Moments, Positive Polynomials and Their Applications by Jean-Bernard Lasserre

By Jean-Bernard Lasserre

Many very important difficulties in worldwide optimization, algebra, chance and information, utilized arithmetic, keep an eye on concept, monetary arithmetic, inverse difficulties, and so on. will be modeled as a specific example of the Generalized second challenge (GMP). This booklet introduces, in a unified handbook, a brand new common method to unravel the GMP while its facts are polynomials and uncomplicated semi-algebraic units. this technique combines semidefinite programming with fresh effects from actual algebraic geometry to supply a hierarchy of semidefinite relaxations converging to the specified optimum worth. utilized on acceptable cones, usual duality in convex optimization well expresses the duality among moments and confident polynomials. within the moment a part of this useful quantity, the technique is particularized and defined intimately for varied purposes, together with international optimization, chance, optimum context, mathematical finance, multivariate integration, etc., and examples are supplied for every specific program.

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Extra resources for Moments, Positive Polynomials and Their Applications (Imperial College Press Optimization Series)

Example text

The set P (f1 , . . 8) is called (by algebraic geometers) a preordering. It is also a convex cone of R[x]; see Appendix A. 11 (Stengle). Let k be a real closed field, and let F := (fi )i∈I1 , G := (gi )i∈I2 , H := (hi )i∈I3 ⊂ k[x] be finite families of polynomials. Let (a) P (F ) be the preordering generated by the family F , (b) M (G) be the set of all finite products of the gi ’s, i ∈ I2 (the empty product being the constant polynomial 1), and (c) I(H) be the ideal generated by H. Consider semi-algebraic set K = {x ∈ k n : fi (x) ≥ 0, hi (x) = 0, ∀ i ∈ I1 ; ∀ i ∈ I3 }.

S. polynomials. For any two real symmetric matrices A, B, recall that A, B stands for trace(AB). Finally, n for a multi-index α ∈ Nn , let |α| := i=1 αi . Consider the vector vd (x) = (xα )|α|≤d = (1, x1 , . . , xn , x21 , x1 x2 , . . , xn−1 xn , x2n , . . , xd1 , . . , xdn ) , of all the monomials xα of degree less than or equal to d, which has dimension s(d) := n+d d . Those monomials form the canonical basis of the vector space R[x]d of polynomials of degree at most d. 1. ) if and only if there exists a real symmetric and positive semidefinite matrix Q ∈ Rs(d)×s(d) such that g(x) = vd (x) Qvd (x), for all x ∈ Rn .

0 ≤ f 2r (x)→f (x)). 4(b), observe that in addition to the l1 -norm convergence f − f 2r 1 →0, the convergence is also uniform on compact sets. s. s. approximation f 2r . s. approximation f ≈ f 1r is not uniform on compact sets, and is really more appropriate for polynomials nonnegative on [−1, 1]n only (and indeed the approximation f ≈ f 1r is uniform on [−1, 1]n ). 4(a) the integer r 1 does not depend on the explicit choice of the polynomial f but only on: (a) and the dimension n, (b) the degree and the size of the coefficients of f .

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