Ebook: Nondifferentiable Optimization and Polynomial Problems

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.

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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.

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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.
Content:
Front Matter....Pages i-xvii
Elements of Convex Analysis, Linear Algebra, and Graph Theory....Pages 1-33
Subgradient and ?-Subgradient Methods....Pages 35-70
Subgradient-Type Methods with Space Dilation....Pages 71-112
Elements of Information and Numerical Complexity of Polynomial Extremal Problems....Pages 113-140
Decomposition Methods Based on Nonsmooth Optimization....Pages 141-167
Algorithms for Constructing Optimal on Volume Ellipsoids and Semidefinite Programming....Pages 169-225
The Role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems....Pages 227-263
Semidefinite Programming Bounds for Extremal Graph Problems....Pages 265-298
Global Minimization of Polynomial Functions and 17-th Hilbert Problem....Pages 299-333
Back Matter....Pages 335-396

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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.
Content:
Front Matter....Pages i-xvii
Elements of Convex Analysis, Linear Algebra, and Graph Theory....Pages 1-33
Subgradient and ?-Subgradient Methods....Pages 35-70
Subgradient-Type Methods with Space Dilation....Pages 71-112
Elements of Information and Numerical Complexity of Polynomial Extremal Problems....Pages 113-140
Decomposition Methods Based on Nonsmooth Optimization....Pages 141-167
Algorithms for Constructing Optimal on Volume Ellipsoids and Semidefinite Programming....Pages 169-225
The Role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems....Pages 227-263
Semidefinite Programming Bounds for Extremal Graph Problems....Pages 265-298
Global Minimization of Polynomial Functions and 17-th Hilbert Problem....Pages 299-333
Back Matter....Pages 335-396
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