A PTAS FOR THE MINIMIZATION OF POLYNOMIALS OF FIXED DEGREE OVER THE SIMPLEX

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1 A PTAS FOR THE MINIMIZATION OF POLYNOMIALS OF FIXED DEGREE OVER THE SIMPLEX ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO Abstract. We consider the problem of computing the imum value p taken by a polynomial px of degree d over the standard simplex. This is an NP-hard problem already for degree d = 2. For any integer k 1, by imizing px over the set of rational points in with denoator k, one obtains a hierarchy of upper bounds p k converging to p as k. These upper approximations are intimately linked to a hierarchy of lower bounds for p constructed via Pólya s theorem about representations of positive forms on the simplex. Revisiting the proof of Pólya s theorem allows us to give estimates on the quality of these upper and lower approximations for p. Moreover, we show that the bounds p k yield a polynomial time approximation scheme for the imization of polynomials of fixed degree d on the simplex, extending an earlier result of Bomze and De Klerk for degree d = 2. March 30, Introduction 1.1. A grid approximation. We consider the problem of imizing a polynomial px of degree d on the standard simplex n := {x IR n + x i = 1}; that is, the problem of computing 1.1 p := x px. One may assume w.l.o.g. that px is a homogeneous polynomial form. Indeed, as observed in [5], if px = d l=0 p lx, where p l x is homogeneous of degree l, then imizing px over is equivalent to imizing the degree d form p x := d l=0 p lx n x i d l. Problem 1.1 is an NP-hard problem, already for forms of degree d = 2, as it contains the maximum stable set problem. Indeed, for a graph G with adjacency matrix A, the maximum size αg of a stable set in G can be expressed as 1 αg = x xt I + Ax 1991 Mathematics Subject Classification. 90C22, 90C26, 49M20. Key words and phrases. Semidefinite programg, global optimization, positive polynomial, sum of squares of polynomials, approximation algorithm. Supported by the Netherlands Organisation for Scientific Research grant NWO Supported by the Netherlands Organisation for Scientific Research grant NWO

2 2 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO by the theorem of Motzkin and Straus [7]. Given an integer k 1, let 1.2 k := {x kx Z n } denote the set of rational points in with denoator k and define 1.3 p k := px s.t. x k. Thus, p p k for any k 1. As k = n+k 1 k, one can compute the bound p k in polynomial time for any fixed k. Set 1.4 p max := max x px. When px is a form of degree d = 2, Bomze and De Klerk [3] show that the following inequality holds: 1.5 p k p 1 k p max p for any k 1. Nesterov [9] subsequently showed: 1.6 p k p 1 k max,...,n p 2e i p, where e 1,...,e n denote the standard unit vectors. This inequality is stronger since, by evaluating px at e i, one finds that pe i = p 2ei p max. Bomze and De Klerk s result is based on an intimate link existing between the upper approximations p k and a hierarchy of lower bounds for p constructed using a result of Pólya about the representation of positive forms on the simplex. This approach also permits, in fact, to establish the stronger inequality 1.6. Nesterov s approach is different and uses a probabilistic argument for justifying the estimation 1.6. We describe both approaches in Section 2.1 below. Using his probabilistic approach, Nesterov [9] proves the following result for polynomials of higher degree. Assume that px is a form of degree d which is a sum of square-free monomials; that is, a monomial x α appears with a nonzero coefficient in px only if α i 1 for all i = 1,...,n. Then, for k d, 1.7 p k p 1 k! k d!k d p dd 1 p. 2k In this paper, we prove some estimates for the approximation p k for general degree d forms. For a polynomial px = α p αx α, define as in [13] the parameter: 1.8 L p := max α p α α 1! α n! α! and the following parameter, introduced in the next subsection: 1.9 p 0 max := max α p α α 1! α n!. α! Then, p max p 0 max L p. We show the inequalities: 1.10 p k p 1 k! k d!k d 1.11 p 0 max p p 0 max p 2d 1 d d d p max p, dd 1 p 0 max p, 2k

3 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 3 which imply: 1.12 p k p dd k d 2 2d 1 d p max p. Our argument for 1.10 follows closely the proof given by Powers and Reznick [13] for Pólya s theorem, and our proof for 1.11 uses some tools of Reznick [15] about powers of linear forms. As an application of 1.12, the approximations p k yield a polynomial time approximation scheme PTAS for the problem of imizing a form of degree d on the simplex, for any fixed d. A more detailed analysis permits to show sharper estimates in some cases. For instance, when d = 2, relations 1.5 and 1.6 hold and, in the case d = 3, we can show that 1.13 p k p 4 k p max p Pólya s representation theorem for positive forms on the simplex. We recall Pólya s result about positive forms on the simplex. Theorem 1.1. Let p be a form of degree d which is positive on the simplex, i.e., p > 0. Then, the polynomial n x i r px has nonnegative coefficients for all r satisfying 0 d p max 1.14 r d. 2 p Pólya [12] proved that n x i r px has nonnegative coefficients for r large enough; Powers and Reznick [13] proved that this holds for any r d Lp 2 p d; we observe here that this holds for any r satisfying the weaker condition 1.14 with L p replaced by p 0 max. We review the proof of Theorem 1.1 in Section 2.1, as it permits, moreover, to estimate the quality of the bound p k for p. For now, let us indicate how Pólya s result can be used for constructing an asymptotically converging hierarchy of lower bounds for p. Observe first that p can alternatively be formulated as the maximum scalar λ for which px λ 0 for all x. Equivalently, n d 1.15 p = max λ such that px λ x i 0 x IR n +. For any integer r 0, define the parameter: 1.16 p r := max λ s.t. the polynomial n x i r px λ n x i d has nonnegative coefficients. Obviously, p r pr+1 p. Moreover, p r 0 if and only if the polynomial n x i r px has nonnegative coefficients. Hence, Theorem 1.1 asserts that p r 0 for any r satisfying The bound p r can be computed in polynomial time for any fixed r, as it can be expressed as the imum over the grid r + d of a perturbation of the polynomial px; see 2.3. As a consequence of Pólya s theorem, the bounds p r converge asymptotically to p as r. This idea of using Pólya s result for

4 4 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO constructing converging approximations goes back to the work of Parrilo [10, 11], who used it for constructing hierarchies of conic relaxations for the cone of copositive matrices corresponding to degree 2 positive semidefinite forms. The construction was extended to general positive semidefinite forms by Faybusovich [5], Zuluaga et al. [18]. For the problem 1.4 of maximizing px over, one can analogously define the bounds: p max r := λ s.t. the polynomial n 1.17 x i r λ n x i d px has nonnegative coefficients, satisfying: p max p r+1 max As we see in Section 2.1, the following holds: 1.18 p 0 = α p α p r max for r 0. α 1! α n!, p 0 max = max d! α which justifies the definition of p 0 max given earlier in 1.9. p α α 1! α n!, d! 1.3. Quality of the upper and lower approximations. Let p be a form of degree d and let p k, p max, L p, p r, and pr max, be as defined in 1.3, 1.4, 1.8, 1.16, and Therefore, Finally, define the parameter 1.19 w r d := One can verify that p r p p p max p r max. d 1 r + d! r!r + d d = 1 i. r + d d 1 2 r + d w rd 1, which implies that lim r w r d = 1. For this, note that m 1 x i 1 m x i if 0 x i 1 for i = 1,...,m. Use induction on m 1. The following result estimates the quality of the approximations p r and p for p. Faybusovich [5] proved the weaker version of the inequality 1.21 where the parameter p 0 max is replaced by L p ; 1.22 coincides with 1.10 with k = r + d. Theorem 1.2. Let p be a form of degree d and r 0 an integer. Then, 1.21 p p r 1 p0 max p w r d 1, 1.22 p p p 0 max p 1 w r d. The next two results give analogous inequalities involving the parameter p max instead of p 0 max in the case when d = 2,3.

5 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 5 Theorem 1.3. Let p be a form of degree d = 2 and r 0 an integer. Then, 1.23 p p r 1 max r + 1 p 2e i p 1,...,n r + 1 p max p, 1.24 p r+2 p 1 max r + 2 p 2e i p,...,n 1 r + 2 p max p. The upper bounds in 1.23, 1.24 involving p max are proved by Bomze and De Klerk [3] and the upper bound in 1.24 involving max i p 2ei is proved by Nesterov [9]. We show in Section 2.1 the following extension for forms of degree 3. Theorem 1.4. Let p be a form of degree d = 3 and r 0 an integer. Then, 1.25 p p r 4r + 3 r + 1r + 2 p max p, 1.26 p r+3 p 4 r + 3 p max p. The results from Theorems 1.2, 1.3 and 1.4 can be proved by a close inspection of the proof of Pólya s theorem; see Section 2.1. For forms of arbitrary degree d, we can show the following result, whose proof needs an additional argument and will be given in Section 2.2. Theorem 1.5. The following holds for a form px of degree d p 0 max p 0 2d 1 d d p max p. d Combined with Theorem 1.2, this implies: Theorem 1.6. Let px be a form of degree d and r 0 an integer. Then, 1.28 p p r 1 2d 1 w r d 1 d d p max p, d 2d p p 1 w r d d d p max p. d As an application, the grid approximations p k k d provide a polynomial time approximation scheme for the problem of imizing a form of degree d on the simplex. See Section 3 for details. 2. Approximating Forms on the Simplex 2.1. Estimating the upper and lower approximations p and p r for p via Pólya s theorem. We will use the following notation. Given α N n, set α! := α 1! α n! and, following [13], given scalars x,t and a nonnegative integer m, set x m t := xx t x m 1t = x it. m 1 i=0

6 6 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO Then, 1 d 1 = w r d, the parameter defined in Define n In,m := {α N n : α = α i = m}. We use the multinomial identity: n m 2.1 x i = α In,m m! α! xα and its generalization, known as the Vandermonde-Chu identity: n 2.2 = m! α! x 1 α1 t x n αn t. x im t α In,m See [13] for a proof. Alternatively, use induction on m 1. In what follows, px is a form of degree d and r 0 is an integer. By definition, p r is the maximum scalar λ for which the polynomial i x i r px λ i x i has nonnegative coefficients. We begin with evaluating the coefficients of this polynomial. We have: n r + d! x i = x β, β! where A β := α In,d, α β n r px x i = β In, β In, r! β α! p r!r + dd α = β! A β x β α In,d p α n αi βi. r + d 1 Therefore, p r is the maximum λ for which A β λ! β! 0 for all β In,; that is, 2.3 As the point x := β 2.4 p r = p r = β In, = β In, β! r + d! A β 1 w r d α In,d p α n belongs to r + d, it follows that x 1 w r d As in [13], define the polynomial φx := px 2.5 = α In,d α In,d α In,d αi βi. r + d 1 p α x 1 α1 1 x n αn 1. p α x 1 α1 1 x n αn 1 p α x α x 1 α1 1 x n αn 1

7 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 7 and set Then, φ max := max φx. x 2.6 p r = 1 px φx. w r d x This implies: and thus, as p p p r, p r 1 w r d p φ max 2.7 p r 1 w r d p φ max, 2.8 p w r dp + φ max. Therefore, 2.9 p p r 1 1 p + 1 w r d w r d φ max, 2.10 p p w r d 1 1 p + 1 w r d w r d φ max. We can now prove the results from Theorems 1.1, 1.2, 1.3, 1.4, and relation Proving Theorem 1.2. As x α n x i αi 1 and p α p 0 max d! α! by the definition 1.9 of p 0 max, we find that φx p 0 max α In,d d! x α α! n x i αi 1. In view of relations 2.1 and 2.2, the right hand side is equal to d d p 0 max x i x i = p 0 max1 w r d. i i 1 Therefore, 2.11 φ max p 0 max1 w r d. This inequality, combined with 2.9 and 2.10, gives the inequalities 1.21 and 1.22 from Theorem Proving Theorem 1.1. Assume that p > 0 and r d max 2 p d. Then, relation 2.11 combined with the bound on w r d from 1.20 implies that φ max p 0 max d 2 1. Now, 2.7 implies that pr 1 w rd nonnegative by our assumption on r. This shows that p r Theorem 1.1 holds. p 0 p p 0 max d 2 1, which is 0 for such r; that is,

8 8 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO Proving the inequality 1.7 for polynomials that are sums of square-free monomials. Assume that p α = 0 whenever α i 2 for some i = 1,...,n. Then the polynomial φx from 2.5 is identically zero. Thus φ max = 0 and the estimate 1.7 follows directly from 2.10 with k = r + d and using Proving Theorem 1.3 for degree 2 forms. By looking more closely at the form of the function φx, one can give an upper bound for φ max depending on p max and p only. Indeed, when d = 2, one can verify that φx = 1 p 2ei x i. r + 2 Therefore, 2.12 φ max = 1 r + 2 max,...,n p 2e i 1 r + 2 p max. As w r 2 = r+1 r+2, together with 2.9 and 2.10, this implies that the inequalities 1.23 and 1.24 from Theorem 1.3 hold. Moreover, p r 0 for r maxi p2e i p 2. That is, for degree 2 forms, Theorem 1.1 remains valid for such r instead of Proving Theorem 1.4 for degree 3 forms. When d = 3, one can verify that n 2.13 φx = p 3ei 3tx 2 i 2t 2 x i + p 2ei+e j + p ei+2e j tx i x j, i 1 i<j n after setting t := 1 r+3. Evaluating p at the simplex points e i and 1 2 e i + e j yields, respectively, the relations: 2.14 p pe i = p 3ei p max, 2.15 p 3ei + p 3ej + p 2ei+e j + p ei+2e j 8p max. Using 2.15, we can bound the second sum in 2.13: i<j p 2e i+e j + p ei+2e j x i x j i<j 8p max p 3ei p 3ej x i x j = 8p max i<j x ix j i p 3e i x i 1 x i. Therefore, φx 8tp max x i x j + 4t i<j i p 3ei x 2 i t1 + 2t i p 3ei x i. Using the fact that p 3ei p max and i x i = 1, the sum of the first two terms can be bounded by 4tp max. Using the fact that p 3ei p, the third term can be bounded by t1 + 2tp = r+5 r+3 2 p. This shows: 2.16 φ max 4 r + 3 p max r + 5 r p. Together with 2.9, 2.10, and the fact that w r 3 = r+1r+2 r+3, this implies that 2 the relations 1.25 and 1.26 from Theorem 1.4 hold. To conclude, let us mention that it is not clear whether this type of argument for bounding φ max in terms of p max and p extends for forms of degree 4. We use in the next subsection a different argument for dealing with the general case d 4.

9 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX Estimating the maximum coefficient range of a polynomial - Proof of Theorem 1.5. In this section we prove the estimate of p 0 max p 0 in terms of p max p given in Theorem 1.5. Following Reznick [15], let us introduce the following definitions that will be useful for us. Recall that In,d = {α Z n + : α = d} and let F n,d denote the set of forms of degree d in n variables. For p F n,d, write px = p α x α = ap,α d! α! xα, after setting α In,d α! ap,α := p α d! For α R n, define the degree d form: α In,d for α In,d. P α x := α T x d for x R n. Define the inner product on F n,d : p,q := ap,αaq,α d! α! α In,d for p,q F n,d. As P α x = β In,d d! β! αβ x β, it follows that, for any p F n,d, 2.17 p,p α = pα for α R n. Moreover, 2.18 p,x α = ap,α for α In,d. Finally, given α In,d, define the polynomials: 2.19 h α x := n α j 1 j=1 l j=0 dx j l j x x n, h αx := 1 α!d d h αx. Lemma 2.1. [15] For α,α In,d, h α,p α = 1 if α = α and 0 otherwise. Proof. Direct verification. Corollary 2.2. Biermann s theorem, see [15, 2] The set {P α α In,d} is a basis of the vector space F n,d. Let A be the In, d In, d matrix permitting to express the monomial basis {x α α In,d} in terms of the basis {P β β In,d}. That is, 2.20 x α = Aα,βP β x. β In,d For p F n,d, by taking the inner product in 2.20 with p and using 2.17 and 2.18, we find: 2.21 ap,α = Aα, βpβ for α In, d. β In,d Taking the inner product in 2.20 with h β, we find: 2.22 ah β,α = h β,x α = Aα,β for α,β In,d.

10 10 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO In view of 1.18, our parameters p 0 max and p 0 are given by: p 0 max = max ap,α, p0 = ap,α. α In,d α In,d Define the vectors x := pα α In,d and y := ap,α α In,d. In view of 2.9, they are related by the relation: 2.23 y = Ax. Denote by x max resp., x the largest entry of x; similarly for y. Thus, For α In,d, pα = pα/dd d. Thus, y max y = p 0 max p x max x d d p max p. Our strategy for proving Theorem 1.5 consists of showing the following two results: Proposition 2.3. Let A be an N N matrix satisfying Ae = µe for some scalar µ, where e denotes the all-ones vector, and set A := max,...,n k=1 N Ai, k. If y = Ax, then y max y A x max x. Proposition 2.4. Let A be the matrix defined in 2.22; that is, A = Aα,β := ah β,α α,β In,d. Then, Ae = 1 e and A d d 2d 1 d Proof of Proposition 2.3. Assume y = Ax where A = a ik N i,k=1. At row i, y i = N k=1 a ikx k. Thus, y i x max a ik x = r + i x max r i x, after setting Similarly, k a ik 0 a ik r + i := k a ik 0 Therefore, for any i,j = 1,...,N, k a ik 0 a ik, r i := k a ik 0 y j r + j x r j x max. a ik. y i y j r + i + r j x max r i + r + j x. Note that ax max bx a+b 2 x max x if and only if b ax max +x 0. Here, a = r + i + r j, b = r i + r + j, b a = k a jk k a ik = 0, and b + a = k a ik + a jk. Therefore, y i y j 1 N a ik + a jk x max x 2 k=1

11 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 11 and y j y i 1 N a ik + a jk x max x. 2 k=1 This implies that, for any i,j, y i y j 1 N a ik + a jk x max x max a ik x max x ; 2 i that is, k=1 y max y A x max x Proof of Proposition 2.4. Let us first prove that Ae = d d e. For this consider the polynomial px := n x i d. Then, ap,α = 1 and pα = d d for all α In,d. Thus, x := pα α In,d = d d e and y := ap,α α In,d = e. As y = Ax by 2.23, it follows that Ae = d d e. Recall that Aα,β = ah β,α = ah β,α 1. Thus, Aα,β is equal to the β!d d coefficient of x α in h β x scaled by the factor α! 1 d!. We proceed in two steps for β!d d proving that 2d 1 A = max Aα,β. α In,d d β In,d 1 First, we show that each entry of A is bounded in absolute value by 1. 2 Second, we show that there are at most 2d 1 d nonzero entries in any row of A. Those two facts imply obviously the desired result. Step 1. Bounding the entries of A. By definition, h β x is defined as the product of d linear forms: f 1 x = n f 1ix i,..., f d x = n f dix i. Thus, h β x =... f 1i1 f did x i1 x id =: s α x α, i 1=1,...,n i d =1,...,n k α In,d where s α = f 1i1 f did and the summation is over all d-tuples i 1,...,i d {1,...,n} d containing 1 exactly α 1 times, 2 exactly α 2 times,..., n exactly α n times. First of all, each product f 1i1 f did is bounded in absolute value by d d. Indeed, the linear forms f j x are of the form: d lx 1 lx 2... lx n ; thus their coefficients belong to { d, d + 1,...,0,1,...,d}. Let us now count the number of terms in the summation defining s α. It is equal to d α 1 d α1 α 2 d α1... α n 1 α n, which is equal to d! α!. Summarizing, we find that s α d d d! α!. Hence, Aα,β = s α α! 1 d! 1 β!d d β! 1. Step 2. Bounding the number of nonzero entries in a row of A. Write h β x = n j=1 P jx, where P j x = β j 1 l j=0 d l j x j,...,n, i j l j x i. Fix α In,d and consider the α-th row of A. We want to bound the number of β s for which Aα,β 0; that is, the number of β s for which x α occurs with a nonzero coefficient in h β x.

12 12 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO Consider some β for which Aα,β 0. Say, suppβ = {1,...,t}; that is, β 1 1,...,β t 1,β t+1 =... = β n = 0. Then, P j x = 1 for j = t + 1,...,n and β j 1 P j x = dx j d l j x j l j x i for j = 1,...,t. Therefore, h β x = d t t j=1 x j l j=1 t β j 1 j=1 l j=1 d l j x j,...,n, i j,...,n, i j l j x i. Hence, if x α has a nonzero coefficient in h β x, then necessarily α 1 1,...,α t 1; that is, the support of β is contained in the support of α. Therefore, the number of β s for which Aα,β 0 is bounded by the number of sequences β In,d with suppβ suppα, which is equal to s+d 1 d, setting s := suppα. As α = d, s d and thus s+d 1 d 2d 1 d. 3. A PTAS for the imization of forms on the simplex Consider the generic optimization problem: φ max := max {fx : x S}, for some continuous f : IR n IR and compact convex set S, and let φ := {fx : x S}. Definition 3.1 Nesterov et al. [8]. A value ψ µ approximates φ with relative accuracy µ 0, 1] if ψ µ φ µφ max φ. Then one also says that ψ µ is a µ-approximation of p. The approximation is called implementable if ψ µ = fx for some x S. Definition 3.2 PTAS. If a problem allows an implementable approximation ψ µ = fx µ for each µ 0,1], such that x µ S can be computed in time polynomial in n and the bit size required to represent f, we say that the problem allows a polynomial time approximation scheme PTAS. For the problem of imizing a form over the simplex, this definition may be summarized as follows. Definition 3.3. Consider the problem 1.1 of imizing a degree d form p on the standard simplex. A PTAS for this problem exists if, for every ǫ > 0, there is an algorithm that returns a solution x satisfying 3.1 px p p max p ǫ in time polynomial in n and the bit size of the coefficients of p. Note that the approximation results in Theorem 1.2 do not constitute a PTAS, since it is not clear how to bound p 0 max p in terms of p max p. One reason for adopting Definition 3.3 is that, in general, nothing is known about the signs of p and p max. Let us mention a variation of this definition that could be considered as well.

13 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 13 Definition 3.4. A PTAS of type 2 for the problem 1.1 of imizing a form px on the standard simplex exists if, for every ǫ > 0, there is an algorithm that returns a solution x satisfying 3.2 px p ǫ p, i.e., px p 1 + ǫ signp. in time polynomial in n and the bit size of the coefficients of p. This new definition is stronger, i.e., relation 3.2 implies relation 3.1, when p 0 p max and when 0 2p p max. Note that Definition 3.4 corresponds to the classic definition used, e.g., for combinatorial optimization problems, where the signs of the imum and maximum are often known in advance. Consider, for instance, the unweighted max-cut problem: 3.3 mcg := max x {±1} n p 0x := x i x j for a graph G = V,E V = n. Then, a PTAS for the max-cut problem should, for every ǫ > 0, return a point x {±1} n for which ij E 3.4 px 1 ǫmcg in time polynomial in n. Similarly, a PTAS for the maximum stable set problem should, for every ǫ > 0, return a stable set S satisfying S 1 ǫαg. As an application of Theorem 1.5 or Theorems 1.3 and 1.4 in the case d = 2,3, we find: Theorem 3.5. There is a PTAS using Definition 3.3 for the problem of imizing a form of degree d 2 over the simplex. On the other hand, if we use Definition 3.4 instead of Definition 3.3, the above result does not remain valid. Theorem 3.6. There is no PTAS of type 2 using Definition 3.4 for the problem of imizing a form of degree d 2 over the simplex. Proof. It suffices to show the result for degree d = 2. The proof is based on a reduction from the maximum stable set problem. Given a graph G = V, E with adjacency matrix A, consider the quadratic polynomial px := x T I + Ax. By Motzkin-Straus theorem, the imum of px over is 1 αg, where αg is the maximum cardinality of a stable set in G. Thus, p = 1 αg > 0 and p max 2p if αg 2, since p max 1 = pe i. Lemma 3.7. Given x, one can construct a stable set S for which 1 S px in time polynomial in n. Proof. The proof is based on the same argument used for proving Motzkin-Straus theorem. Let T denote the support of x. First we construct another point x whose support is stable and such that px px. If T is stable, let x := x. Suppose that T contains two adjacent nodes, say, nodes 1 and 2. Consider the function fx 1,x 2 := px 1,x 2,x 3,...,x n in two variables x 1,x 2. For any point x 1,x 2 0 := {x 1,x 2 x 1,x 2 0, x 1 + x 2 = 1 i 3 x i }, fx 1,x 2 has the form ax 1 + bx 2 + c where a,b,c are constants depending on x 3,...,x n. As f is linear, it attains its imum on the segment 0 at one of its extremities, i.e.,

14 14 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO at x 1,x 2 with x 1 = 0 or x 2 = 0. Thus one can construct a new point x such that px px, whose support is contained in T and does not contain {1,2}. Iterating, we find a point x whose support S is stable and such that px px. Now, px = i S x2 i which, using Cauchy-Schwartz inequality, implies that px 1 S. Assume that there is a PTAS of type 2, using Definition 3.4, for the problem of imizing a quadratic form on the simplex. Let ǫ be given such that 0 < ǫ < 1 and set ǫ := ǫ 1 ǫ. Then one can construct in polynomial time a point x satisfying 3.2, i.e., px 1 αg 1+ǫ. By Lemma 3.7, one can construct in polynomial time a stable set S such that 1 S px 1 αg 1+ǫ, which implies S αg1 ǫ. This shows therefore the existence of a PTAS for the maximum stable set problem, contradicting the inapproximability result of Arora et al. [1]. 4. Concluding Remarks 4.1. A probabilistic approach for estimating the grid bounds p k. Nesterov [9] proposes an alternative probabilistic argument for estimating the quality of the bounds p k. He introduces a random walk on the simplex, which generates a sequence of random points x k k 0 in the simplex with the property that x k k. Thus the expected value Epx k of the evaluation of the polynomial px at x k satisfies: Epx k p k. Hence upper bounds for p k can be obtained by bounding Epx k. Fix a point q to be chosen later as a global imizer of the polynomial px over. Let ζ be a discrete random variable with values in {1,...,n} distributed as follows: 4.1 Probζ = i = q i i = 1,...,n. Consider the random process: y 0 = 0 IR n, y k+1 = y k + e ζk k 0 where ζ k are random independent variables distributed according to 4.1. In other words, y k+1 is y k + e i with probability q i. Finally, define x k = 1 k y k k 1. Thus all x k lie in the set k. In order to evaluate Epx k, one needs to compute Ex β k for any monomial with β = d. One can verify that Ex β k = 1 Ey β k d k, with Ey β k = Proby k 1 = α 1,...,y k n = α n α β = k! α! qα α β. α N n, α =k α =k Here, y k i is the i-th coordinate of y k and recall that q α = q α1 1 qαn α β1 1 αβn n. The following computations are given in [9]: Ex k i = q i, Ex k i 2 = 1 k q i + Ex k ix k j = 1 1 k 1 1 q i q j i j, k q 2 i, n and α β =

15 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 15 Ex β k = k! k d!k d qβ if β {0,1} n with β = d. Assume now that q is a global imizer of px over. If px is a sum of square-free monomials of degree d, then Epx k = p β Ex β k = pq k! k d!k d = p w r d, β =d with k = r + d. This gives the estimate 1.7 [9], Lemma 3. If px is a form of degree 2, then Epx k = 1 p 2ei q i pq = w r 2p + 1 n p 2ei q i, k k r + 2 i which gives the estimate 1.6 [9], Theorem 2. Remark 4.1. In these two cases sum of square-free monomials, or degree 2, it turns out that Epx k = w r dp + φq. Hence, the upper bound w r dp + φ max for p recall 2.8 remains an upper bound for Epx k. The identity Epx k = w r dp + φq is not true when d = 3. When d = 3, one can verify that Epx k = w r 3p + φq + φ q, where 2 1 φ q := 3 p 3ei q i 1 q i 2ei+e r + 3 j + p ei+2e j q i q j. i i<jp One can verify that φ q 4 r+3 p 2 max p. Combined with 2.16, this implies that 4 Epx k p + r r p max p Approximating polynomials over polytopes. As observed by Nesterov [9], some results for the simplex can be extended to the problem of imizing a degree d form px over a polytope P := convu 1,...,u N where u 1,...,u N IR n. Indeed, if U denote the n N matrix with columns u 1,...,u N, then imizing the polynomial in n variables px over P is equivalent to imizing the polynomial in N variables px := pux over the standard simplex in IR N. Thus, p,p := px = px x P x and, for an integer k 1, one can define the grid approximation: N p Pk := p k = p x i u i. x k The bounds obtained earlier for p k translate into bounds for p Pk. For instance, when px has degree 2, p Pk p,p 1 max k pu i p,p.,...,n

16 16 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO When px is a sum of square-free monomials, p Pk p,p dd 1 p,p. 2k Let us point out, however, that the complexity of computing the parameter p Pk depends on the number N of vertices, which can be exponentially large in terms of the number n of variables. Observe that the problem of maximizing a quadratic form over the cube [ 1,1] n is NP-hard and no PTAS can exist, since it contains the max-cut problem. Indeed, given a graph G = V,E, define its Laplacian matrix L as the V V matrix with entries L ii := degi i V and L ij := 1 if i j are adjacent, L ij := 0 otherwise. Then, 1 1 mcg = max x {±1} n 4 xt Lx = max x [ 1,1] n 4 xt Lx, where the last equality follows from the fact that L Semidefinite approximations. Stronger semidefinite bounds can be defined for the imum p of a degree d form px over the standard simplex. For this, if px = α p αx α, consider the even polynomial px := α p α x 2α. The problem of imizing px over the simplex is equivalent to the problem of imizing px over the unit sphere S := {x IR n n x2 i = 1}; that is, p = x S px. Let Σ 2 denote the set of polynomials in IR[x 1,...,x n ] that can written as a sum of squares of polynomials. Given an integer r 0, define the parameter: n r 4.2 p r,sos := max λ for which n d x 2 i px λ x 2 i Σ 2. If the polynomial i x i r px λ n x i d has nonnegative coefficients, then the polynomial i x2 i r px λ n d i x2 is obviously a sum of squares. Therefore, p r pr,sos p for all r 0. The bound p r,sos can be computed in polynomial time with an arbitrary precision for any fixed r. This follows from the well known fact see, e.g., [14] that testing whether a polynomial can be written as a sum of squares of polynomials can be formulated as a semidefinite program. As a consequence of Pólya s theorem, the semidefinite bounds p r,sos converge to p as r. Schmüdgen [17] proved in a more general context that every polynomial which is positive on the unit sphere S has a representation of the form s 0 x + 1 i x2 i s 1x, where s 0 x Σ 2 and s 1 x IR[x 1,...,x n ]. This fact motivates the definition of the following alternative semidefinite lower bound for p, for any integer r 0: 4.3 max λ such that px λ = s 0 x + 1 n x2 i s 1x where s 0 Σ 2, s 1 IR[x 1,...,x n ], degs 0 2r + d

17 APPROXIMATING POLYNOMIALS OVER THE SIMPLEX 17 It follows from Schmüdgen s theorem that these bounds also converge to p as r. In fact, De Klerk et al. [4] show that the bounds 4.3 coincide with the semidefinite bounds p r,sos. That is, both approaches based on Pólya s result and on Schmüdgen s result yield the same hierarchies of semidefinite bounds for the problem of imizing a form on the simplex Optimizing polynomials over the unit sphere. We group here a few observations about the complexity of optimizing a form over the sphere. As is well-known, imizing a quadratic form over the unit sphere is an easy problem, as it amounts to computing the imum eigenvalue of a matrix, a problem for which efficient algorithms exist. As we saw in the previous subsection, the problem of imizing an even form on the unit sphere can be reformulated as the problem of imizing an associated form on the simplex. Hence upper and lower bounds are available as well as good estimates on their quality. On the other hand, Nesterov [9] shows that maximizing a cubic form on the unit sphere is a NP-hard problem, using a reduction from the maximum stable set problem. Let us finally mention a result of Faybusovich [5] about the quality of the semidefinite bounds for the optimization of forms on the unit sphere. Let px be a form of even degree 2d, let S denote the unit sphere, and set p,s := x S px, p max,s := max x S px. For an integer r 0, define the parameter n r n d p r S := max λ s.t. x 2 i px λ x 2 i Σ 2. Thus, p r S p,s for all r 0. Using a result of Reznick [16], Faybusovich [5] shows that, for r 2nd2d 1 4 ln 2 n 2 d, p,s p r S 2nd2d 1 2ln 22r + n + 2d 2nd2d 1 p max,s p,s. This does not yield a PTAS, since this estimate holds only for r = On. It remains an open problem whether optimization of a fixed degree form over the unit sphere allows a PTAS. References [1] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedi. Proof verification and intractability of approximations problems. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, IEEE Computer Science Press, Los Alamitos, CA, pp 14 23, [2] P. Berman and M. Karpinski. On some tighter inapproximability results, further improvements. Electronic Colloquium on Computational Complexity, Report TR98-065, [3] I. Bomze and E. De Klerk. Solving standard quadratic optimization problems via linear, semidefinite and copositive programg. Global Optimization, 242: , [4] E. De Klerk, M. Laurent, and P. Parrilo. On the equivalence of algebraic approaches to the imization of forms on the simplex. In Positive Polynomials in Control, D. Henrion and A. Garulli, eds., LNCIS, Springer, forthcog. [5] L. Faybusovich. Global optimization of homogeneous polynomials on the simplex and on the sphere. In C. Floudas and P. Pardalos, editors, Frontiers in Global Optimization. Kluwer Academic Publishers, 2003.

18 18 ETIENNE DE KLERK, MONIQUE LAURENT, AND PABLO PARRILO [6] J. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J.Optim., 113: , [7] T.S. Motzkin and E.G. Straus. Maxima for graphs and a new proof of a theorem of Túran. Canadian J. Math., 17: , [8] Yu. Nesterov, H. Wolkowicz, and Y. Ye. Semidefinite programg relaxations of nonconvex quadratic optimization. In H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors, Handbook of semidefinite programg, pages Kluwer Academic Publishers, Norwell, MA, [9] Yu. Nesterov. Random walk in a simplex and quadratic optimization over convex polytopes. CORE Discussion Paper 2003/71, CORE-UCL, Louvain-La-Neuve, September [10] P.A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, May [11] P.A. Parrilo. Semidefinite programg relaxations for semialgebraic problems. Mathematical Programg, Ser. B, 96: , [12] G. Pólya. Collected Papers, volume 2, pages MIT Press, Cambridge, Mass., London, [13] V. Powers and B. Reznick. A new bound for Pólya s theorem with applications to polynomials positive on polyhedra. Journal of Pure and Applied Algebra, 164: , [14] V. Powers and T. Wörmann. An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Algebra, 127:99 104, [15] B. Reznick. Sums of even powers of real linear forms. Memoirs of the American Mathematical Society, Number 463, [16] B. Reznick. Uniform denoators in Hilbert s seventeenth problem. Mathematische Zeitschrift, 220:75 97, [17] K. Schmüdgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289: , [18] L.F. Zuluaga, J.C. Vera, and J.F. Peña. LMI approximations for cones of positive semidefinite forms. Preprint, E. De Klerk, University of Waterloo address: edeklerk@math.uwaterloo.ca M. Laurent, CWI, Amsterdam address: M.Laurent@cwi.nl P. Parrilo, Swiss Federal Institute of Technology, Zürich address: parrilo@control.ee.ethz.ch