Using results from the generalised moment problem and duality from convex optimization the author shows that given a nonnegative polynomial on simple sum of squares (sos) perturbations arbitrarily near to can be constructed. This is important since sos representations can be found relatively fast by semidefinite programming and are a surrogate for deciding the NP hard problem of nonnegativity of a polynomial. Furthermore, certain minimization problems for polynomials can be related to problems for positive semidefinite polynomials by observing that where is suitable.
For define and denote by P the problem and by the problem Here the space of probability measures on Let be the optimal value of The multiindices are considered suitably endowed with a natural linear order.
P3.2: Assume Then as Next the author introduces for a semidefinite programming problem that is a relaxation of Let its dual be In one has to minimize given certain semidefiniteness constraints on the reals
T3.3: Assume has a global minimum Then for all admissible as Let be an optimal solution of and embed it naturally into Banach space Then every pointwise accumulation point y of the sequence of optimal solutions admits a representation with a unique that itself is an optimal solution to As a consequence one can approximate the optimal value of as closely as desired by solving SDP relaxations for sufficiently large values of and can be fixed whenever whenever a global minimizer of can be shown to have -norm This method is simpler than the procedure proposed in J. B. Lasserre [SIAM J. Optim. 11, No. 3, 796–817 (2001; Zbl 1010.90061)].
Another important result is T4.1: Assume For every there is some such that is sum of squares. In particular as Define the semialgebraic set
C4.3: If the are concave (hence convex) and satisfy Slater’s condition, is convex, and compact, then with some sos and nonnegative In particular this shows a simplified Putinar representation for see M. Putinar [Indiana Univ. Math. J. 42, No. 3, 969–984 (1993; Zbl 0796.12002)] or A. Prestel and C. N. Delzell [Positive polynomials. From Hilbert’s 17th problem to real algebra. Springer Monographs in Mathematics. Berlin: Springer (2001; Zbl 0987.13016)].
In a footnote the reader is informed that this well written and interestingly motivated paper also appeared in SIAM J. Optim. 16, 751–765 (2006), see above.