# Nondifferentiable Optimization and Polynomial Problems

Springer Science & Business Media, Mar 31, 1998 - Mathematics - 396 pages
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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### 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 Copyright

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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.

### References to this book

 Solving Systems of Polynomial Equations, Issue 97Bernd SturmfelsNo preview available
 Deterministic Global OptimizationChristodoulos A. FloudasLimited preview - 2000
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