1.Elements of convex analysis linear algebra and graft theory. 2.Subgradient and 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 constructions optimal on volume ellipsoids and semidefinite programming. 7.The role of ellipsoid method for complexity analysis of combinatorial graph problems. 8.Semidefinite programming bounds for extremal graph problems. 9.Global minimization of polynomial functions and 17, TH Hilbert problem