## Nondifferentiable Optimization and Polynomial ProblemsPolynomial 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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### Contents

II | 1 |

III | 8 |

IV | 19 |

V | 22 |

VI | 25 |

VII | 30 |

VIII | 35 |

IX | 37 |

XXV | 169 |

XXVI | 178 |

XXVII | 187 |

XXVIII | 193 |

XXIX | 196 |

XXX | 200 |

XXXI | 210 |

XXXII | 220 |

X | 51 |

XI | 57 |

XII | 68 |

XIII | 71 |

XIV | 74 |

XV | 88 |

XVI | 100 |

XVII | 113 |

XVIII | 118 |

XIX | 121 |

XX | 133 |

XXI | 141 |

XXII | 145 |

XXIII | 147 |

XXIV | 152 |

XXXIII | 227 |

XXXIV | 232 |

XXXV | 239 |

XXXVI | 254 |

XXXVII | 257 |

XXXVIII | 265 |

XXXIX | 279 |

XL | 293 |

XLI | 299 |

XLII | 308 |

XLIII | 320 |

XLIV | 330 |

XLV | 335 |

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approximation arbitrary aw(G calculate called clique coefficients combinatorial complexity compute Consider constraints convergence convex programming convex programming problem convex set corresponding decomposition denote descent differentiable dimensional direction dual bound edges eigenvalues eigenvectors ellipsoid method equal estimate exists formula function f given global minimum gradient graph G hyperplane inscribed ellipsoids integer interior point methods iteration Lagrange function Lagrange multipliers linear programming LP problem Math Mathematical matrix max-cut max-cut problem maximal maximum minimization minimum point n x n nodes nonempty nonlinear nonnegative nonsmooth objective function obtain optimal point optimal solution optimization problems orthogonal parameter penalty function perfect graphs polytope positive definite positive semidefinite Proof quadratic function quadratic-type r-algorithm satisfies semidefinite programming sequence Shor solving space dilation squares of polynomials step subgradient method subgradient-type subset sum of squares symmetric matrices system of linear Theorem upper bound variables vector vertex vertices weight

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Page 360 - JL GOFFIN, Convergence results in a class of variable metric subgradient methods, in Nonlinear Programming 4, Editors: OL Mangasarian, RR Meyer and SM Robinson, pp.