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Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on
convex bodies
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Università degli Studi di Padova
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Markov inequalities, Dubiner distance,
norming meshes and polynomial
optimization on convex bodies
∗
Federico Piazzon and Marco Vianello
1
Department of Mathematics, University of Padova, Italy
November 22, 2018
Abstract
We construct norming meshes for polynomial optimization by the clas-
sical Markov inequality on general convex bodies in R
d
, and by a tangen-
tial Markov inequality via an estimate of the Dubiner distance on smooth
convex bodies. These allow to compute a (1−ε)-approximation to the min-
imum of any polynomial of degree not exceeding n by O
(n/
√
ε)
αd
sam-
ples, with α = 2 in the general case, and α = 1 in the smooth case. Such
constructions are based on three cornerstones of convex geometry, Bieber-
bach volume inequality and Leichtweiss inequality on the affine breadth
eccentricity, and the Rolling Ball Theorem, respectively.
2010 AMS subject classification: 41A17, 65K05, 90C26.
Keywords: polynomial optimization, norming mesh, Markov inequality, tangential
Markov inequality, Dubiner distance, convex bodies.
1 Introduction
Sampling methods, typically on suitable grids, are one of the possible approaches
in the vast literature on polynomial optimization theory, cf., e.g., [10, 11, 36]
with the references therein. In this paper we extend in the general framework
of convex bodies our previous work on sampling methods for polynomial opti-
mization, based on the multivariate approximation theory notions of norming
mesh and Dubiner distance, cf. [28, 27, 32, 33, 34].
Polynomial inequalities based on the notion of norming mesh mesh have been
recently playing a relevant role in multivariate approximation theory, as well in
its computational applications. We recall that a polynomial (norming) mesh of
a polynomial determining compact set K ⊂ R
d
(i.e., a polynomial vanishing on
K vanishes everywhere), is a sequence of finite subsets A
n
⊂ K such that
kpk
K
≤ C kpk
A
n
, ∀p ∈ P
d
n
, (1)
∗
Work partially supported by the DOR funds and the biennial project BIRD163015 of the
University of Padova, and by the GNCS-INdAM. This research has been accomplished within
the RITA “Research ITalian network on Approximation”.
1
corresponding author: marcov@math.unipd.it
1
for some C > 1 independent of p and n, where card(A
n
) = O(n
s
), s ≥ d. Here
and below we denote by P
d
n
the subspace of d-variate real polynomials of total
degree not exceeding n, and by kfk
X
the sup-norm of a bounded real function
on a discrete or continuous compact set X ⊂ R
d
.
Observe that A
n
is P
d
n
-determining, consequently card(A
n
) ≥ dim(P
d
n
) =
n+d
d
∼ n
d
/d! (d fixed, n → ∞). A polynomial mesh is termed optimal when
s = d. All these notions can be given more generally for K ⊂ C
d
but we restrict
here to real compact sets.
Polynomial meshes were formally introduced in the seminal paper [9] as a
tool for studying the uniform convergence of discrete least squares polynomial
approximation, and then studied from both the theoretical and the computa-
tional point of view throughout a series of papers. Among their features, we
recall for example that the property of being a polynomial mesh is stable under
invertible affine transformations and small perturbations (see [13, 25]). Also,
given the polynomial meshes A
1
n
and A
2
n
for the compact sets K
1
and K
2
re-
spectively, the sequence of sets A
1
n
∪A
2
n
and A
1
n
×A
2
n
are polynomial meshes for
K
1
∪K
2
and K
1
×K
2
, with the constants being the maximum and the product
of the constants of K
1
and K
2
respectively. Moreover, if T : R
2
→ R
d
is a
polynomial map of degree not greater than k and A
n
is a polynomial mesh for
the compact set K ⊂ R
d
, then T (A
kn
) is an admissible mesh for the compact
set T (K).
Polynomial meshes have been constructed by different analytical and ge-
ometrical techniques on various classes of compact sets, such as Markov and
subanalytic sets, polytopes, convex and starlike bodies; we refer the reader,
e.g., to [3, 9, 16, 23, 25, 29] and the references therein, for a comprehensive view
of construction methods.
Since polynomial meshes have first been introduced in the framework of
discrete least squares, their most direct application is in the approximation
of functions and data. As a consequence, polynomial meshes can be used as
a tool for spectral methods for the solution of PDEs, see [37, 38]. Perhaps
more surprisingly, near optimal interpolation arrays can be extracted from an
admissible mesh by standard numerical linear algebra tools [3]. Note that the
problem of finding unisolvent interpolation arrays with slowly increasing (e.g.,
polynomial in the degree) Lebesgue constant on a given compact set K ⊂ R
d
is very hard to attack numerically, even for small values of d > 1. Lastly, we
mention that polynomial meshes are the key ingredient for the approximation
algorithms proposed in [24], where the numerical approximation of the main
quantities of pluripotential theory (a non linear potential theory in C
d
, d > 1)
is studied.
In many instances, by suitably increasing the mesh cardinality it is possible
to let C → 1, where C is the “constant” of the polynomial mesh in (1). This
opens the way for a computational use of polynomial meshes in the framework of
polynomial optimization, in view of the general elementary estimate given below.
It is however worth to mention that, in view of the exponential dependence
on d of the cardinality of the meshes, this approach is attractive only for low
dimensional problems, e.g. d = 2, 3.
Proposition 1. (cf. [32]). Let {A
n
} be a polynomial mesh of a compact set
2
K ⊂ R
d
. Then, the following polynomial minimization error estimate holds
min
x∈A
n
p(x) − min
x∈K
p(x) ≤ (C − 1)
max
x∈K
p(x) − min
x∈K
p(x)
. (2)
Proof. Consider the polynomial q(x) = p(x) − max
x∈K
p(x) ∈ P
d
n
, which is
nonpositive in K. We have that kqk
K
= |min
x∈K
p(x) − max
x∈K
p(x)| =
max
x∈K
p(x) − min
x∈K
p(x), and kqk
A
n
= |min
x∈A
n
p(x) − max
x∈K
p(x)| =
max
x∈K
p(x) − min
x∈K
p(x). Then by (1)
min
x∈A
n
p(x) − min
x∈K
p(x) = kqk
K
− kqk
A
n
≤ (C − 1) kqk
A
n
≤ (C − 1) kqk
K
= (C − 1)
max
x∈K
p(x) − min
x∈K
p(x)
Notice that the error estimate in (2) is relative to the range of p, a usual
requirement in polynomial optimization; cf., e.g., [10]. Clearly, by the arbitrarity
of the polynomial, taking −p instead of p we can obtain the same estimate for
the discrete approximation to the maximum of p.
The discrete optimization suggested by Proposition 1 has been already used
in special instances, for example on Chebyshev-like grids with (mn + 1)
d
points
in d-dimensional boxes. Such grids (that are nonuniform) turn out to be poly-
nomial meshes for total degree polynomials, with C =
1
cos(π/(2m))
, as it has
been shown in [28] resorting to the notion of Dubiner distance [6], so that
C − 1 = O(1/m
2
). A similar approach, though essentially in a tensor-product
framework, was adopted also in [36]. In [33, 34], the method is applied to
polynomial optimization on 2-dimensional sphere and torus.
On the other hand, polynomial optimization on uniform rational grids is a
well-known procedure on standard compact sets (hypercube, simplex), cf. e.g.
[10, 11] with the references therein.
In Sections 2 and 3 we present a general approach to polynomial optimiza-
tion on norming meshes of Markov compact sets and then of general convex
bodies, constructed starting from sufficiently dense uniform grids. To this pur-
pose, we adapt and refine an approximation theoretic construction of Calvi and
Levenberg [9], based on the fulfillement of a classical Markov polynomial in-
equality, and we resort to some deep results of convex geometry, Bieberbach
volume inequality and Leichtweiss inequality on affine breadth eccentricity. We
get a (1 − ε)-approximation to the minimum of a polynomial of degree not
exceeding n, by O
(n
2
/ε)
d
samples.
In Section 3, we modify and improve the construction on convex bodies
with C
2
-boundary via the approximation theoretic notion of Dubiner distance,
providing an original estimate for such a distance by another cornerstone of
convex geometry, the Rolling Ball Theorem, together with a recent deep result
by Totik on the Szeg¨o-version of Bernstein-like inequalities. In such a way we
obtain a (1 − ε)-approximation by O
(n/
√
ε)
d
nonuniform samples.
2 Markov compact sets
Following [9], we’ll now show a general discretization procedure, that allows to
construct a polynomial mesh on any compact set admitting a Markov polynomial
3
inequality (often called Markov compact sets). Given positive scalars r, M > 0,
a compact set K is said to admit a Markov Inequality of exponent r and constant
M if, for every n ∈ N , we have
k∇pk
K
≤ M n
r
kpk
K
, ∀p ∈ P
d
n
, (3)
where k∇pk
K
= max
x∈K
k∇p(x)k
2
, k · k
2
denoting the euclidean norm of d-
dimensional vectors. For example, with d = 1 and K = [−1, 1] we have r = 2
and M = 1. The Markov exponent can be r = 1 only on real algebraic manifolds
without boundary [7], for example on the sphere S
d−1
. The exponent is r = 2
on compact domains with Lipschitz boundary, or more generally satisfying a
uniform interior cone condition; cf. [12, §6.4]. In the special case of a convex
body, we have
r = 2 , M = 4/w(K) , (4)
where w(K) is the width of the convex body (the minimal distance between
parallel supporting hyperplanes); on centrally symmetric bodies the numerator
4 can be replaced by 2, cf. [35]. We refer the reader, e.g., to [2, 9] with the
references therein for a general view on Markov polynomial inequalities.
For the reader’s convenience, we state and prove the following result which
is, in the real case, essentially Theorem 5 of [9].
Proposition 2. Let K ⊂ R
d
a compact set satisfying (3), and L be the maximal
length of the convex hulls of its projections on the Cartesian axes.
Then, for any fixed ε ∈ (0, 1), K possesses a polynomial mesh {A
n
(ε)}
n∈N
such that, for any n ∈ N,
kpk
K
≤ (1 + ε) kpk
A
n
(ε)
, ∀p ∈ P
d
n
, (5)
with
card(A
n
(ε)) ≤
&
√
dLMn
r
g(ε)
'!
d
, (6)
where g(ε) = σ(ε) =
ε
1+ε
for K convex, and g(ε) = σ(ε) exp(−
√
d σ(ε)) for K
non convex.
Before proving Proposition 2, we observe that by Proposition 1 we get im-
mediately
min
x∈A
n
(ε)
p(x) − min
x∈K
p(x) ≤ ε
max
x∈K
p(x) − min
x∈K
p(x)
. (7)
The usual way to express an inequality like (7), is to say that min
x∈A
n
(ε)
p(x)
is a (1 − ε)-approximation to min
x∈K
p(x); see, e.g., [10].
Proof of Proposition 2. We first assume K to be convex. Let us pick, for any
n ∈ N and ∈ (0, 1), a uniform coordinate grid on R
d
of step
σ()
√
dMn
r
. Let us
denote by B
i
, i ∈ I := {1, 2, . . . , S(n, )} the (clearly finite) collection of the
boxes of the grid intersecting K and let us pick y
i
∈ K ∩ B
i
, ∀i ∈ I. We set
A
n
() = {y
i
}
i∈I
. The estimate (6) immediately follows by K ⊆ v + [0, L]
d
for a
suitable vector v ∈ R
d
.
4
![Figure 4: The construction of the proof of Proposition 6. The continuous curve represent the boundary of K, while the dashed one represent the level set φK ≡ tj(x). The small dots are the points of An(ε). In order to find y(x) one first finds a geodesically closest point z(x) in Zn(ε) to the radial projection x ′ of x onto ∂K. The the index j(x) is determined as one for which tj(x) is a closest point of {tj}j=0,...,mn to φK(x) in the arcosine metric. Finally y(x) is the point in [0, z(x)] such that φK(y(x)) = tj(x).](/figures/figure-4-the-construction-of-the-proof-of-proposition-6-the-ffxidp9k.webp)
