# Global Minimization of Polynomial Functions and 17-th Hilbert Problem

Chapter

## Abstract

Consider one of the simplest on formulation problem of nonconvex polynomial programming: find infinum
where

*ρ** (*A*) of$$K\left( x \right) = \left( {Ax,x} \right)\,on\,the\,cone:x \geqslant 0,x = \left( {{x_1} \ldots ,{x_n}} \right) \in {E^n},$$

(9.1)

*A*is an arbitrary symmetric matrix*n*×*n*,*A*= {*a*_{ ij }}_{ i }^{ n }_{ ,j =1}.*ρ** (*A*) may accept only two values: 0 or − ∞. If*ρ** (*A*) = 0, then*x** = 0 is the optimal point of the problem (9.1). It is known that class of the problems: “*ρ** (*A*)= 0? ” for arbitrary integer (rational) symmetric matrices*A*of arbitrary size*n*×*n*is NP-complete (see [GJ 79]. Note that for convex*K*(*x*) the answer is trivial:*ρ** (*A*) = 0.## Keywords

Global Minimum Polynomial Function Total Degree Quadratic Problem Global Minimum Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 1998