# Global Optimization of Polynomials over Real Algebraic Sets

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## Abstract

Let *f*, *g*_{1},..., *g*_{s} be polynomials in R[*X*_{1},..., *X*_{n}]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum *f** of *f* over an arbitrary given real algebraic set *V* = {*x* ∈ R^{n} | *g*_{1}(*x*) = 0,..., *g*_{s}(*x*) = 0}, where *V* is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing *f* over a basic closed semi-algebraic set *S* = {*x* ∈ R^{n} | *g*_{1}(*x*) ≥ 0,..., *g*_{s}(*x*) ≥ 0}.

## Keywords

Polynomial optimization real algebraic set generalized critical value## Preview

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## Notes

### Acknowledgement

We would like to thank Mohab Safey El Din for valuable comments and suggestions on the topic of the paper.

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