Murray Marshall

Positivity, sums of squares and the multi-dimensional moment problem

Let K be the basic closed semi-algebraic set in R n defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f a polynomial in n variables nonnegative on K and real e > 0, we have that f + ∈ E T. In particular, the K-Moment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the K-Moment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n > 2, we show that the K-Moment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (+) ∀ f ∈ R[X], f ≥ 0 on K ⇒ ∃q ∈ T such that ∀ real ∈ > 0, f + eq ∈ T. This property is strictly weaker than the one given in Schmudgen (1991), but in turn it implies a positive solution to the K-Moment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which () holds. Finally, we provide a list of 8 open problems.

REPRESENTATIONS OF NON-NEGATIVE POLYNOMIALS, DEGREE BOUNDS AND APPLICATIONS TO OPTIMIZATION

Natural su-cient conditions for a polynomial to have a local min- imum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the flnal section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

Optimization of Polynomial Functions

Recently progress has been made in the development of algorithms for optimizing polynomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing methods. The idea traces back to work of Shor [16][17] and is further developed by Parrilo [10] and by Parrilo and Sturmfels [11] and by Lasserre [7][8].

Orderings on a ring with involution

The object of the paper is to extend part of the theory of *-orderings on a skewfield with involution to a general ring with involution. The valuation associated to a *-ordering is examined. Every *-ordering is shown to extend. *-orderings are shown to form a space of signs as defined by Brocker and Marshall. In case the involution is the identity, the ring under consideration is commutative and the *-orderings are just the usual orderings making up the usual real spectrum of a commutative ring as defined by Coste and Roy.

Lower Bounds for Polynomials Using Geometric Programming

We make use of a result of Hurwitz [J. Reine Angew. Math., 108 (1891), pp. 266–268] and Reznick [Math. Ann., 283 (1989), pp. 431–464], and a consequence of this result due to Fidalgo and Kovacec [Math. Z., 269 (2011), pp. 629–645], to establish, in Theorem 2.3, a new sufficient condition for a polynomial

Approximating Positive Polynomials Using Sums of Squares

f\in\mathbb{R}[X_1,\dots,X_n]

Polynomials non-negative on a strip

of even degree to be a sum of squares. Theorem 2.3 yields as special cases the results of Ghasemi and Marshall in [Arch. Math. (Basel), 95 (2010), pp. 343–353] and, consequently, also those of Fidalgo and Kovacec and Lasserre [Arch. Math. (Basel), 89 (2007), pp. 390–398]. We apply Theorem 2.3 to obtain a new lower bound

Embedding ordered fields in formal power series fields

f_{\textrm{gp}}

Extending the Archimedean Positivstellensatz to the Non-Compact Case

for f, and we explain how

Orderings and real places on commutative rings

f_{\textrm{gp}}