# Nondifferentiable Optimization and Polynomial Problems Book

Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 24)

1. Front Matter
Pages i-xvii
2. Naum Z. Shor
Pages 1-33
3. Naum Z. Shor
Pages 35-70
4. Naum Z. Shor
Pages 71-112
5. Naum Z. Shor
Pages 113-140
6. Naum Z. Shor
Pages 141-167
7. Naum Z. Shor
Pages 169-225
8. Naum Z. Shor
Pages 227-263
9. Naum Z. Shor
Pages 265-298
10. Naum Z. Shor
Pages 299-333
11. Back Matter
Pages 335-396

### Introduction

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.

### Keywords

Mathematica algebra algorithms calculus complexity graph theory optimization programming

#### Authors and affiliations

1. 1.V.M. Glushkov Institute of CyberneticsUkrainian National Academy of SciencesKievUkraine

### Bibliographic information

• Book Title Nondifferentiable Optimization and Polynomial Problems
• Authors N.Z. Shor
• Series Title Nonconvex Optimization and Its Applications
• DOI https://doi.org/10.1007/978-1-4757-6015-6
• Copyright Information Springer-Verlag US 1998
• Publisher Name Springer, Boston, MA
• eBook Packages
• Hardcover ISBN 978-0-7923-4997-6
• Softcover ISBN 978-1-4419-4792-5
• eBook ISBN 978-1-4757-6015-6
• Series ISSN 1571-568X
• Edition Number 1
• Number of Pages XVII, 396
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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