In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is
full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.
We consider the problem of the semidefinite representation of a class of non-compact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modif
ied hierarchies of nested theta bodies and Lasserres relaxations converge to the closure of the convex hull of $S$. Moreover, if the PP-BDR property is satisfied, our theta body and Lasserres relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies convergent uniformly to the closure of the convex hull of $S$ restricted to every fixed ball centered at the origin. We illustrate through a set of examples that the conditions of pointedness and closedness are essential to ensure the convergence. Finally, we provide some strategies to deal with cases where the conditions of pointedness and closedness are violated.
For an ideal I with a positive dimensional real variety, based on moment relaxations, we study how to compute a Pommaret basis which is simultaneously a Groebner basis of an ideal J generated by the kernel of a truncated moment matrix and nesting bet
ween I and its real radical ideal. We provide a certificate consisting of a condition on coranks of moment matrices for terminating the algorithm. For a generic delta-regular coordinate system, we prove that the condition is satisfiable in a large enough order of moment relaxations.
In this paper, we generalize the algorithm described by Rump and Graillat, as well as our previous work on certifying breadth-one singular solutions of polynomial systems, to compute verified and narrow error bounds such that a slightly perturbed sys
tem is guaranteed to possess an isolated singular solution within the computed bounds. Our new verification method is based on deflation techniques using smoothing parameters. We demonstrate the performance of the algorithm for systems with singular solutions of multiplicity up to hundreds.
We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials with real
coefficients must contain polynomials in the denominator of degree no less than a given input lower bound. By Artins solution to Hilberts 17th problems, such representations always exist for some denominator degree. Our certificates of infeasibility are based on the generalization of Farkass Lemma to semidefinite programming. The literature has many famous examples of impossibility of SOS representability including Motzkins, Robinsons, Chois and Lams polynomials, and Reznicks lower degree bounds on uniform denominators, e.g., powers of the sum-of-squares of each variable. Our work on exact certificates for positive semidefiniteness allows for non-uniform denominators, which can have lower degree and are often easier to convert to exact identities. Here we demonstrate our algorithm by computing certificates of impossibilities for an arbitrary sum-of-squares denominator of degree 2 and 4 for some symmetric sextics in 4 and 5 variables, respectively. We can also certify impossibility of base polynomials in the denominator of restricted term structure, for instance as in Landaus reduction by one less variable.
