# On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

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## Abstract

Consider a finite system of non-strict polynomial inequalities with solution set \(S\subseteq \mathbb R^n\). Its Lasserre relaxation of degree *d* is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most *d*. It defines a spectrahedron that projects down to a convex semialgebraic set containing *S*. In the best case, the projection equals the convex hull of *S*. We show that this is very often the case for sufficiently high *d* if *S* is compact and “bulges outwards” on the boundary of its convex hull. Now let additionally a polynomial objective function *f* be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree *d* is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if *S* is compact and *d* exceeds some bound that depends on the description of *S* and certain characteristics of *f* like the mutual distance of its global minimizers on *S*.

## Keywords

Moment relaxation Lasserre relaxation Pure state Basic closed semialgebraic set Positive polynomial Sum of squares Polynomial optimization Semidefinite programming Linear matrix inequality Spectrahedron Semidefinitely representable set## Mathematics Subject Classification

Primary: 13J30 14P10 46L30 52A20 Secondary: 12D15 52A20 52A41 90C22 90C26## Notes

### Acknowledgements

Both authors were supported by the DFG Grant SCHW 1723/1-1.

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