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Selecting a Monomial Basis for Sums of Squares Programming Over a Quotient Ring

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Selecting a Monomial Basis for Sums of Squares Programming Over a Quotient Ring Selecting a Monomial Basis for Sums of Squares Programming over a Quotient Ring Frank Permenter1 and Pablo A. Parrilo2 Abstract— In this paper we describe a method for choosing for λi(x) and s(x), this feasibility problem is equivalent to a “good” monomial basis for a sums of squares (SOS) program an SDP. This follows because the sum of squares constraint formulated over a quotient ring. It is known that the monomial is equivalent to a semidefinite constraint on the coefficients basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible of s(x), and the equality constraint is equivalent to linear to use a reduced subset of standard monomials by combining equations on the coefficients of s(x) and λi(x). Groebner basis techniques with the well-known Newton poly- Naturally, the complexity of this sums of squares program tope reduction. This reduced subset of standard monomials grows with the complexity of the underlying variety, as yields a smaller semidefinite program for obtaining a certificate does the size of the corresponding SDP. It is therefore of non-negativity of a polynomial on an algebraic variety. natural to explore how algebraic structure can be exploited I. INTRODUCTION to simplify this sums of squares program. Consider the Many practical engineering problems require demonstrat- following reformulation [10], which is feasible if and only ing non-negativity of a multivariate polynomial over an if (2) is feasible: algebraic variety, i.e. over the solution set of polynomial Find s(x) equations. This problem arises, for example, in local Lya- subject to punov analysis of a polynomial dynamical system. (3) Unfortunately, certifying non-negativity over a variety is s(x) is a sum of squares in general a hard computational problem. An alternative is to s(x) = f(x): demonstrate a polynomial is equal to a sum of squares over the variety by solving a sums of squares program. A sums Here, s(x) denotes the normal form of s(x) with respect to of squares program optimizes a linear function of polyno- a Groebner basis for the ideal I = hh1; h2; : : : ; hmi. Two mial coefficients subject to constraints that polynomials are polynomials have the same normal form if and only if they sums of squares. If the polynomials in the program are of share an equivalence class in the quotient ring R[x]=I. In bounded degree, a sums of squares program is equivalent to other words, two polynomials have the same normal form a semidefinite program (SDP) and hence efficiently solved if and only if they differ by a polynomial of the form Pm [7]. Consider the following sums of squares program, which i λi(x)hi(x). Note if one specifies a vector of monomials demonstrates non-negativity of the polynomial f(x) on the ~m(x) one can solve (3) with the SDP: set n Find Q 2 S V = fx : hi(x) = 0; i = 1; : : : ; mg; (1) subject to (4) ~m(x)T Q~m(x) = f(x) where x is a vector of indeterminates and each hi is a polynomial in R[x]: Q 0: Find s(x) and λi(x) 2 R[x] This is an SDP since the constraint matching normal forms subject to is linear in the entries of Q. Formulation (3) has two practical advantages over (2). s(x) is a sum of squares (2) λ (x) m First, the polynomials i have been eliminated from the X search. Second, one can build ~m(x) in the SDP formulation s(x) − f(x) = λi(x)hi(x): (4) using only standard monomials, which are defined with i respect to a Groebner basis as all monomials not divisible by Feasibility of (2) is sufficient to conclude non-negativity of an initial term of a polynomial in the Groebner basis. Here, f(x) V f(x) on . To see this, note that feasibility implies initial term means the unique maximal term of a polynomial and a sum of squares polynomial differ by an expression that with respect to a term ordering. Constructing ~m(x) from V vanishes everywhere on . If one specifies a monomial basis standard monomials is justified by the fact that any sum 1F. Permenter is with the Computer Science and Artificial Intelligence of squares polynomial is congruent modulo I to a sum of Laboratory (CSAIL), Massachusetts Institute of Technology, Cambridge, squares of polynomials supported by standard monomials MA 02139. [email protected] (see, for example, Lemma 2 of Section III). 2P.A. Parrilo is with the Laboratory For Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA Systematic procedures for selecting ~m(x), however, have 02139. [email protected] not been thoroughly addressed. If we consider a total degree γ ordering ≥α, a term bound x , and define inα(s(x)) to be A. Polynomial Ideals, Term Orderings, and Groebner Bases the maximal term of s(x) with respect to ≥α, an “optimal” The ideal I generated by a set of polynomials monomial vector is one of minimum dimension solving the following problem: H = fh1(x); h2(x); : : : ; hm(x)g Problem 1: Given a total degree ordering ≥ and term α is denoted bound xγ , find a vector of monomials ~m(x) such that the semidefinite program (4) is feasible whenever the sums of I = hh1(x); h2(x); : : : ; hm(x)i squares program (3) is feasible for some s(x) satisfying γ and is equal to the set inα(s(x)) ≤α x : Note the total degree ordering ensures the set of all ( m ) γ X polynomials satisfying the term bound x can be expressed I = λi(x)hi(x): λi(x) 2 R[x] : (5) with a finite set of monomials, which in turn implies an ~m(x) i=1 of finite dimension exists that solves Problem 1. In particular, In other words, the ideal generated by a set of polynomials H one can construct an ~m(x) solving Problem 1 from the set of is the set of all polynomials that can be obtained by summing Term Bounded Standard Monomials, which we denote Mγ scaled versions of polynomials in H, where the scale factors and define below: can be any polynomial in R[x]. Definition 1: Term Bounded Standard Monomials. For a n A term ordering ≥α is a relation on Z+ (i.e. monomial γ given x and total degree ordering ≤α, let Mγ be the set exponents) satisfying of all exponents β such that xβ is a standard monomial and • ≥ 2β γ The relation α is a total ordering. x ≤α x : n • If γ >α β then for any ζ 2 Z+, γ + ζ >α β + ζ. A monomial vector constructed from all elements of Mγ • The relation ≥α is a well-ordering, which means every solves Problem 1 for a general polynomial f(x) and a n nonempty subset of Z+ has a smallest element. general ideal I. In this paper, we develop an algorithm for We say a term ordering is a total degree ordering whenever constructing ~m(x) using just a subset of Mγ by exploiting n n problem specific structure. Given a Groebner basis for I X X γi ≥ βi ) γ ≥α β: with respect to a total degree ordering ≤α, the algorithm we i=1 i=1 present constructs a subset of Mγ using the structure induced γ by multi-divisor polynomial division and well known Newton We will abuse notation and write for a monomial that x ≥α β polytope arguments. As we illustrate with examples, this x whenever γ ≥α β. subset is often a strict subset of Mγ which enables one The initial term inα(f(x)) of a polynomial is the unique to build smaller SDP formulations for the sums of squares maximal term of f(x) with respect to ≥α. Given a term program (3). ordering ≥α, one can consider the set of initial terms of polynomials in I. These initial terms generate an ideal, called A. Prior Work the initial ideal, which we denote inα(I). Many authors have investigated ways of exploiting alge- For a particular term ordering, we say a set of polynomials braic in sums of squares programs. General methods are G = fg1(x); g2(x); : : : ; gn(x)g introduced in [10] and [8] that exploit sparsity and symmetry. Other techniques for exploiting sparsity are further discussed is a Groebner basis for an ideal I if and only if it generates in [12], [6], [5]. Symmetry methods are discussed in [3]. I and Practical implementations of these types of techniques are in (I) = hin (g ); in (g ); : : : ; in (g )i: (6) discussed in [4]. α α 1 α 2 α n Quotient ring formulations, aside from their introduction Property (6) implies the initial term of any polynomial in I is in [10] and discussion in [8], have received less attention. This divisible by an element of the Groebner basis. This enables a is perhaps due to their reliance on Groebner bases methods, division algorithm equipped with a Groebner basis to decide and their lack of support by sums of squares modeling tools. if an arbitrary polynomial f(x) is a member of the ideal I. The former concern is not always relevant since Groebner We discuss the important properties of this division algorithm bases are easily calculated (or are immediately available) in in the next section. many instances. The latter concern, of course, does not take away from the inherent power of quotient ring formulations, B. Division Algorithm and Normal Forms which we hope to extend with this paper.
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