期刊
SIAM JOURNAL ON OPTIMIZATION
卷 15, 期 3, 页码 805-825出版社
SIAM PUBLICATIONS
DOI: 10.1137/S1052623403431779
关键词
nonconvex optimization; positive polynomial; sum of squares; moment problem; Positivstellensatz; semidefinite programming
A basic closed semialgebraic subset S of R-n is defined by simultaneous polynomial inequalities g(1) >= 0,..., g(m) >= 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the in. mum f* of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x*, we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in R-n converging to x*.
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