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This book presents a smooth and unified transitional framework from
generalised fractional programming, with a finite number of
variables and a finite number of constraints, to semi-infinite
fractional programming, where a number of variables are finite but
with infinite constraints. It focuses on empowering graduate
students, faculty and other research enthusiasts to pursue more
accelerated research advances with significant interdisciplinary
applications without borders. In terms of developing general
frameworks for theoretical foundations and real-world applications,
it discusses a number of new classes of generalised second-order
invex functions and second-order univex functions, new sets of
second-order necessary optimality conditions, second-order
sufficient optimality conditions, and second-order duality models
for establishing numerous duality theorems for discrete minmax (or
maxmin) semi-infinite fractional programming problems. In the
current interdisciplinary supercomputer-oriented research
environment, semi-infinite fractional programming is among the most
rapidly expanding research areas in terms of its multi-facet
applications empowerment for real-world problems, which may stem
from many control problems in robotics, outer approximation in
geometry, and portfolio problems in economics, that can be
transformed into semi-infinite problems as well as handled by
transforming them into semi-infinite fractional programming
problems. As a matter of fact, in mathematical optimisation
programs, a fractional programming (or program) is a generalisation
to linear fractional programming. These problems lay the
theoretical foundation that enables us to fully investigate the
second-order optimality and duality aspects of our principal
fractional programming problem as well as its semi-infinite
counterpart.
This book presents a smooth and unified transitional framework from
generalised fractional programming, with a finite number of
variables and a finite number of constraints, to semi-infinite
fractional programming, where a number of variables are finite but
with infinite constraints. It focuses on empowering graduate
students, faculty and other research enthusiasts to pursue more
accelerated research advances with significant interdisciplinary
applications without borders. In terms of developing general
frameworks for theoretical foundations and real-world applications,
it discusses a number of new classes of generalised second-order
invex functions and second-order univex functions, new sets of
second-order necessary optimality conditions, second-order
sufficient optimality conditions, and second-order duality models
for establishing numerous duality theorems for discrete minmax (or
maxmin) semi-infinite fractional programming problems. In the
current interdisciplinary supercomputer-oriented research
environment, semi-infinite fractional programming is among the most
rapidly expanding research areas in terms of its multi-facet
applications empowerment for real-world problems, which may stem
from many control problems in robotics, outer approximation in
geometry, and portfolio problems in economics, that can be
transformed into semi-infinite problems as well as handled by
transforming them into semi-infinite fractional programming
problems. As a matter of fact, in mathematical optimisation
programs, a fractional programming (or program) is a generalisation
to linear fractional programming. These problems lay the
theoretical foundation that enables us to fully investigate the
second-order optimality and duality aspects of our principal
fractional programming problem as well as its semi-infinite
counterpart.
This monograph presents smooth, unified, and generalized fractional
programming problems, particularly advanced duality models for
discrete min-max fractional programming. In the current,
interdisciplinary, computer-oriented research environment, these
programs are among the most rapidly expanding research areas in
terms of their multi-faceted applications including problems
ranging from robotics to money market portfolio management. The
other more significant aspect of this monograph is in its
consideration of minimax fractional integral type problems using
higher order sonvexity and sounivexity notions. This is significant
for the development of different types of duality models in terms
of weak, strong, and strictly converse duality theorems, which can
be handled by transforming them into generalized fractional
programming problems. Fractional integral type programming is one
of the fastest expanding areas of optimization, which feature
several types of real-world problems. It can be applied to
different branches of engineering (including multi-time
multi-objective mechanical engineering problems) as well as to
economics, to minimize a ratio of functions between given periods
of time. Furthermore, it can be utilized as a resource in order to
measure the efficiency or productivity of a system. In these types
of problems, the objective function is given as a ratio of
functions. For example, we consider a problem that deals with
minimizing a maximum of several time-dependent ratios involving
integral expressions.
This monograph is aimed at presenting "Next Generation Newton-Type
Methods", which outperform most of the iterative methods and offer
great research potential for new advanced research on iterative
computational methods. This monograph provides readers with a
unique presentation on the subject that can be used for
interdisciplinary research for the world scientific community at
large. The methods presented therein are of great importance and
significance since these can be extended, generalised and applied
to solving equations defined not only on the real line but on
abstract spaces as well. This monograph is a must-read for
undergraduate students, graduate students, professors, researchers,
and research scientists at all universities and colleges.
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