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The focus of this book is on bilevel programming which combines
elements of hierarchical optimization and game theory. The basic
model addresses the problem where two decision-makers, each with
their individual objectives, act and react in a noncooperative
manner. The actions of one affect the choices and payoffs available
to the other but neither player can completely dominate the other
in the traditional sense. Over the last 20 years there has been a
steady growth in research related to theory and solution
methodologies for bilevel programming. This interest stems from the
inherent complexity and consequent challenge of the underlying
mathematics, as well as the applicability of the bilevel model to
many real-world situations. The primary aim of this book is to
provide a historical perspective on algorithmic development and to
highlight those implementations that have proved to be the most
efficient in their class. A corollary aim is to provide a sampling
of applications in order to demonstrate the versatility of the
basic model and the limitations of current technology. What is
unique about this book is its comprehensive and integrated
treatment of theory, algorithms and implementation issues. It is
the first text that offers researchers and practitioners an
elementary understanding of how to solve bilevel programs and a
perspective on what success has been achieved in the field.
Audience: Includes management scientists, operations researchers,
industrial engineers, mathematicians and economists.
The analysis and design of engineering and industrial systems has
come to rely heavily on the use of optimization techniques. The
theory developed over the last 40 years, coupled with an increasing
number of powerful computational procedures, has made it possible
to routinely solve problems arising in such diverse fields as
aircraft design, material flow, curve fitting, capital expansion,
and oil refining just to name a few. Mathematical programming plays
a central role in each of these areas and can be considered the
primary tool for systems optimization. Limits have been placed on
the types of problems that can be solved, though, by the difficulty
of handling functions that are not everywhere differentiable. To
deal with real applications, it is often necessary to be able to
optimize functions that while continuous are not differentiable in
the classical sense. As the title of the book indicates, our chief
concern is with (i) nondifferentiable mathematical programs, and
(ii) two-level optimization problems. In the first half of the
book, we study basic theory for general smooth and nonsmooth
functions of many variables. After providing some background, we
extend traditional (differentiable) nonlinear programming to the
nondifferentiable case. The term used for the resultant problem is
nondifferentiable mathematical programming. The major focus is on
the derivation of optimality conditions for general
nondifferentiable nonlinear programs. We introduce the concept of
the generalized gradient and derive Kuhn-Tucker-type optimality
conditions for the corresponding formulations.
The analysis and design of engineering and industrial systems has
come to rely heavily on the use of optimization techniques. The
theory developed over the last 40 years, coupled with an increasing
number of powerful computational procedures, has made it possible
to routinely solve problems arising in such diverse fields as
aircraft design, material flow, curve fitting, capital expansion,
and oil refining just to name a few. Mathematical programming plays
a central role in each of these areas and can be considered the
primary tool for systems optimization. Limits have been placed on
the types of problems that can be solved, though, by the difficulty
of handling functions that are not everywhere differentiable. To
deal with real applications, it is often necessary to be able to
optimize functions that while continuous are not differentiable in
the classical sense. As the title of the book indicates, our chief
concern is with (i) nondifferentiable mathematical programs, and
(ii) two-level optimization problems. In the first half of the
book, we study basic theory for general smooth and nonsmooth
functions of many variables. After providing some background, we
extend traditional (differentiable) nonlinear programming to the
nondifferentiable case. The term used for the resultant problem is
nondifferentiable mathematical programming. The major focus is on
the derivation of optimality conditions for general
nondifferentiable nonlinear programs. We introduce the concept of
the generalized gradient and derive Kuhn-Tucker-type optimality
conditions for the corresponding formulations.
The use of optimization techniques has become integral to the
design and analysis of most industrial and socio-economic systems.
Great strides have been made recently in the solution of
large-scale problems arising in such areas as production planning,
airline scheduling, government regulation, and engineering design,
to name a few. Analysts have found, however, that standard
mathematical programming models are often inadequate in these
situations because more than a single objective function and a
single decision maker are involved. Multiple objective programming
deals with the extension of optimization techniques to account for
several objective functions, while game theory deals with the
inter-personal dynamics surrounding conflict. Bilevel programming,
the focus of this book, is in a narrow sense the combination of the
two. It addresses the problern in which two decision makers, each
with their individual objectives, act and react in a
noncooperative, sequential manner. The actions of one affect the
choices and payoffs available to the other but neither player can
completely dominate the other in the traditional sense.
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