Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization

M. Marshall
2009 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Natural sufficient conditions for a polynomial to have a local minimum 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
more » ... large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty. Fix an algebraic set V in R n , where R is a real closed field. Let A denote the coordinate ring of V , i.e., where I(V ) denotes the ideal of polynomials vanishing on V . The reader may assume, for simplicity, that V = R n , so A = R [x]. Fix a quadratic module M in A, i.e., a subset M of A satisfying M + M ⊆ M , 1 ∈ M , and f 2 M ⊆ M for all f ∈ A, and let We often assume, in addition, that M M ⊆ M , i.e., that M is a quadratic preordering. One is especially interested in the case where M is finitely generated (as a quadratic module or as a quadratic preordering). In this case K 1991 Mathematics Subject Classification. Primary 13J30; Secondary 12Y05, 13P99, 14P10, 90C22. This research was supported in part by NSERC of Canada and in part by the Trimester on Real Geometry, Sept. 12 -Dec. 16, 2005, Institut Henri Poincaré, Univ. Paris. Typeset by One is especially interested in the case R = R. The quadratic module M is said to be archimedean if for each f ∈ A there exists an integer k ≥ 1 such that k − f ∈ M . Results of Putinar [14] and Jacobi [2] show that if R = R and the quadratic module M is archimedean then, for all f ∈ A, When M is a quadratic preordering which is finitely generated, the arithmetic hypothesis 'M is archimedean' is equivalent to the geometric hypothesis 'K is compact ' [21]. This result extends to quadratic modules in various ways [3] . In [20] Scheiderer shows that if M is archimedean, See [9] for another proof of this. Applications of this result are given in [17] [19] [20]. The proof of [9, Th. 2.3] shows that if R = R, V is irreducible, M is archimedean, the zeros of f in K are non-singular points of V , and certain 'boundary hessian conditions' hold at each zero of f in K, then f ∈ M + (f 2 ) (and consequently, if we also assume f ≥ 0 on K, then f ∈ M ). We prove that the above stated version of [9, Th. 2.3] continues to hold when the hypothesis 'R = R and M is archimedean' is replaced by the hypothesis 'M is a finitely generated preordering'. The proof of this result is, in fact, simpler than the proof of [9, Th. 2.3]. Using standard ideas from model theory, this yields degree bounds for the presentation of f as an element of M +(f 2 ) in this case. The result has application to global optimization, yielding a new class of polynomials f such that f − f * is contained in R[x] 2 + I, where f * is the minimum value of f on R n and I is the gradient ideal of f , compare to [10] , and again we obtain degree bounds for the presentation. Exploiting other degree bounds in [13] and [22], we show that if R = R, M is a finitely generated preordering, K is compact, and f ≥ 0 on K, then there are degree bounds for the presentation of f as an element of M in terms of the presentation of f as an element of M + (f 2 ). This has application to the optimization algorithm in [5] . We also consider the likelihood of the boundary hessian conditions holding in case the algebraic set V and the boundary of K in V are sufficently well behaved. The conclusion is that, in a suitable sense, these conditions hold with probability 1. In the final section, we determine concrete degree bounds for the algorithms in [5] and [10] , which ensure that the feasible set of solutions obtained is not the empty set.
doi:10.4153/cjm-2009-010-4 fatcat:vnhiom7icrcttayy3b2hhdrwgi