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

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1 Selectng a Monomal Bass for Sums of Squares Programmng over a Quotent Rng Frank Permenter 1 and Pablo A. Parrlo 2 Abstract In ths paper we descrbe a method for choosng a good monomal bass for a sums of squares (SOS) program formulated over a quotent rng. It s known that the monomal bass need only nclude standard monomals wth respect to a Groebner bass. We show that n many cases t s possble to use a reduced subset of standard monomals by combnng Groebner bass technques wth the well-known Newton polytope reducton. Ths reduced subset of standard monomals yelds a smaller semdefnte program for obtanng a certfcate of non-negatvty of a polynomal on an algebrac varety. I. INTRODUCTION Many practcal engneerng problems requre demonstratng non-negatvty of a multvarate polynomal over an algebrac varety,.e. over the soluton set of polynomal equatons. Ths problem arses, for example, n local Lyapunov analyss of a polynomal dynamcal system. Unfortunately, certfyng non-negatvty over a varety s n general a hard computatonal problem. An alternatve s to demonstrate a polynomal s equal to a sum of squares over the varety by solvng a sums of squares program. A sums of squares program optmzes a lnear functon of polynomal coeffcents subject to constrants that polynomals are sums of squares. If the polynomals n the program are of bounded degree, a sums of squares program s equvalent to a semdefnte program (SDP) and hence effcently solved [7]. Consder the followng sums of squares program, whch demonstrates non-negatvty of the polynomal f(x) on the set V = {x : h (x) = 0, = 1,..., m}, (1) where x s a vector of ndetermnates and each h s a polynomal n R[x]: Fnd s(x) and λ (x) R[x] subject to s(x) s a sum of squares m s(x) f(x) = λ (x)h (x). Feasblty of (2) s suffcent to conclude non-negatvty of f(x) on V. To see ths, note that feasblty mples f(x) and a sum of squares polynomal dffer by an expresson that vanshes everywhere on V. If one specfes a monomal bass 1 F. Permenter s wth the Computer Scence and Artfcal Intellgence Laboratory (CSAIL), Massachusetts Insttute of Technology, Cambrdge, MA fperment@mt.edu 2 P.A. Parrlo s wth the Laboratory For Informaton and Decson Systems (LIDS), Massachusetts Insttute of Technology, Cambrdge, MA parrlo@mt.edu (2) for λ (x) and s(x), ths feasblty problem s equvalent to an SDP. Ths follows because the sum of squares constrant s equvalent to a semdefnte constrant on the coeffcents of s(x), and the equalty constrant s equvalent to lnear equatons on the coeffcents of s(x) and λ (x). Naturally, the complexty of ths sums of squares program grows wth the complexty of the underlyng varety, as does the sze of the correspondng SDP. It s therefore natural to explore how algebrac structure can be exploted to smplfy ths sums of squares program. Consder the followng reformulaton [10], whch s feasble f and only f (2) s feasble: Fnd s(x) subject to s(x) s a sum of squares s(x) = f(x). Here, s(x) denotes the normal form of s(x) wth respect to a Groebner bass for the deal I = h 1, h 2,..., h m. Two polynomals have the same normal form f and only f they share an equvalence class n the quotent rng R[x]/I. In other words, two polynomals have the same normal form f and only f they dffer by a polynomal of the form m λ (x)h (x). Note f one specfes a vector of monomals m(x) one can solve (3) wth the SDP: Fnd Q S n subject to m(x) T Q m(x) = f(x) Q 0. Ths s an SDP snce the constrant matchng normal forms s lnear n the entres of Q. Formulaton (3) has two practcal advantages over (2). Frst, the polynomals λ (x) have been elmnated from the search. Second, one can buld m(x) n the SDP formulaton (4) usng only standard monomals, whch are defned wth respect to a Groebner bass as all monomals not dvsble by an ntal term of a polynomal n the Groebner bass. Here, ntal term means the unque maxmal term of a polynomal wth respect to a term orderng. Constructng m(x) from standard monomals s justfed by the fact that any sum of squares polynomal s congruent modulo I to a sum of squares of polynomals supported by standard monomals (see, for example, Lemma 2 of Secton III). Systematc procedures for selectng m(x), however, have not been thoroughly addressed. If we consder a total degree (3) (4)

2 orderng α, a term bound x γ, and defne n α (s(x)) to be the maxmal term of s(x) wth respect to α, an optmal monomal vector s one of mnmum dmenson solvng the followng problem: Problem 1: Gven a total degree orderng α and term bound x γ, fnd a vector of monomals m(x) such that the semdefnte program (4) s feasble whenever the sums of squares program (3) s feasble for some s(x) satsfyng n α (s(x)) α x γ. Note the total degree orderng ensures the set of all polynomals satsfyng the term bound x γ can be expressed wth a fnte set of monomals, whch n turn mples an m(x) of fnte dmenson exsts that solves Problem 1. In partcular, one can construct an m(x) solvng Problem 1 from the set of Term Bounded Standard Monomals, whch we denote M γ and defne below: Defnton 1: Term Bounded Standard Monomals. For a gven x γ and total degree orderng α, let M γ be the set of all exponents β such that x β s a standard monomal and x 2β α x γ. A monomal vector constructed from all elements of M γ solves Problem 1 for a general polynomal f(x) and a general deal I. In ths paper, we develop an algorthm for constructng m(x) usng just a subset of M γ by explotng problem specfc structure. Gven a Groebner bass for I wth respect to a total degree orderng α, the algorthm we present constructs a subset of M γ usng the structure nduced by mult-dvsor polynomal dvson and well known Newton polytope arguments. As we llustrate wth examples, ths subset s often a strct subset of M γ whch enables one to buld smaller SDP formulatons for the sums of squares program (3). A. Pror Work Many authors have nvestgated ways of explotng algebrac n sums of squares programs. General methods are ntroduced n [10] and [8] that explot sparsty and symmetry. Other technques for explotng sparsty are further dscussed n [12], [6], [5]. Symmetry methods are dscussed n [3]. Practcal mplementatons of these types of technques are dscussed n [4]. Quotent rng formulatons, asde from ther ntroducton n [10] and dscusson n [8], have receved less attenton. Ths s perhaps due to ther relance on Groebner bases methods, and ther lack of support by sums of squares modelng tools. The former concern s not always relevant snce Groebner bases are easly calculated (or are mmedately avalable) n many nstances. The latter concern, of course, does not take away from the nherent power of quotent rng formulatons, whch we hope to ext wth ths paper. II. BACKGROUND We begn by revewng polynomal deals and Groebner bases. We then dscuss mportant propertes of multvarate, mult-dvsor polynomal long dvson and ts connectons to quotent rngs. A. Polynomal Ideals, Term Orderngs, and Groebner Bases The deal I generated by a set of polynomals s denoted H = {h 1 (x), h 2 (x),..., h m (x)} I = h 1 (x), h 2 (x),..., h m (x) and s equal to the set { m } I = λ (x)h (x) : λ (x) R[x]. (5) =1 In other words, the deal generated by a set of polynomals H s the set of all polynomals that can be obtaned by summng scaled versons of polynomals n H, where the scale factors can be any polynomal n R[x]. A term orderng α s a relaton on Z n + (.e. monomal exponents) satsfyng The relaton α s a total orderng. If γ > α β then for any ζ Z n +, γ + ζ > α β + ζ. The relaton α s a well-orderng, whch means every nonempty subset of Z n + has a smallest element. We say a term orderng s a total degree orderng whenever n n γ β γ α β. =1 =1 We wll abuse notaton and wrte for a monomal that x γ α x β whenever γ α β. The ntal term n α (f(x)) of a polynomal s the unque maxmal term of f(x) wth respect to α. Gven a term orderng α, one can consder the set of ntal terms of polynomals n I. These ntal terms generate an deal, called the ntal deal, whch we denote n α (I). For a partcular term orderng, we say a set of polynomals G = {g 1 (x), g 2 (x),..., g n (x)} s a Groebner bass for an deal I f and only f t generates I and n α (I) = n α (g 1 ), n α (g 2 ),..., n α (g n ). (6) Property (6) mples the ntal term of any polynomal n I s dvsble by an element of the Groebner bass. Ths enables a dvson algorthm equpped wth a Groebner bass to decde f an arbtrary polynomal f(x) s a member of the deal I. We dscuss the mportant propertes of ths dvson algorthm n the next secton. B. Dvson Algorthm and Normal Forms Gven a polynomal f(x), a lst of dvsors H = {h 1 (x), h 2 (x),..., h m (x)} and a monomal orderng α, a dvson algorthm can be defned as n [2] that fnds λ (x) and a remander term f(x) H such that m f(x) = λ (x)h (x) + f(x) H =1

3 wth propertes n α (f(x) H ) s not dvsble by any n α (h (x)) (7) n α (f(x) H ) α n α (f(x)) (8) n α (λ (x)h (x)) α n α (f(x)) If we use a lst of dvsors G that form a Groebner bass for I = h 1, h 2,..., h m, the remander term f(x) G equals f(x), the normal form of f(x) wth respect to the Groebner bass. The normal form of a polynomal s unque, meanng the lst of dvsors G can be permuted wthout changng the remander term f(x) G. Normal forms also obey the followng propertes: Ideal membershp Arthmetc denttes Congruence modulo I f(x) I f(x) = 0 g(x) f(x) = g(x) f(x) (9) g(x) + f(x) = g(x) + f(x) (10) s(x) f(x) mod I f(x) = s(x) (11) Note property (7) mples f(x) contans only standard monomals,.e. monomals not dvsble by an ntal term of a polynomal n the Groebner bass. Property (11) gves the correspondence between normal forms and equvalence classes n R[x]/I. III. APPROACH We now present an algorthm for computng a vector of standard monomals satsfyng Problem 1 gven a term bound x γ, a total degree term orderng α, and a correspondng Groebner bass. Frst, we calculate a set of monomals S γ consstent wth the structure nduced by the Groebner bass when t s used as the dvsor set n the dvson algorthm. We then use S γ to compute a sutable set of monomals appearng n a sum of squares decomposton of s(x) usng Newton polytope arguments. Fnally, we take normal forms of these monomals to arrve at a reduced set N γ of standard monomals. We prove N γ s a subset of the term bounded standard monomals M γ and that ths subset can be used to construct a monomal vector solvng Problem 1. The proposed procedure s explctly gven n Algorthm 1. Here, we defne the support supp(w(x)) of a polynomal w(x) to be the unque set gvng w(x) = c β x β β supp(w(x)) for nonzero real valued coeffcents c β. We clam Algorthm 1 has the followng propertes: Property 1: Algorthm 1 returns a set N γ such that the monomal vector m(x) = x β, β N γ, = 1,..., N γ solves Problem 1. Property 2: The set N γ returned by Algorthm 1 s a subset of the Term Bounded Standard Monomals M γ. Input: A polynomal f(x), a total degree term orderng α and correspondng Groebner bass G, a maxmal monomal x γ Output: A set of standard monomal exponents N γ # Compute monomals n normal form of f(x) S γ = supp(f(x)) # Monomals that respect term bound foreach g(x) G do # Total degree orderng ensures ths loop termnates forall x β α x γ do f n α (x β g(x)) α x γ then S γ = S γ supp(x β g(x)) # Normal form of monomals n Newton polytope N γ = {} foreach β conv( 1 2 S γ) Z n do N γ = N γ supp(x β ) Algorthm 1: Computes a reduced subset of term bounded standard monomals. To understand the mechancs of Algorthm 1 and how Propertes 1 and 2 can be demonstrated, frst note that any polynomal s(x) satsfyng the condtons n α (s(x)) α x γ (12) s(x) f(x) mod I (13) can be decomposed usng a Groebner bass G for I wth respect to the term orderng α as where G s(x) = λ (x)g (x) + f(x) =1 n α (λ (x)g (x)) α n α (s(x)) α x γ, and g G and λ (x) R[x]. Exstence of ths decomposton s mpled by propertes of dvson by a Groebner bass and motvates the constructon n Algorthm 1 of the set S γ, whch has the obvous property: s(x) satsfes (12) and (13) supp(s(x)) S γ. Remark 1: The total degree orderng ensures that the set S γ constructed by Algorthm 1 s fnte. The next phase of the algorthm explots the structure of sums of squares polynomals. If a polynomal wth support n S γ s equal to a sum of squares p (x) 2, then Newton polytope arguments gven n [1] descrbe the support of p (x). Specfcally, s(x) = ( ) 1 p (x) 2 supp(p ) conv 2 S γ Z n, where conv denotes convex hull. Ths gves the followng mmedate result:

4 Lemma 1: If a sums of squares polynomal p (x) 2 satsfes (12) and (13), then supp(p (x)) s contaned n conv( 1 2 S γ) Z n. Lemma 1 mples a monomal vector constructed from conv( 1 2 S γ) Z n solves Problem 1. We are nterested, however, not n conv( 1 2 S γ) Z n but n a set of standard monomals. The algorthm therefore bulds the set N γ by takng normal forms of monomals constructed from conv( 1 2 S γ) Z n. Ths s motvated by the followng lemma: Lemma 2: If a polynomal s(x) = p (x) 2 satsfes s(x) f(x) mod I, then there exsts a (possbly dfferent) sum of squares polynomal ŝ(x) = ˆp (x) 2 satsfyng ŝ(x) f(x) mod I where supp(ˆp ) = supp(p ). Proof: See Appx. Combnng Lemma 1, Lemma 2, and (10) demonstrates a monomal vector constructed from N γ solves Problem 1 and hence demonstrates Property 1. To demonstrate Property 2, we need the followng basc fact relatng term orderngs and polytopes: Lemma 3: If α s a vald term orderng and S γ s a fnte set of exponents, then there s a unque maxmal element x ζ of {x β : β conv(s γ )} wth respect to α, and ζ s an element of S γ. Proof: See Appx. To show that N γ M γ, we frst note that for any σ S γ, x σ α x γ by constructon of S γ. We then use Lemma 3 to conclude for any pont β n conv( 1 2 S γ) Z n, x 2β α x γ. Fnally, we use (8) to conclude that for any ζ n supp(x β ), x 2ζ α x γ. That s, all ζ n N γ are also n M γ and the desred result follows. IV. EXAMPLES We llustrate the results wth a few examples. A. Example 1 Consder provng non-negatvty of f(x, y) = x 3 xy 2 on the varety V = {(x, y) : x 3 + y 2 = 0}. Take I to be the deal x 3 + y 2 and α to be the graded lex orderng wth x > y. Snce the deal I has one generator, a Groebner bass for I n any term orderng s just the generator x 3 + y 2. Consder a quotent rng formulaton that searches for sum of squares polynomals s(x, y) satsfyng n α (s(x, y)) α y 8. For ths choce of term bound, the set of all term bounded standard monomals s gven by Ths example llustrates ths s not necessarly the case. Consder provng f(x) = x 3 s nonnegatve on the { 1 x y x 2 x y y 2 x 2 y x y 2 y 3 x 2 y 2 x y 3 y 4 varety V }. The correspondng set of exponents M γ s shown n Fgure V = {(x, y) : x 3 + y 6 = 0, x 2 y 5 = 0}. 1d. Let I equal the deal generated by the defnng polynomals Applyng Algorthm 1, we ll fnd a subset of exponents of the varety V : from whch a smaller set of monomals can be constructed. Decomposng s(x, y) as I = x 3 + y 6, x 2 y 5. s(x, y) = λ(x, y)(x 3 + y 2 ) + x 3 xy 2 = λ(x, y)(x 3 + y 2 ) + y 2 xy 2, (a) S γ (c) N γ (b) conv( 1 Sγ) Zn 2 (d) M γ Fg. 1: Dfferent sets of monomals arsng n Example IV-A. Fg. 1a-1c show monomal sets calculated at dfferent stages of Algorthm 1. Fgure 1d shows all term bounded standard monomals. Note N γ s a strct subset of M γ, whch mples Algorthm 1 leads to a smaller SDP for the gven example. we seek the set of monomals S γ that can appear n s(x, y) for all λ(x, y) such that n α (λ(x, y)(x 3 + y 2 )) α y 8. Ths set s plotted n Fgure 1a. Plotted n Fgure 1b are the nteger ponts contaned n conv( 1 2 (S γ)). Constructed from these nteger ponts s N γ, the desred subset of M γ. Note from Fgure 1 that N γ s a strct subset of M γ. B. Example 2 The precedng example dealt wth a postve dmensonal varety. We now explore a zero-dmensonal case, for whch the set of standard monomals s always fnte. One may perhaps thnk that as the degree of s(x) n our search ncreases, all standard monomals must eventually be ncluded and there s no beneft to runnng the proposed algorthm. Takng α to agan be the graded lex orderng wth x > y, a Groebner bass for I s {x 2 y 5, x 3 + x 2 y}.

5 (a) S γ (b) conv( 1 Sγ) Zn 2 x γ N γ M γ dm S n dm S n % Reducton n = N γ n = M γ x θ x θ x θ x θ x TABLE I: Comparatve savngs for dfferent term bounds on s(x) for Example IV-C. The M γ column shows the number of term bounded standard monomals. The N γ column shows the subset of monomals returned by Algorthm 1. Columns also show the dmenson of the vector space of symmetrc matrces appearng n the correspondng SDPs, a quantty relevant for SDP solvers. The percent reducton n dmenson s shown n the last column. (c) N γ (d) M γ Fg. 2: Dfferent sets of monomals arsng n Example IV- B. Fg. 2a-2c show monomal sets calculated at dfferent stages of Algorthm 1. Fgure 1d shows all term bounded standard monomals. Note N γ s a strct subset of M γ, whch mples Algorthm 1 leads to a smaller SDP for the gven zero dmensonal example. We wll consder all s(x, y) such that n α (s(x, y)) α x 12. For ths term bound, M γ contans the exponents of all standard monomals. We see n Fgure 2 that even n ths case, the proposed technque leads to a set of monomals N γ that s a strct subset of M γ. Ths wll always happen for the gven f(x), even as n α (s(x, y)) s allowed to grow arbtrarly large. C. Example 3 As a fnal example, we explore a varety V relevant to stablty analyss of a cart-pole, a 4 th order mechancal system common n robotcs. Here V takes the form V = {x : s 2 θ + c 2 θ 1 = 0, ẍ g(x) = 0, Ẇ (x) = 0}, where x = (x, ẋ, ẍ, θ, c θ, s θ ). Notce here that all varables are treated as ndepent formal ndetermnates,.e., no algebrac relatonshp s assumed to hold between x and ẋ (and smlarly for all other varables). The frst two equatons encode a polynomal representaton of the cart-pole dynamcs. The frst equaton s a unt crcle constrant for trgonometrc varables. The second equaton encodes an mplct dynamcal relatonshp between an acceleraton and the other system varables. The 3rd equaton Ẇ (x) s the tme dervatve of a quadratc Lyapunov functon W (x, ẋ, θ, s θ ). The hypersurface where Ẇ (x) vanshes defnes a verfable regon of stablty. If α s the graded lex orderng wth x > ẋ > ẍ > θ > c θ > s θ, a Groebner bass for I, the deal generated by the defnng polynomals of V, contans 8 polynomals n ndetermnates (x, ẋ, ẍ, θ, c θ, s θ ) n degrees rangng from 2 to 8 (for a partcular choce of Lyapunov functon and system parameters). Local stablty can be estmated by fndng the largest scalar ρ such that f(x) = (s 2 θ + θ 2 + ẋ 2 + x 2 )(W ρ) s congruent to a sum of squares polynomal s(x) modulo I. For the same choce of Lyapunov functon and system parameters, the ntal monomal of f(x) s x 4. Thus, we consder s(x) wth n α (s(x)) bounded below by x 4. Table I compares the use of all term bounded standard monomals versus the subset returned by the proposed algorthm for varous upper bounds on n α (s(x)). Compared are the number of monomals and the dmenson of the correspondng vector space of symmetrc matrces S n. The latter quantty s relevant for SDP solvers such as SeDuM [11] snce t equals the number of scalar decson varables ntroduced by the solver. V. CONCLUSIONS AND FUTURE WORK A. Conclusons We have proposed a technque for selectng a reduced set of monomals n semdefnte program formulatons for sums of squares programs formulated over quotent rngs. The proposed method can lead to smaller, better condtoned semdefnte programs for provng non-negatvty of polynomals over algebrac varetes.

6 B. Future Work For the zero-dmensonal case, the set of standard monomals always has fnte cardnalty. Ths fact provdes the justfcaton for a degree bound n [9] for sums of squares representatons of polynomals nonnegatve on zero dmensonal varetes. Can the proposed method be used to mprove ths bound? Another open queston addresses the ssue of strct contanment of N γ n M γ. Is t possble to characterze the cases when strct contanment holds, and under what condtons t fals? In other words, n what cases s the proposed method guaranteed to reduce the sze of the resultng semdefnte programs? Fnally, what role do the dfferent possble term orderngs play n producng structure useful for complexty reducton? APPENDIX Proof: (of Lemma 2) Snce s(x) = p (x) 2 s congruent modulo I to f(x) we can wrte that f(x) = p (x) 2. Usng propertes of normal form arthmetc gven by (10) and (9), we see that p (x) 2 = p (x) 2 = = = p (x) p (x) ˆp (x) 2 ˆp (x) 2, where we have taken ˆp (x) = p (x). Thus, ŝ(x) = ˆp (x) 2 s congruent modulo I to f(x) and supp(ˆp ) = supp(p ). Proof: (of Lemma 3) To prove ths clam, we frst recall a fact about vald term orderngs: monomals can be compared wth respect to the orderng by takng nner products of ther exponents wth the rows of a sutable weght matrx. Suppose we want to show x β > α x σ. We frst check that w1 T β w1 T σ, where w1 T s the frst row of the weght matrx. If w1 T β = w1 T σ, we move to the next row of the matrx w2 T and check f w2 T β w2 T σ. Ths process contnues down the rows of the matrx untl the nequalty s strct. For any vald term orderng, ths process termnates n fntely many steps. From ths we note f α s a vald term orderng, there s a unque maxmal element of {x β : β conv(s γ )} and t can be found by solvng a seres of lnear programs (LPs) over faces of conv(s γ ). The cost vector of the th lnear program s w and the feasble set of each LP s the face of conv(s γ ) on whch the ( 1) th LP acheves ts optmal soluton (the ntal feasble set s just conv(s γ )). We terate untl an LP has a unque optmum, whch occurs only at extreme ponts of conv(s γ ). Snce the extreme ponts are contaned n the set S γ, we must have that the maxmal element x ζ satsfes ζ S γ. REFERENCES [1] M. D. Cho, T. Y. Lam, and B. Reznck. Sums of squares of real polynomals. Proceedngs of Symposa n Pure Mathematcs, 58(2): , [2] D. A. Cox, J. Lttle, and D. O Shea. Usng Algebrac Geometry. Graduate Texts n Mathematcs. Sprnger-Verlag, Berln-Hedelberg- New York, March [3] K. Gatermann and P. A. Parrlo. Symmetry groups, semdefnte programs, and sums of squares. Journal of Pure and Appled Algebra, 192(1 3):95 128, [4] J. Löfberg. Pre- and post-processng sum-of-squares programs n practce. IEEE Transactons on Automatc Control, 54(5): , [5] J. Ne. Sum of squares method for sensor network localzaton. Computatonal Optmzaton and Applcatons, 43: , /s z. [6] J. Ne and J. Demmel. Sparse SOS relaxatons for mnmzng functons that are summatons of small polynomals. SIAM Journal on Optmzaton, 19(4): , [7] P. A. Parrlo. Structured Semdefnte Programs and Semalgebrac Geometry Methods n Robustness and Optmzaton. PhD thess, Calforna Insttute of Technology, Pasadena, CA, [8] P. A. Parrlo. Explotng algebrac structure n sum of squares programs. In D. Henron and A. Garull, edtors, Postve Polynomals n Control, volume 312 of Lecture Notes n Control and Informaton Scences, pages Sprnger Berln / Hedelberg, / [9] P.A. Parrlo. An explct constructon of dstngushed representatons of polynomals nonnegatve over fnte sets. Techncal report, March [10] P.A. Parrlo. Explotng structure n sum of squares programs. In IEEE Conference on Decson and Control, Mau, Hawa, December [11] J. F. Sturm. Usng SeDuM 1.02, a MATLAB toolbox for optmzaton over symmetrc cones. Optmzaton Methods and Software, 11 12: , [12] H. Wak, S. Km, M. Kojma, and M. Muramatsu. Sums of squares and semdefnte programmng relaxatons for polynomal optmzaton problems wth structured sparsty. SIAM Journal on Optmzaton, 17: , 2006.