Abstract
A basic closed semialgebraic subset of {mathbb {R}}^{n} is defined by simultaneous polynomial inequalities p_{1}ge 0,ldots ,p_{m}ge 0. We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.
Highlights
B María López QuijornaFor α ∈ Nn0, we use the standard notation :
Let polynomials f, p1, . . . , pm ∈ R[X ] with m ∈ N0 be given
We give a general new definition of Gaussian quadrature rule for a positive semidefinite linear form and we get a result concerning commutativity of the truncated GNS multiplication operators associated to this linear form and existence of Gaussian quadrature rule. Providing these operators commute we are able to get the factorization (7) or in other words we find a Gaussian quadrature rule representation for this positive semidefinite linear form, that is to say a quadrature rule for L on R[X ]2d−1 which number of nodes is the rank of ML|R[X]2d−2, see the Notation 3.5
Summary
For α ∈ Nn0, we use the standard notation :. For a polynomial p ∈ R[X ] we denote p = α pα X α (aα ∈ R). For d ∈ N0, by the notation R[X ]d := { |α|≤d aα X α | aα ∈ R} we will refer to the vector space of polynomials with degree less or equal to d. Polynomials all of whose monomials have exactly the same degree d ∈ N0 are called d-forms. They form a finite dimensional vector space that we will denote by:. We the will denote dual space by of sk := dim R[X ]k and by rk R[X ]d i.e. the set of linear.
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