, natural and medical sciences
etc. It has become very common to use mathematical tools to solve,
study the behavior and different aspects of a system and its different subsystems.
Because of various uncertainties arising in real world situations,
methods of classical mathematics may not be successfully applied to solve
them. Thus, new mathematical theories such as probability theory and fuzzy
set theory have been introduced by mathematicians and computer scientists
to handle the problems associated with the uncertainties of a model. But
there are certain deficiencies pertaining to the parametrization in fuzzy set
theory. Soft set theory aims to provide enough tools in the form of parameters
to deal with the uncertainty in a data and to represent it in a useful
way. The distinguishing attribute of soft set theory is that unlike probability
theory and fuzzy set theory, it does not uphold a precise quantity. This
attribute has facilitated applications in decision making, demand analysis,
forecasting, information sciences, mathematics and other disciplines.
In this thesis we will discuss several algebraic and topological properties
of soft sets and fuzzy soft sets. Since soft sets can be considered as setvalued
maps, the study of fixed point theory for multivalued maps on soft
topological spaces and on other related structures will be also explored.
The contributions of the study carried out in this thesis can be summarized
as follows:
i) Revisit of basic operations in soft set theory and proving some new
results based on these modifications which would certainly set a new
dimension to explore this theory further and would help to extend its
limits further in different directions. Our findings can be applied to
develop and modify the existing literature on soft topological spaces
ii) Defining some new classes of mappings and then proving the existence
and uniqueness of such mappings which can be viewed as a positive
contribution towards an advancement of metric fixed point theory
iii) Initiative of soft fixed point theory in framework of soft metric spaces
and proving the results lying at the intersection of soft set theory and
fixed point theory which would help in establishing a bridge between
these two flourishing areas of research.
iv) This study is also a starting point for the future research in the area of
fuzzy soft fixed point theory.Abbas, M. (2014). Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48470TESISCompendi
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