By Jean-Bernard Lasserre

Many vital difficulties in international optimization, algebra, chance and information, utilized arithmetic, regulate thought, monetary arithmetic, inverse difficulties, and so on. will be modeled as a selected example of the Generalized second challenge (GMP). This booklet introduces, in a unified handbook, a brand new basic method to resolve the GMP whilst its info are polynomials and easy semi-algebraic units. this technique combines semidefinite programming with fresh effects from genuine algebraic geometry to supply a hierarchy of semidefinite relaxations converging to the specified optimum price. utilized on applicable cones, regular duality in convex optimization properly expresses the duality among moments and optimistic polynomials. within the moment a part of this priceless quantity, the technique is particularized and defined intimately for varied functions, together with worldwide optimization, chance, optimum context, mathematical finance, multivariate integration, etc., and examples are supplied for every specific program.

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Example text

4 hold and in addition, assume that for every k = 1, . . , p − 1, Ik+1 ∩ ∪kj=1 Ij ⊆ Is If f ∈ R[x] is strictly positive on K, then  p f = k=1  qk + j∈Jk for some s ≤ k. 33) for some sums of squares polynomials (qk , qjk ) ⊂ R[x(Ik )], k = 1, . . , p. 33) is called the running intersection property. 33). 14. 28 with an example. 4. Let x = (x1 , . . , x5 ) and x → g1 (x) = 1 − x21 − x22 x → g2 (x) = 1 − x42 x43 − x23 x → g3 (x) = 1 − x23 − x24 − x25 x → f (x) = 1 + x21 x22 + x22 x23 + x23 − x23 x24 − x23 x25 K = {x : g1 (x1 , x2 ), g2 (x2 , x3 ), g3 (x3 , x4 , x5 ) ≥ 0}.

Fix j ∈ Γ arbitrary so that hj ∈ Γm for all m sufficiently large. As hj is continuous, K is compact and µmi ⇒ µ, we have hj dµmi hj dµ = lim K i→∞ γj , K and so, as j was arbitrary, µ is feasible for the moment problem. 1. 18 in Hern´ andez-Lerma and Lasserre (2003). This proves that µ is a primal i solution of the moment problem (with Γ countable), and so ρm mom ↓ ρmom , m m which also implies ρmom ↓ ρmom because the sequence (ρmom ) is monotone. m m Finally, as ρm pop = ρmom , we also get ρpop ↓ ρmom .

10) is compact. If f ∈ R[x] is strictly positive on K then f ∈ P (g1 , . . s. 11) gj . 11) is exponential in the number of polynomials that define the set K. However, a major improvement is possible under a relatively weak assumption on the polynomials that define the compact set K. Associated with the finite family (gj ) ⊂ R[x], the subset of P (g1 , . . 12) Q(g) = Q(g1 , . . , gm ) := q0 + qj gj : (qj )m ⊂ Σ[x] j=0   j=1 is a convex cone called the quadratic module generated by the family (g j ).