Skip to main content
SpringerLink
Log in

Nondifferentiable Optimization and Polynomial Problems

  • Book
  • © 1998

Overview

Authors:
  1. Naum Z. Shor

    1. V.M. Glushkov Institute of Cybernetics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    View author publications

    You can also search for this author in PubMed   Google Scholar

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

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (9 chapters)

  1. Front Matter

    Pages i-xvii
    Download chapter PDF

  2. Elements of Convex Analysis, Linear Algebra, and Graph Theory

    • Naum Z. Shor
    Pages 1-33

  3. Subgradient and ε-Subgradient Methods

    • Naum Z. Shor
    Pages 35-70

  4. Subgradient-Type Methods with Space Dilation

    • Naum Z. Shor
    Pages 71-112

  5. Elements of Information and Numerical Complexity of Polynomial Extremal Problems

    • Naum Z. Shor
    Pages 113-140

  6. Decomposition Methods Based on Nonsmooth Optimization

    • Naum Z. Shor
    Pages 141-167

  7. Algorithms for Constructing Optimal on Volume Ellipsoids and Semidefinite Programming

    • Naum Z. Shor
    Pages 169-225

  8. The Role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems

    • Naum Z. Shor
    Pages 227-263

  9. Semidefinite Programming Bounds for Extremal Graph Problems

    • Naum Z. Shor
    Pages 265-298

  10. Global Minimization of Polynomial Functions and 17-th Hilbert Problem

    • Naum Z. Shor
    Pages 299-333

  11. Back Matter

    Pages 335-396
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • V.M. Glushkov Institute of Cybernetics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    Naum Z. Shor

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation