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...me index k. For instance this is the case for boolean (or 0-1) optimization problems, and more generally, bounded discrete optimization problems. For more details the interested reader is referred to =-=[13, 14, 15, 17]-=-. 2.2 Extraction algorithm Assume that the LMI relaxation Qk in (3) has been solved, producing a vector y ∗ . Assume further that the rank condition (4) is satisfied. Then the main steps of the extrac...

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...be found by SOS relaxation f ∗ sos = max γ γ∈R φi(x)∈R[x] f ∗ = min f(x) (5.1) s.t. gi(x) = 0, i = 1, . . . , m. (5.2) subject to f(x) − γ − ∑ gi(x)φi(x) ≽sos 0. (5.3) There are several recent papers =-=[13, 14, 21]-=- on solving this kind of constrained problem using SOS or moment matrix techniques. The convergence of their methods is based on the assumption that the real variety V R (g1, . . . , gm) is compact or...

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...)). We also show how theta bodies can be handled computationally, by directly computing SOS decompositions in the quotient ring R[x]/I [18], or dually, by using the combinatorial moment matrices from =-=[8]-=-.THETA BODIES FOR POLYNOMIAL IDEALS 5 2.1. Lasserre’s hierarchy and theta bodies. Definition 2.1. Let G = {g1,...,gm} be a set of polynomials and I an ideal in R[x]. The quadratic module of G (mod I)...

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...eal radical of a zero-dimensional ideal. Moreover there are links between moment matrices and the Hermite quadratic forms used in [31] for computing the number of real roots, that were pointed out in =-=[26]-=-. Our approach with its specificity is best illustrated on the task of computing the real radical ideal I(VR(I)) that we now briefly describe. 3sGiven a sequence y = (yα)α∈Nn ∈ RNn, consider the momen...

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...sing Putinar’s Positivstellensatz (cf. Theorem 2.1). When fk = fmin for some k, we say Lasserre’s hierarchy has finite convergence. When h(x) = 0 defines a finite set in the complex space Cn, Laurent =-=[13]-=- proved that Lasserre’s hierarchy has finite convergence. Indeed, when h(x) = 0 defines a finite set in Rn, the sequence {fk} also has finite convergence to fmin, as shown in [22]. There exist example...

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...ds (see [10] and references therein). We explain the details of the algorithm, which we call NulLA, in Section 2. Our method can be seen as a general-field variation of work by Lasserre [11], Laurent =-=[13]-=- and Parrilo [16] and many others, who studied the problem of minimizing a general polynomial function f(x) over a real algebraic variety with finitely many points. Laurent proved that when the variet...

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... its gradient ideal or an ideal arising from the KKT system. There is some related work in SOS representations of positive polynomials modulo certain ideals, such as Hanzon and Jibetean [10], Laurent =-=[16]-=-, Parrilo [26], Jibetean and Laurent [14]. Nie, Demmel and Sturmfels [19] proposed using SOS representations of positive polynomials modulo their gradient ideals, i.e., the ideals generated by all the...

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... allow for polynomial equalities in the description of K, either by replacing them with pairs of inequalities, extending the SDP formulations as in (2), or by factoring out by the ideal they generate =-=[37]-=-.) Assuming K is explicitly compact, Lasserre [34] showed that the SOS SDP’s value tends to the optimal value as the degree increases. If furthermore K has a nonempty interior then there is no duality...