Mathematical Programming (2022) 193:665–685

https://doi.org/10.1007/s10107-019-01465-1

FULL LENGTH PAPER

Series B

Convergence analysis of a Lasserre hierarchy of upper

bounds for polynomial minimization on the sphere

Etienne de Klerk1 · Monique Laurent1,2

Received: 18 April 2019 / Accepted: 31 December 2019 / Published online: 21 January 2020
© The Author(s) 2020

Abstract
We study the convergence rate of a hierarchy of upper bounds for polynomial mini-
mization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), for
the special case when the feasible set is the unit (hyper)sphere. The upper bound at
level r ∈ N of the hierarchy is defined as the minimal expected value of the poly-
nomial over all probability distributions on the sphere, when the probability density
function is a sum-of-squares polynomial of degree at most 2r with respect to the sur-
face measure. We show that the rate of convergence is O(1/r 2 ) and we give a class of
polynomials of any positive degree for which this rate is tight. In addition, we explore
the implications for the related rate of convergence for the generalized problem of
moments on the sphere.

Keywords Polynomial optimization on sphere · Lasserre hierarchy · Semidefinite

programming · Generalized eigenvalue problem

Mathematics Subject Classification 90C22 · 90C26 · 90C30

1 Introduction

We consider the problem of minimizing an n-variate polynomial f : Rn → R over a

compact set K ⊆ Rn , i.e., the problem of computing the parameter:

f min,K := min f (x). (1)

x∈K

B Etienne de Klerk
E.deKlerk@uvt.nl
Monique Laurent
monique@cwi.nl

1 Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands

2 Centrum Wiskunde & Informatica (CWI), Postbus 94079, 1090 GB Amsterdam, The Netherlands

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666 E. de Klerk, M. Laurent

In this paper we will focus on the case when K is the unit sphere: K = Sn−1 =
{x ∈ Rn : x = 1}. Here and throughout, x denotes the Euclidean norm for real
vectors. When considering K = Sn−1 , we will omit the subscript K and simply write
f min = min x∈Sn−1 f (x).
Problem (1) is in general a computationally hard problem, already for simple sets
K like the hypercube, the standard simplex, and the unit ball or sphere. For instance,
the problem of finding the maximum cardinality α(G) of a stable set in a graph
G = ([n], E) can be expressed as optimizing a quadratic polynomial over the standard
simplex [18], or a degree 3 polynomial over the unit sphere [19]:
 n 
1
= minn x (I + A G )x : x ≥ 0,
T
xi = 1
α(G) x∈R
i=1
⎛ ⎞
 
= min ⎝ yi2 y 2j + yi4 ⎠ ,
y∈Sn−1
i= j:{i, j}∈E i∈[n]
√
2 1 
√ 1− = max yi y j z i j ,
3 3 α(G) (y,z)∈Sn+m−1
i j∈E

where A G is the adjacency matrix of G, E is the set of non-edges of G and m = |E|.

Other applications of polynomial optimization over the unit sphere include deciding
whether homogeneous polynomials are positive semidefinite. Indeed, a homogeneous
polynomial f is defined as positive semidefinite precisely if

f min = min f (x) ≥ 0,

x∈Sn−1

and positive definite if the inequality is strict; see e.g. [22]. As special case, one may
decide if a symmetric matrix A = (ai j ) ∈ Rn×n is copositive, by deciding if the
associated form f (x) = i, j∈[n] ai j xi2 x 2j is positive semidefinite; see, e.g. [20].
Another special case is to decide the convexity of a homogeneous polynomial f ,
by considering the parameter

min y T ∇ f (x)y,
(x,y)∈S2n−1

which is nonnegative if and only if f is convex. This decision problem is known to be

NP-hard, already for degree 4 forms [1].
As shown by Lasserre [16], the parameter (1) can be reformulated via the infinite
dimensional program

f min,K = inf h(x) f (x)dμ(x) s.t. K h(x)dμ(x) = 1, (2)

h∈Σ[x] K

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Convergence analysis of a Lasserre hierarchy of upper… 667

where Σ[x] denotes the set of sums of squares of polynomials, and μ is a given Borel
measure supported on K . Given an integer r ∈ N, by bounding the degree of the
polynomial h ∈ Σ[x] by 2r , Lasserre [16] defined the parameter:

(r )
f K := min h(x) f (x)dμ(x) s.t. h(x)dμ(x) = 1, (3)
h∈Σ[x]r K K

where Σ[x]r consists of the polynomials in Σ[x] with degree at most 2r . Here we use
the ‘overline’ symbol to indicate that the parameters provide upper bounds for f min,K ,
in contrast to the parameters f (r ) in (9) below, which provide lower bounds for it.
Since sums of squares of polynomials can be formulated using semidefinite pro-
gramming, the parameter (3) can be expressed via a semidefinite program. In fact,
since this program has only one affine constraint, it even admits an eigenvalue refor-
mulation [16], which will be mentioned in (12) in Sect. 2.2 below. Of course, in order
to be able to compute the parameter (3) in practice, one needs to know explicitly
(or via some computational procedure) the moments of the reference measure μ on
K . These moments are known for simple sets like the simplex, the box, the sphere,
the ball and some simple transforms of them (they can be found, e.g., in Table 1 in
[9]).
(r )
As a direct consequence of the formulation (2), the bounds f K converge asymp-
totically to the global minimum f min,K when r → ∞. How fast the bounds converge
to the global minimum in terms of the degree r has been investigated
√ in the papers
[7,8,11], which show, respectively, a convergence rate in O(1/ r ) for general compact
K (satisfying a minor geometric condition, implying a nonempty interior), a conver-
gence rate in O(1/r ) when K is a convex body, and a convergence rate in O(1/r 2 )
when K is the box [−1, 1]n . In these works the reference measure μ is the Lebesgue
measure, except for the box [−1, 1]n where more general measures are considered
(see Theorem 3 below for details).
The convergence rates in [7,11] are established by constructing an explicit sum of
squares h ∈ Σ[x]r , obtained by approximating the Dirac delta at a global minimizer
a of f in K by a suitable density function and considering a truncation of its Taylor
expansion. Roughly speaking, a Gaussian density of the form exp(−x − a2 /σ 2 )
(with σ ∼ 1/r ) is used in [11], and a Boltzman density of the form exp(− f (x)/T )
(with T ∼ 1/r ) is used in [7] (and relying on a result of [14] about simulated annealing
for convex bodies). For the box K = [−1, 1]n , the stronger analysis in [8] relies on an
eigenvalue reformulation of the bounds and exploiting links to the roots of orthogonal
polynomials (for the selected measure), as will be briefly recalled in Sect. 2.2 below.
These results do not apply to the sphere, which has an empty interior and is not a convex
body. Nevertheless, as we will see in this paper, one may still derive information for
the sphere from the analysis for the interval [−1, 1].
In this paper we are interested in analyzing the worst-case convergence of the
bounds (3) in the case of the unit sphere K = Sn−1 , when selecting as reference
measure the surface (Haar) measure dσ (x) on Sn−1 . We let σn−1 denote the surface
measure of Sn−1 , so that dσ (x)/σn−1 is a probability measure on Sn−1 , with

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668 E. de Klerk, M. Laurent

n
2π 2
σn−1 := dσ (x) = . (4)
Sn−1 Γ n2

(See, e.g., [6, relation (2.2.3)].) To simplify notation we will throughout omit the
subscript K = Sn−1 in the parameters (1) and (3), which we simply denote as

f min = min f (x),

x∈Sn−1
 
(r )
f = inf h(x) f (x)dσ (x) : h(x)dσ (x) = 1 . (5)
h∈Σ[x]r Sn−1 Sn−1

Example 1 Consider the minimization of the Motzkin form

f (x1 , x2 , x3 ) = x36 + x14 x22 + x12 x24 − 3x12 x22 x32

on S2 . This form has 12 minimizers on the sphere, namely √1 (±1, ±1, ±1) as well
3
as (±1, 0, 0) and (0, ±1, 0), and one has f min = 0.
(r )
In Table 1 we give the bounds f for the Motzkin form for r ≤ 9. In Fig. 1 we
show contour plots of the optimal density function for r = 3, r = 6, and r = 9. In the
figure, the red end of the spectrum denotes higher function values.
When r = 3 and r = 6, the modes of the optimal density are at the global min-
imizers (±1, 0, 0) and (0, ±1, 0) (one may see the contours of two of these modes
in one hemisphere). On the other hand, when r = 9, the mass of the distribution is
concentrated at the 8 global minimizers √1 (±1, ±1, ±1) (one may see 4 of these
3
in one hemisphere), and there are no modes at the global minimizers (±1, 0, 0) and
(0, ±1, 0).

Table 1 Upper bounds for the Motzkin form

r 0 1 2 3 4 5 6 7 8 9

(r )
f 0.1714 0.0952 0.0519 0.0457 0.0287 0.0283 0.0193 0.0177 0.0139 0.0122

Fig. 1 Contour plots of the optimal density for r = 3, r = 6, and r = 9

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Convergence analysis of a Lasserre hierarchy of upper… 669

Fig. 2 Plots of the optimal density for r = 3 (top left), r = 6 (top right), and r = 9 (bottom), in spherical
coordinates

It is also illustrative to do the same plots using spherical coordinates:

x1 = sin θ sin φ
x2 = sin θ cos φ
x3 = cos θ
θ ∈ [0, π ]
φ ∈ [0, 2π ].

In Fig. 2 we plot the optimal density function that corresponds to r = 3 (top right),
r = 6 (bottom left), and r = 9 (bottom right). For example, when r = 9 one can
see the 8 modes (peaks) of the density that correspond to the 8 global minimizers
√1 (±1, ±1, ±1). (Note that the peaks at φ = 0 and φ = 2π correspond to the same
3
mode of the density, due to periodicity.) Likewise when r = 3 and r = 6 one may see
4 modes corresponding to (±1, 0, 0) and (0, ±1, 0).
(r )
The convergence rate of the bounds f was investigated by Doherty and Wehner
[4], who showed
 
(r ) 1
f − f min = O (6)
r

when f is a homogeneous polynomial. As we will briefly recap in Sect. 2.1, their

result follows in fact as a byproduct of their analysis of another Lasserre hierarchy of
bounds for f min , namely the lower bounds (9) below.

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670 E. de Klerk, M. Laurent

Our main contribution in this paper is to show that the convergence rate of the
(r )
bounds f is O(1/r 2 ) for any polynomial f and, moreover, that this analysis is
tight for any (nonzero) linear polynomial f (and some powers). This is summarized
in the following theorem, where we use the usual Landau notation: for two functions
f 1 , f 2 : N → R+ , then
f 1 (r )
f 1 = Ω( f 2 ) ⇐⇒ lim inf > 0.
r →∞ f 2 (r )

Theorem 1 (i) For any polynomial f we have

 
(r ) 1
f − f min = O . (7)
r2

(ii) For any polynomial f (x) = (−1)d−1 (c T x)d , where c ∈ Rn \ {0} and d ∈ N,
d ≥ 1, we have
 
(r ) 1
f − f min = Ω . (8)
r2

Let us say a few words about the proof technique. For the first part (i), our analysis
relies on the following two basic steps: first, we observe that it suffices to consider the
case when f is linear (which follows using Taylor’s theorem), and then we show how
to reduce to the case of minimizing a linear univariate polynomial over the interval
[−1, 1], where we can rely on the analysis completed in [8]. For the second part (ii),
by exploiting a connection recently mentioned in [17] between the bounds (3) and
cubature rules, we can rely on known results for cubature rules on the unit sphere to
show tightness of the bounds.
Organization of the paper In Sect. 2 we recall some previously known results that
are most relevant to this paper. First we give in Sect. 2.1 a brief recap of the approach
of Doherty and Wehner [4] for analysing bounds for polynomial optimization over the
unit sphere. After that, we recall our earlier results about the quality of the bounds (3)
in the case of the interval K = [−1, 1]. Section 3 contains our main results about the
convergence analysis of the bounds (3) for the unit sphere: after showing in Sect. 3.1
that the convergence rate is in O(1/r 2 ) we prove in Sect. 3.2 that the analysis is tight
for nonzero linear polynomials (and their powers).

2 Preliminaries

2.1 The approach of Doherty and Wehner for the sphere

Here we briefly sketch the approach followed by Doherty and Wehner [4] for showing
the convergence rate O(1/r ) mentioned above in (6). Their approach applies to the
case when f is a homogeneous polynomial, which enables using the tensor analysis
framework. A first and nontrivial observation, made in [4, Lemma B.2], is that we

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Convergence analysis of a Lasserre hierarchy of upper… 671

may restrict to the case when f has even degree, because if f is homogeneous with
odd degree d then we have

d d/2
max f (x) = max xn+1 f (x).
x∈Sn−1 (d + 1)(d+1)/2 (x,xn+1 )∈Sn

So we now assume that f is homogeneous with even degree d = 2a.

The approach in [4] in fact also permits to analyze the following hierarchy of lower
bounds on f min :

f (r ) := sup λ s.t. f (x) − λ ∈ Σ[x]r + (1 − x2 )R[x], (9)

λ∈R

which are the usual sums-of-squares bounds for polynomial optimization (as intro-
duced in [15,21]).
One can verify that (9) can be reformulated as

f (r ) = sup λ s.t. ( f (x) − λx2a )x2r −2a ∈ Σ[x]r + (1 − x2 )R[x]

λ∈R
(10)
= sup λ s.t. f (x)x2r −2a − λx2r ∈ Σ[x]
λ∈R

(see [10]). For any integer r ∈ N we have

(r )
f (r ) ≤ f min ≤ f .

(r )
The following error estimate is shown on the range f − f (r ) in [4].

Theorem 2 [4] Assume n ≥ 3 and f is a homogeneous polynomial of degree 2a.

There exists a constant Cn,a (depending only on n and a) such that, for any integer
r ≥ a(2a 2 + n − 2) − n/2, we have

(r ) Cn,a
f − f (r ) ≤ ( f max − f min ),
r

where f max is the maximum value of f taken over Sn−1 .

The starting point in the approach in [4] is reformulating the problem in terms
of tensors. For this we need the following notion of ‘maximally symmetric matrix’.
Given a real symmetric matrix M = (Mi, j ) indexed by sequences i ∈ [n]a , where
[n] := {1, . . . , n}, M is called maximally symmetric if it is invariant under action of the
permutation group Sym(2a) after viewing M as a 2a-tensor acting on Rn . This notion
is the analogue of the ‘moment matrix’ property, when expressed in the tensor setting.
To see this, for a sequence i = (i 1 , . . . , i a ) ∈ [n]a , define α(i) = (α1 , . . . , αn ) ∈ Nn
by letting α denote the number of occurrences of within the multi-set {i 1 , . . . , i a }
n
for each ∈ [n], so that a = |α| = i=1 αi . Then, the matrix M is maximally
symmetric if and only if each entry Mi, j depends only on the n-tuple α(i) + α( j).

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672 E. de Klerk, M. Laurent

Following [4] we let MSym((Rn )⊗a ) denote the set of maximally symmetric matrices
acting on (Rn )⊗a .
It is not difficult to see that any degree 2a homogeneous polynomial f can be
represented in a unique way as

f (x) = (x ⊗a )T Z f x ⊗a ,

where the matrix Z f is maximally symmetric.

Given an integer r ≥ a, define the polynomial fr (x) = f (x)x2r −2a , thus homo-
geneous with degree 2r . The parameter (10) can now be reformulated as

f (r ) = sup{Z fr , M : M ∈ MSym((Rn )⊗r ), M  0, Tr(M) = 1}. (11)

The approach in [4] can be sketched as follows. Let M be an optimal solution to

the program (11) (which exists since the feasible region is a compact set). Then the
polynomial Q M (x) := (x ⊗r )T M x ⊗r is a sum of squares since M  0. After scaling,
we obtain the polynomial

h(x) = Q M (x)/ Q M (x)dσ (x) ∈ Σ[x]r ,

Sn−1

which defines a probability density function on Sn−1 , i.e., Sn−1 h(x)dσ (x) = 1. In
(r )
this way h provides a feasible solution for the program defining the upper bound f .
This thus implies the chain of inequalities

(r )
Z fr , M = f (r ) ≤ f min ≤ f ≤ f (x)h(x)dσ (x).
Sn−1

The main contribution in [4] is their analysis for bounding the range between the two
extreme values in the above chain and showing Theorem 2, which is done by using,
in particular, Fourier analysis on the unit sphere.
Using different techniques we will show below a rate of convergence in O(1/r 2 )
(r )
for the upper bounds f , thus stronger than the rate O(1/r ) in Theorem 2 above
and applying to any polynomial (not necessarily homogeneous). On the other hand,
while the constant involved in Theorem 2 depends only on the degree of f and the
dimension n, the constant in our result depends also on other characteristics of f (its
first and second order derivatives). A key ingredient in our analysis will be to reduce
to the univariate case, namely to the optimization of a linear polynomial over the
interval [−1, 1]. Thus we next recall the relevant known results that we will need in
our treatment.

2.2 Convergence analysis for the interval [− 1, 1]

We start with recalling the following eigenvalue reformulation for the bound (3),
which holds for general K compact and plays a key role in the analysis for the case
K = [−1, 1]. For this consider the following inner product

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Convergence analysis of a Lasserre hierarchy of upper… 673

( f , g) → f (x)g(x)dμ(x)
K

on the space of polynomials on K and let {bα (x) : α ∈ Nn } denote a basis of this
polynomial space that is orthonormal with respect to the above inner product; that is,
K bα (x)bβ (x)dμ(x) = δα,β . Then the bound (2) can be equivalently rewritten as
 
(r )
f = λmin (A f ), where A f = f (x)bα (x)bβ (x)dμ(x) (12)
K α,β∈Nn
|α|,|β|≤r

(see [8,16]). Using this reformulation we could show in [8] that the bounds (3) have
a convergence rate in O(1/r 2 ) for the case of the interval K = [−1, 1] (and as an
application also for the n-dimensional box [−1, 1]n ).
This result holds for a large class of measures on [−1, 1], namely those which
admit a weight function w(x) = (1 − x)a (1 + x)b (with a, b > −1) with respect to
the Lebesgue measure. The corresponding orthogonal polynomials are known as the
Jacobi polynomials Pda,b (x) where d ≥ 0 is their degree. The case a = b = −1/2
(resp., a = b = 0) corresponds to the Chebychev polynomials (resp., the Legendre
polynomials), and when a = b = λ − 1/2, the corresponding polynomials are the
Gegenbauer polynomials Cdλ (x) where d is their degree. See, e.g., [6, Chapter 1] for
a general reference about orthogonal polynomials.
The key fact is that, in the case of the univariate polynomial f (x) = x, the matrix A f
in (12) has a tri-diagonal shape, which follows from the 3-term recurrence relationship
satisfied by the orthogonal polynomials. In fact, A f coincides with the so-called Jacobi
matrix of the orthogonal polynomials in the theory of orthogonal polynomials and its
eigenvalues are given by the roots of the degree r + 1 orthogonal polynomial (see, e.g.
[6, Chapter 1]). This fact is key to the following result.

Theorem 3 [8] Consider the measure dμ(x) = (1 − x)a (1 + x)b d x on the interval
[−1, 1], where a, b > −1. For the univariate polynomial f (x) = x, the parameter
(r )
f is equal to the smallest root of the Jacobi polynomial Pra,b
+1 (with degree r + 1). In
(r )
 
π
particular, f = − cos 2r +2 when a = b = −1/2. For any a, b > −1 we have

(r ) (r )
1
f − f min = f +1=Θ .
r2

3 Convergence analysis for the unit sphere

(r )
In this section we analyze the quality of the bounds f when minimizing a polynomial
(r )
f over the unit sphere Sn−1 . In Sect. 3.1 we show that the range f − f min is in
O(1/r 2 ) and in Sect. 3.2 we show that the analysis is tight for linear polynomials.

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674 E. de Klerk, M. Laurent

3.1 The bound O(1/r2 )

We first deal with the n-variate linear (coordinate) polynomial f (x) = x1 and after
that we will indicate how the general case can be reduced to this special case. The
key idea is to get back to the analysis in Sect. 2.2, for the interval [−1, 1] with an
appropriate weight function. We begin with introducing some notation we need.
To simplify notation we set d = n − 1 (which also matches the notation customary
in the theory of orthogonal polynomials where d usually is the number of variables).
We let Bd = {x ∈ Rd : x ≤ 1} denote the unit ball in Rd . Given a scalar λ > −1/2,
define the d-variate weight function

wd,λ (x) = (1 − x2 )λ−1/2 (13)

(well-defined when x < 1) and set

 
π d/2 Γ λ + 21
Cd,λ := wd,λ (x1 , . . . , xd )d x1 · · · d xd =   (14)
Bd Γ λ + d+12

−1
so that Cd,λ wd,λ (x1 , . . . , xd )d x1 · · · d xd is a probability measure over the unit ball
B . See, e.g., [6, Section 2.3.2] or [2, Section 11].
d

We will use the following simple lemma, which indicates how to integrate the
d-variate weight function wd,λ along d − 1 variables.

Lemma 1 Fix x1 ∈ [−1, 1] and let d ≥ 2. Then we have:

d−2
wd,λ (x1 , . . . , xd )d x2 · · · d xd = Cd−1,λ (1 − x12 )λ+ 2 ,
{(x2 ,...,xd ):x22 +···+xd2 ≤1−x12 }

which is thus equal to Cd−1,λ w1,λ+(d−1)/2 (x1 ).


Proof Change variables and set u j = x j / 1 − x12 for 2 ≤ j ≤ d. Then we have
1 1 1
wd,λ (x) = (1 − x12 − x22 + · · · − xd2 )λ− 2 = (1 − x12 )λ− 2 (1 − u 22 − · · · − u 2d )λ− 2 and
d−1
d x2 · · · d xd = (1 − x12 ) 2 du 2 · · · du d . Putting things together and using relation (14)
we obtain the desired result. 


We also need the following lemma, which relates integration over the unit sphere
Sd ⊆ Rd+1 and integration over the unit ball Bd ⊆ Rd and can be found, e.g., in [6,
Lemma 3.8.1] and [2, Lemma 11.7.1].

Lemma 2 Let g be a (d + 1)-variate integrable function defined on S d and d ≥ 1.

Then we have:
    dx · · · dx
1 d
g(x)dσ (x) = g(x, 1 − x2 ) + g(x, − 1 − x2 )  .
Sd Bd 1 − x2

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Convergence analysis of a Lasserre hierarchy of upper… 675

By combining these two lemmas we obtain the following result.

Lemma 3 Let g(x1 ) be a univariate polynomial and d ≥ 1. Then we have:

1
σd−1 −1
g(x1 )dσ (x1 , . . . , xd+1 ) = C1,ν g(x1 )w1,ν (x1 )d x1 ,
Sd −1

where we set ν = 2 .
d−1

Proof Applying Lemma 2 to the function x ∈ Rd+1 → g(x1 ) we get

σd−1 g(x1 )dσ (x1 , . . . , xd+1 ) = 2σd−1 g(x1 )wd,0 (x)d x1 · · · d xd . (15)
Sd Bd

If d = 1 then ν = 0 and the right hand side term in (15) is equal to

1 1
2σ1−1 −1
g(x1 )w1,0 (x1 )d x1 = C1,0 g(x1 )w1,0 (x1 )d x1 ,
−1 −1

−1
√ desired, since 2σ1 C1,0 = 1 using σ1 = 2π and C1,0 = π (by (14) and Γ (1/2) =
as
π ). Assume now d ≥ 2. Then the right hand side in (15) is equal to
 
1
2σd−1 g(x1 ) wd,0 (x1 , . . . , xd )d x2 · · · d xd d x1
−1 x22 +···+xd2 ≤1−x12
1
= 2σd−1 Cd−1,0 g(x1 )(1 − x12 )(d−2)/2 d x1
−1
1
= 2σd−1 Cd−1,0 g(x1 )w1,ν (x1 )d x1 ,
−1

where we have used Lemma 1 for the first equality. Finally we verify that the constant
2σd−1 Cd−1,0 C1,ν is equal to 1:
 d−1  1 
Γ d+1
π 2 Γ 1
π2Γ d
2σd−1 Cd−1,0 C1,ν =2 2
 2 2
 =1
Γ d
Γ d+1
d+1
2π 2 2 2

[using relations (4) and (14)], and thus we arrive at the desired identity. 


We can now complete the convergence analysis for the minimization of x1 on the
unit sphere.

Lemma 4 For the minimization of the polynomial f (x) = x1 over Sd with d ≥ 1, the
order r upper bound (3) satisfies
 
(r ) 1
f = −1 + O .
r2

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676 E. de Klerk, M. Laurent

Proof Let h(x1 ) be an optimal univariate sum-of-squares polynomial of degree 2r for

the order r upper bound corresponding to the minimization of x1 over [−1, 1], when
−1
using as reference measure on [−1, 1] the measure with weight function w1,ν (x1 )C1,ν
and ν = (d − 1)/2 (thus ν > −1). Applying Lemma 3 to the univariate polynomials
h(x1 ) and x1 h(x1 ), we obtain
1
σd−1 −1
h(x1 )dσ (x) = C1,ν h(x1 )w1,ν (x1 )d x1 = 1
Sd −1

and
1
(r )
f ≤ σd−1 −1
x1 h(x1 )dσ (x) = C1,ν x1 h(x1 )w1,ν (x1 )d x1 .
Sd −1

Since the function x1 has the same global minimum −1 over [−1, 1] and over the
sphere Sd , we can apply Theorem 3 to conclude that

(r )
1 1
−1
f + 1 ≤ 1 + C1,ν x1 h(x1 )w1,ν (x1 )d x1 = O .
−1 r2




We now indicate how the analysis for an arbitrary polynomial f reduces to the
case of the linear coordinate polynomial x1 . To see this, suppose a ∈ Sn−1 is a global
minimizer of f over Sn−1 . Then, using Taylor’s theorem, we can upper estimate f as
follows:

f (x) ≤ f (a) + ∇ f (a)T (x − a) + 21 C f x − a2 ∀x ∈ Sn−1

(16)
= f (a) + ∇ f (a) (x − a) + C f (1 − a x) =: g(x) ∀x ∈ Sn−1 ,
T T

where C f = max x∈Sn−1 ∇ 2 f (x)2 , and we have used the identity

x − a2 = x2 + a2 − 2a T x = 2 − 2a T x for a, x ∈ Sn−1 .

Note that the upper estimate g(x) is a linear polynomial, which has the same minimum
value as f (x) on Sn−1 , namely f (a) = f min = gmin . From this it follows that
(r )
f − f min ≤ g (r ) − gmin and thus we may restrict to analyzing the bounds for a linear
polynomial.
Next, assume f is a linear polynomial, of the form f (x) = c T x with (up to scaling)
c = 1. We can then apply a change of variables to bring f (x) into the form x1 .
Namely, let U be an orthogonal n × n matrix such that U c = e1 , where e1 denotes
the first standard unit vector in Rn . Then the polynomial g(x) := f (U T x) = x1 has
the desired form and it has the same minimum value −1 over Sn−1 as f (x). As the
(r )
sphere is invariant under any orthogonal transformation it follows that f = g (r ) =
−1 + O(1/r 2 ) (applying Lemma 4 to g(x) = x1 ).
Summarizing, we have shown the following.

123
Convergence analysis of a Lasserre hierarchy of upper… 677

Theorem 4 For the minimization of any polynomial f (x) over Sn−1 with n ≥ 2, the
order r upper bound (3) satisfies
 
(r ) 1
f − f min = O .
r2

Note the difference to Theorem 2 where the constant depends only on the degree
of f and the number n of variables; here the constant in O(1/r 2 ) does also depend on
the polynomial f , namely it depends on the norm of ∇ f (a) at a global minimizer a
of f in Sn−1 and on C f = max x∈Sn−1 ∇ 2 f (x)2 .

3.2 The analysis is tight for some powers of linear polynomials

In this section we show—through  a class of examples—that the convergence rate

cannot be better than Ω 1/r 2 for general polynomials. The class of examples is
simply minimizing some powers of linear functions over the sphere Sn−1 . The key
(r )
tool we use is a link between the bounds f and properties of some known cubature
rules on the unit sphere. This connection, recently mentioned in [17], holds for any
compact set K . It goes as follows.
Theorem 5 [17] Assume that the points x (1) , . . . , x (N ) ∈ K and the weights
w1 , . . . , w N > 0 provide a (positive) cubature rule for K for a given measure μ,
which is exact up to degree d + 2r , that is,


N
g(x)dμ(x) = wi g(x (i) )
K i=1

for all polynomials g with degree at most d + 2r . Then, for any polynomial f with
degree at most d, we have
(r )
f ≥ min f (x (i) ).
1≤i≤N

The argument is simple: if h ∈ Σ[x]r is an optimal sum-of-squares density for the

(r )
parameter f , then we have


N
1= h(x)dμ(x) = wi h(x (i) ),
K i=1

(r ) 
N
f = f (x)h(x)dμ(x) = wi f (x (i) )h(x (i) ) ≥ min f (x (i) ).
K 1≤i≤N
i=1

As a warm-up we first consider the case n = 2, where we can use the cubature
rule in Theorem 6 below for the unit circle. We use spherical coordinates (x1 , x2 ) =
(cos θ, sin θ ) to express a polynomial f in x1 , x2 as a polynomial g in cos θ, sin θ .

123
678 E. de Klerk, M. Laurent

Theorem 6 [2, Proposition 6.5.1] For each d ∈ N, the cubature formula

d−1  
1 2π 1 2π j
g(θ )dθ = g
2π 0 d d
j=0

is exact for all g ∈ span{1, cos θ, sin θ, . . . , cos(dθ ), sin(dθ )}, i.e. for all polynomials
of degree at most d, restricted to the unit circle.
(r )
Using this cubature rule on S1 we can lower bound the parameters f for the
minimization of the coordinate polynomial f (x) = x1 over S1 . Namely, by setting
x1 = cos θ , we derive directly from Theorems 5 and 6 that

(r )
 2π j   2πr  1
f ≥ min cos = cos = −1 + Ω 2 .
0≤ j≤2r 2r + 1 2r + 1 r

This reasoning extends to any dimension n ≥ 2, by using product-type cubature

formulas on the sphere Sn−1 . In particular we will use the cubature rule described in
[2, Theorem 6.2.3], see Theorem 8 below.
We will need the generalized spherical coordinates given by
⎫
x1 = r sin θn−1 · · · sin θ3 sin θ2 sin θ1 ⎪
⎪
⎪
x2 = r sin θn−1 · · · sin θ3 sin θ2 cos θ1 ⎪
⎪
⎬
x3 = r sin θn−1 · · · sin θ3 cos θ2 (17)
.. ⎪
⎪
. ⎪
⎪
⎪
⎭
xn = r cos θn−1 ,

where r ≥ 0 (r = 1 on Sn−1 ), 0 ≤ θ1 ≤ 2π , and 0 ≤ θi ≤ π (i = 2, . . . , n − 1).

To define the nodes of the cubature rule on Sn−1 we need the Gegenbauer polyno-
mials Cdλ (x), where λ > −1/2. Recall that these are the orthogonal polynomials with
respect to the weight function

w1,λ (x) = (1 − x 2 )λ−1/2 , x ∈ (−1, 1)

on [−1, 1]. We will not need the explicit expressions for the polynomials Cdλ (x), we
only need the following information about their extremal roots, shown in [7] (for
general Jacobi polynomials, using results of [3,5]). It is well known that each Cdλ (x)
has d distinct roots, lying in (−1, 1).
(λ) (λ)
Theorem 7 Denote the roots of the polynomial Cdλ (x) by t1,d < · · · < td,d . Then,
(λ) λ = Θ(1/d 2 ).
t1,d + 1 = Θ(1/d 2 ) and 1 − td,d

The cubature rule we will use may now be stated.

123
Convergence analysis of a Lasserre hierarchy of upper… 679

Theorem 8 [2, Theorem 6.2.3] Let f : Sn−1 → R be a polynomial of degree at most

2k − 1, and let

g(θ1 , . . . , θn−1 ) := f (x1 , . . . , xn ),

be the expression of f in the generalized spherical coordinates (17). Then

f (x)dσ (x)
Sn−1

π  
2k−1 k 
k  ((i−1)/2)  π j1 (1/2)
n−1
((n−2)/2)

= ··· μi,k g , θ j2 ,k , . . . , θ jn−1 ,k ,
k k
j1 =0 j2 =1 jn−1 =1 i=2
(18)
 
(λ) (λ) ((i−1)/2)
where cos θ j,k := t j,k and the parameters μi,k are positive scalars as in
relation (6.2.3) of [2].

We can now show the tightness of the convergence rate Ω(1/r 2 ) for the minimiza-
tion of a coordinate polynomial on Sn−1 .

Theorem 9 Consider the problem of minimizing the coordinate polynomial f (x) = xn

on the unit sphere Sn−1 with n ≥ 2. The convergence rate for the parameters (3)
satisfies
 
(r ) (r ) 1
f − f min = f +1=Ω .
r2

Proof We have f (x1 , . . . , xn ) = xn , so that g(θ1 , . . . , θn−1 ) = cos θn−1 . Using The-
orem 5 combined with Theorem 8 (applied with 2k − 1 = 2r + 1, i.e., k = r + 1) we
obtain that

(r )
1
((n−2)/2) ((n−2)/2) ((n−2)/2)
f ≥ min cos θ j,r +1 = min t j,r +1 = t1,r +1 = −1 + Ω ,
1≤ j≤r +1 1≤ j≤r +1 r2

(λ)
where we use the fact that t1,r +1 + 1 = Θ(1/r 2 ) (Theorem 7). 


This reasoning extends to some powers of linear forms.

Theorem 10 Given an integer d ≥ 1 and a nonzero c ∈ Rn , the following holds for

the polynomial f (x) = (−1)d−1 (c T x)d :
 
(r ) 1
f − f min = Ω .
r2

Proof Up to scaling we may assume c = 1 and, up to applying an orthogonal

transformation, we may assume that f (x) = (−1)d−1 xnd , so that f min = −1. Again we

123
680 E. de Klerk, M. Laurent

use Theorem 5, as well as Theorem 8, now with 2k −1 = 2r +d, i.e., k = r +(d +1)/2,
and we obtain

(r ) ((n−2)/2) ((n−2)/2) d
f ≥ min (−1)d−1 cosd θ j,k = min (−1)d−1 (t j,k ) .
1≤ j≤k 1≤ j≤k

We can now conclude using Theorem 7. For d odd, the right hand side is equal
((n−2)/2) d
to (t1,k ) = −1 + Θ( r12 ) and, for d even, the right hand side is equal to
((n−2)/2) d
−(tk,k ) = −1 + Θ( r12 ). 


4 Some extensions

Here we mention some possible extensions of our results. First we consider the general
problem of moments and its application to the problem of minimizing a rational
function. Thereafter we mention that the rate of convergence in O(1/r 2 ) extends to
some other measures on the unit sphere.

4.1 Implications for the generalized problem of moments

In this section, we describe the implications of our results for the generalized problem
of moments (GPM), defined as follows for a compact set K ⊂ Rn :
 
val := inf f 0 (x)dν(x) : f i (x)dν(x) = bi ∀i ∈ [m] , (19)
ν∈M(K )+ K K

where

– the functions f i (i = 0, . . . , m) are continuous on K ;

– M(K )+ denotes the convex cone of probability measures supported on the set K ;
– the scalars bi ∈ R (i ∈ [m]) are given.

As before, we are interested in the special case where K = Sn−1 . This special case
is already of independent interest, since it contains the problem of finding cubature
schemes for numerical integration on the sphere, see e.g. [9] and the references therein.
Our main result in Theorem 4 has the following implication for the GPM on the sphere,
as a corollary of the following result in [12] (which applies to any compact K , see
also [9] for a sketch of the proof in the setting described here).

Theorem 11 (De Klerk-Postek-Kuhn [12]) Assume that f 0 , . . . , f m are polynomials,

K is compact, μ is a Borel measure supported on K , and the GPM (19) has an optimal
solution. Given r ∈ N, define the parameter

 
Δ(r ) = min max  f i (x)h(x)dμ(x) − bi ,
h∈Σr i∈{0,1,...,m} K

123
Convergence analysis of a Lasserre hierarchy of upper… 681

setting b0 = val. Assume ε : N → R+ is such that limr →∞ ε(r ) = 0, and that, for
any polynomial f , we have

(r )
f K − f min = O(ε(r )).
√
Then the parameters Δ(r ) satisfy: Δ(r ) = O( ε(r )).
As a consequence of our main result in Theorem 4, combined with Theorem 11,
we immediately obtain the following corollary.
Corollary 1 Assume that f 0 , . . . , f m are polynomials, K = Sn−1 , and the GPM (19)
has an optimal solution. Then, for any integer r ∈ N, there exists a polynomial h r ∈ Σr
such that
 
 
 
 n−1 f 0 (x)h r (x)dσ (x) − val  = O(1/r ),
S
 
 
 f i (x)h r (x)dσ (x) − bi  = O(1/r ) ∀i ∈ [m].

Sn−1

Minimization of a rational function on K is a special case of the GPM where we

may prove a better rate of convergence. In particular, we now consider the global
optimization problem:

p(x)
val = min , (20)
x∈K q(x)

where p, q are polynomials such that q(x) > 0 ∀ x ∈ K , and K ⊆ Rn is compact.

It is well-known that one may reformulate this problem as the GPM with m = 1
and f 0 = p, f 1 = q, and b1 = 1, i.e.:
 
val = min p(x)dν(x) : q(x)dν(x) = 1 .
ν∈M(K )+ K K

Analogously to (3), we now define the hierarchy of upper bounds on val as follows:

(r )
p/q K := min p(x)h(x)dμ(x) s.t. K q(x)h(x)dμ(x) = 1, (21)
h∈Σ[x]r K

where μ is a Borel measure supported on K .

Theorem 12 Consider the rational optimization problem (20). Assume ε : N → R+
is such that limr →∞ ε(r ) = 0, and that, for any polynomial f , we have

(r )
f K − f min = O(ε(r )).

(r ) (r )
Then one also has p/q K − val = O(ε(r )). In particular, if K = Sn−1 , then p/q K −
val = O(1/r 2 ).

123
682 E. de Klerk, M. Laurent

Proof Consider the polynomial

f (x) = p(x) − val · q(x).

Then f (x) ≥ 0 for all x ∈ K , and f min,K = 0, with global minimizer given by the
minimizer of problem (20).
(r )
Now, for given r ∈ N, let h ∈ Σr be such that f K = K f (x)h(x)dμ(x), and
K h(x)dμ(x) = 1, where μ is the reference measure for K . Setting

1
h∗ = h,
K h(x)q(x)dμ(x)

one has h ∗ ∈ Σr and K h ∗ (x)q(x)dμ(x) = 1. Thus h ∗ is feasible for problem (21).

Moreover, by construction,

(r )
fK
p(x)h ∗ (x)dμ(x) − val =
K K h(x)q(x)dμ(x)
(r )
fK
≤ = O(ε(r )).
min x∈K q(x)

The final result for the special case K = Sn−1 and μ = σ (surface measure) now
follows from our main result in Theorem 4. 


4.2 Extension to other measures

Here we indicate how to extend the convergence analysis to a larger class of measures
on the unit sphere Sn−1 of the form dμ(x) = w(x)dσ (x), where w(x) is a positive
bounded weight function on Sn−1 , i.e., w(x) satisfies the condition:

There exist m, M > 0 such that m ≤ w(x) ≤ M for all x ∈ Sn−1 . (22)

(r )
Given a polynomial f we let f μ denote the bound obtained by using the measure μ
instead of the Haar measure σ on Sn−1 . We will show that under the condition (22) the
(r )
bounds f μ converge to f min with the same convergence rate O(1/r 2 ). These results
follow the same line of arguments as in the recent paper [23]. We start with dealing
with the case of linear polynomials.

Lemma 5 Consider an affine polynomial g of the form g(x) = 1 − c T x, where c ∈

Sn−1 . If dμ(x) = w(x)dσ (x) and w satisfies (22) then we have:

M
g (r )
μ ≤g
(r )
.
m

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Convergence analysis of a Lasserre hierarchy of upper… 683

Proof Let H ∈ Σr be an optimal sum of squares density for the Haar measure σ , i.e.,
such that

H (x)dσ (x) = 1 and g(x)H (x)dσ (x) = g (r ) .

Sn−1 Sn−1

Define the polynomial

H
h= ∈ Σr ,
Sn−1 H (x)w(x)dσ (x)

which defines a density for the measure μ on Sn−1 , so that we have

g(x)H (x)w(x)dσ (x)

g (r
μ ≤
)
g(x)h(x)dμ(x) = Sn−1
.
Sn−1 Sn−1 H (x)w(x)dσ (x)

Since m ≤ w(x) ≤ M on Sn−1 the numerator is at most M g (r ) and the denominator

is at least m, which concludes the proof. 


Theorem 13 Consider a weight function w(x) on Sn−1 that satisfies the condition
(22), and the corresponding measure dμ(x) = w(x)dσ (x) on the unit sphere Sn−1 .
Then, for any polynomial f , we have

(r )
1
f μ − f min = O .
r2

Proof Let a ∈ Sn−1 be a global minimizer of f in the unit sphere. We may assume
that f min = f (a) = 0 (else replace f by f − f (a)). As observed in relation (16), we
have

f (x) ≤ ∇ f (a)T (x − a) + C f (1 − a T x) =: g(x) for all x ∈ Sn−1 .

Note that g is affine linear with gmin = g(a) = 0. Hence we may apply Lemma 5
which, combined with Theorem 4 (applied to g), implies that g (r )
μ = O(1/r ). As
2
(r ) (r )
f ≤ g on Sn−1 it follows that f μ ≤ g (r )
μ and thus f μ = O(1/r ) as desired.
2 


5 Concluding remarks

In this paper we have improved on the O(1/r ) convergence result of Doherty and
Wehner [4] for the Lasserre hierarchy of upper bounds (3) for (homogeneous) polyno-
mial optimization on the sphere. Having said that, Doherty and Wehner also showed
that the hierarchy of lower bounds (9) of Lasserre satisfies the same rate of conver-
gence, due to Theorem 2. In view of the fact that we could show the improved O(1/r 2 )
rate for the upper bounds, and the fact that the lower bounds hierarchy empirically
converges much faster in practice, one would expect that the lower bounds (9) also

123
684 E. de Klerk, M. Laurent

converge at a rate no worse than O(1/r 2 ). This has been recently confirmed in the
paper [13].
Another open problem is the exact rate of convergence of the bounds in Theorem 11
for the generalized problem of moments (GPM). In our analysis of the GPM on the
sphere in Corollary 1, we could only obtain O(1/r ) convergence, which is a square
root worse than the special cases for polynomial and rational function minimization.
We do not know at the moment if this is a weakness of the analysis or inherent to the
GPM.
As we showed in Theorem 13, if we pick another reference measure dμ(x) =
w(x)dσ (x), where w is upper and lower bounded by strictly positive constants on the
sphere, then the convergences rates with respect to both measures σ and μ have the
same behaviour. It would be interesting to understand the convergence rate for more
general reference measures.

Acknowledgements We thank two anonymous referees for their useful remarks. This work has been
supported by European Union’s Horizon 2020 Research and Innovation Programme under the Marie
Skłodowska-Curie Grant Agreement 813211 (POEMA).

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