On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields
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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.
KeywordsMoment 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 ClassificationPrimary: 13J30 14P10 46L30 52A20 Secondary: 12D15 52A20 52A41 90C22 90C26
Both authors were supported by the DFG Grant SCHW 1723/1-1.
- 1.J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 36, Springer-Verlag, Berlin, 1998Google Scholar
- 3.K.R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs 20, American Mathematical Society, Providence, RI, 1986Google Scholar
- 7.R.B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24. Springer-Verlag, New York-Heidelberg, 1975Google Scholar
- 11.M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, 157–270, IMA Vol. Math. Appl., 149, Springer, New York, 2009, [updated version available at http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf]
- 14.A. Prestel, Bounds for representations of polynomials positive on compact semi-algebraic sets, Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 253–260, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002Google Scholar
- 15.A. Prestel, C.N. Delzell, Positive polynomials, From Hilbert’s 17th problem to real algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001Google Scholar
- 18.R.T. Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970Google Scholar