Nondifferentiable Optimization and Polynomial Problems (Nonconvex Optimization and Its Applications, 24)

Shor, N.Z.

ISBN 10: 0792349970 / ISBN 13: 9780792349976

Published by Springer, 1998

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Synopsis:

Polynomial extremal problems (PEP) constitute one of the most important subclasses of nonlinear programming models. Their distinctive feature is that an objective function and constraints can be expressed by polynomial functions in one or several variables. Let :e = {:e 1, ... , :en} be the vector in n-dimensional real linear space Rn; n PO(:e), PI (:e), ... , Pm (:e) are polynomial functions in R with real coefficients. In general, a PEP can be formulated in the following form: (0.1) find r = inf Po(:e) subject to constraints (0.2) Pi (:e) =0, i=l, ... ,m (a constraint in the form of inequality can be written in the form of equality by introducing a new variable: for example, P( x) ~ 0 is equivalent to P(:e) + y2 = 0). Boolean and mixed polynomial problems can be written in usual form by adding for each boolean variable z the equality: Z2 - Z = O. Let a = {al, ... ,a } be integer vector with nonnegative entries {a;}f=l. n Denote by R[a](:e) monomial in n variables of the form: n R[a](:e) = IT :ef'; ;=1 d(a) = 2:7=1 ai is the total degree of monomial R[a]. Each polynomial in n variables can be written as sum of monomials with nonzero coefficients: P(:e) = L caR[a](:e), aEA{P) IX x Nondifferentiable optimization and polynomial problems where A(P) is the set of monomials contained in polynomial P.

Synopsis: The book is devoted to investigation of polynomial optimization problems, including Boolean problems which are the most important part of mathematical programming. It is shown that the methods of nondifferentiable optimization can be used for finding solutions of many classes of polynomial problems and for obtaining good dual estimates for optimal objective value in these problems.

Bibliographic Details

Title: Nondifferentiable Optimization and ...
Publisher: Springer
Publication Date: 1998
Binding: Hardcover
Condition: New