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Abstract Convexity and Global Optimization

2013-03-14 Mathematics
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Release : 2013-03-14
Genre : Mathematics
Kind : eBook
Book Rating : 007/5 ( reviews)

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Book Synopsis Abstract Convexity and Global Optimization by : Alexander M. Rubinov

Download or read book Abstract Convexity and Global Optimization written by Alexander M. Rubinov. This book was released on 2013-03-14. Available in PDF, EPUB and Kindle. Book excerpt: Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.

Condition for Global Minimum Through Abstract Convexity

2009-11
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Release : 2009-11
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Book Rating : 059/5 ( reviews)

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Book Synopsis Condition for Global Minimum Through Abstract Convexity by : Evgeny Sharikov

Download or read book Condition for Global Minimum Through Abstract Convexity written by Evgeny Sharikov. This book was released on 2009-11. Available in PDF, EPUB and Kindle. Book excerpt: Subdifferential calculus and separation theorems play a crucial role for applications of classical convex analysis to global optimization. More precisely, they allow the formulation of conditions (necessary or sufficient) for the global minimum of some convex optimization problems. The theory of abstract convexity generalizes ideas of convex analysis by using the notion of global supports and the global definition of subdifferential. In order to apply this theory to optimization, we need to extend subdifferential calculus and separation properties into the area of abstract convexity. This is the main objective of the present thesis. The work should be useful to professionals in generalized convexity and global optimization.

Advances in Convex Analysis and Global Optimization

2013-12-01 Mathematics
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Release : 2013-12-01
Genre : Mathematics
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Book Rating : 79X/5 ( reviews)

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Book Synopsis Advances in Convex Analysis and Global Optimization by : Nicolas Hadjisavvas

Download or read book Advances in Convex Analysis and Global Optimization written by Nicolas Hadjisavvas. This book was released on 2013-12-01. Available in PDF, EPUB and Kindle. Book excerpt: There has been much recent progress in global optimization algo rithms for nonconvex continuous and discrete problems from both a theoretical and a practical perspective. Convex analysis plays a fun damental role in the analysis and development of global optimization algorithms. This is due essentially to the fact that virtually all noncon vex optimization problems can be described using differences of convex functions and differences of convex sets. A conference on Convex Analysis and Global Optimization was held during June 5 -9, 2000 at Pythagorion, Samos, Greece. The conference was honoring the memory of C. Caratheodory (1873-1950) and was en dorsed by the Mathematical Programming Society (MPS) and by the Society for Industrial and Applied Mathematics (SIAM) Activity Group in Optimization. The conference was sponsored by the European Union (through the EPEAEK program), the Department of Mathematics of the Aegean University and the Center for Applied Optimization of the University of Florida, by the General Secretariat of Research and Tech nology of Greece, by the Ministry of Education of Greece, and several local Greek government agencies and companies. This volume contains a selective collection of refereed papers based on invited and contribut ing talks presented at this conference. The two themes of convexity and global optimization pervade this book. The conference provided a forum for researchers working on different aspects of convexity and global opti mization to present their recent discoveries, and to interact with people working on complementary aspects of mathematical programming.

Convex Analysis and Global Optimization

1998-01-31 Business & Economics
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Release : 1998-01-31
Genre : Business & Economics
Kind : eBook
Book Rating : 184/5 ( reviews)

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Book Synopsis Convex Analysis and Global Optimization by : Hoang Tuy

Download or read book Convex Analysis and Global Optimization written by Hoang Tuy. This book was released on 1998-01-31. Available in PDF, EPUB and Kindle. Book excerpt: Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development. Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.

Duality for Nonconvex Approximation and Optimization

2007-03-12 Mathematics
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Release : 2007-03-12
Genre : Mathematics
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Book Rating : 951/5 ( reviews)

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Book Synopsis Duality for Nonconvex Approximation and Optimization by : Ivan Singer

Download or read book Duality for Nonconvex Approximation and Optimization written by Ivan Singer. This book was released on 2007-03-12. Available in PDF, EPUB and Kindle. Book excerpt: The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.

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