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Abstract Convexity and Global Optimization (Hardcover, 2000 ed.)
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Abstract Convexity and Global Optimization (Hardcover, 2000 ed.)
Series: Nonconvex Optimization and Its Applications, 44
Expected to ship within 12 - 17 working days
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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.
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