2009 Lagrange Prize Citation
Jean B. Lasserre,
"A sum of squares approximation of nonnegative polynomials",
SIAM Journal on Optimization 16 (2006), 751765.
(The paper was also reproduced in the SIGEST section of SIAM Review 49
(2007), 651669.)
Lasserre has been a pioneer in the field of global polynomial optimization since his 2001
paper "Global optimization with polynomials and the problem of moments", SIOPT 11,
796817. His approach relies on certificates of positivity for polynomials on compact sets
defined by polynomial inequalities, using representations as sums of squared polynomials  a
deep and powerful technique taken from real algebraic geometry. Using semidefinite
programming to recognize sums of squares, Lasserre constructs a hierarchy of semidefinite
relaxations converging to a solution of the (NP hard!) global optimization p roblem. The
convergence is often fast or even finite in practice, as illustrated in software packages
such as GloptiPoly, SOSTOOLS and SPARSEPOPT. The duality theory for the semidefinite
relaxations corres ponds exactly to the duality between polynomial optimization and
generalized moment problems, a relationship explored in great generality in Lasserre's paper
"A semidefinite programming approach to the gene ralized problem of moments",
Mathematical Programming 112 (2008), 6592.
The winning paper is a particularly striking exemplar of Lasserre's work on polynomial
optimization. He presents an elegant new proof of a classical foundation stone for the
theory: that any nonne gative polynomial can be approximated by sums of squares. Combining
convex duality theory and a momenttheoretic result, he constructs approximating
polynomials that are both simple and explicit. One conse quence is a satisfying
simplification of his original relaxation hierarchy. The paper is a beautiful blend of
modern optimization theory and deep classical mathematics, with striking computational
implicat ions.
