Preface. 1. Elements of Convex Analysis, Linear Algebra, and Graph Theory. 2. Subgradient and epsilon-Subgradient Methods. 3. Subgradient-Type Methods with Space Dilation. 4. Elements of Information and Numerical Complexity of Polynomial Extremal Problems. 5. Decomposition Methods Based on Nonsmooth Optimization. 6. Algorithms for Constructing Optimal on Volume Ellipsoids and Semidefinite Programming. 7. The Role of Ellipsoid Method for Complexity Analysis of Combinatorial Problems. 8. Semidefinite Programming Bounds for Extremal Graph Problems. 9. Global Minimization of Polynomial Functions and 17-th Hilbert Problem. References.

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.