A frank wolfe type theorem and holder type global error bound for generic polynomial systems

¨
A FRANK-WOLFE TYPE THEOREM AND HOLDER-TYPE
GLOBAL
ERROR BOUNDS FOR GENERIC POLYNOMIAL SYSTEMS
‡
ˆ. P D
` † , AND TIE
ˆ´N SO.N PHA
- INH† , HUY VUI HA
S˜I TIE
.M

Abstract. This paper studies generic polynomial systems. More precisely, let f0 and
f1 , . . . , fp : Rn → R be convenient polynomial functions, and let S := {x ∈ Rn | fi (x) ≤
0, i = 1, . . . , p} = ∅. The following results are shown:
(i) A Frank-Wolfe type Theorem: Suppose that the map (f0 , f1 , . . . , fp ) : Rn → Rp+1 is
non-degenerate at infinity. If f0 is bounded from below on S, then f0 attains its infimum
on S;
(ii) A H¨
older-type global error bound: Suppose that the map (f1 , . . . , fp ) : Rn → Rp is
non-degenerate at infinity. Let d := maxi=1,...,p deg fi and H(d, n, p) := d(6d − 3)n+p−1 .
Then there exists a constant c > 0 such that
1

cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+

for all

x ∈ Rn ,

where d(x, S) denotes the Euclidean distance between x and the set S, f (x) := maxi=1,...,p fi (x)

and [f (x)]+ := max{f (x), 0}; and
(iii) For polynomial maps with fixed Newton polyhedra, the property of being nondegenerate at infinity is generic.

1. Introduction
Let f0 and f1 , . . . , fp : Rn → R be polynomial functions in the variable x ∈ Rn . Let
S := {x ∈ Rn | f1 (x) ≤ 0, . . . , fp (x) ≤ 0},
and suppose throughout that S is nonempty. Consider the following constrained optimization
problem
(1)

inf f0 (x)

such that

x ∈ S.

The purpose of this paper is twofold. Firstly, we are concerned with the question of
existence of optimal solutions to the problem (1). In the case when all fi , i = 0, . . . , p,
Date: November 26, 2012.
1991 Mathematics Subject Classification. Primary 32B20; Secondary 14P, 49K40.
Key words and phrases. Error bounds, Frank-Wolfe type theorem, Newton polyhedron, nondegenerate
polynomial maps.
†
These authors were partially supported by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) grant 101.01-2011.44.
‡

This author’s research was partially supported by Vietnam National Foundation for Science and Tech-

nology Development (NAFOSTED) grant 101.01-2010.08.

1


are linear, it is well known that the set of optimal solutions is nonempty provided the
problem is bounded below. In 1956, Frank and Wolfe [20] proved that if fi ’s remain affine
linear functions for i = 1, . . . , p, and f0 is an arbitrary quadratic polynomial, then f0 being
bounded from below over S implies that an optimal solution exists. If the statement holds
with respect to other classes of polynomial functions f0 , . . . , fp we will speak of a Frank-Wolfe
type theorem.
Many other authors generalized the Frank-Wolfe theorem to broader classes of functions.
For example, Perold [55] generalized the Frank-Wolfe theorem to a class of non-quadratic
objective functions and linear constraints. Andronov et al. [2] extended the Frank-Wolfe
theorem to the case of a cubic polynomial objective function f0 under linear constraints.
Luo and Zhang [43] also extended the Frank-Wolfe theorem to various classes of general
convex/non-convex quadratic constraint systems. More recently, Belousov and Klatte in [6]
(see also [5]) showed that this result is still true if f0 , f1 , . . . , fp are convex polynomials of
arbitrary degree.
Secondly, we are interested in the question of whether one can use the residual (constraint
violation) at a point x ∈ Rn to bound the distance from x to the set S. More precisely, we
study if there exist some positive constants c, α, and β such that
(2)

cd(x, S) ≤ [f (x)]α+ + [f (x)]β+

for all

x ∈ Rn ,

where d(x, S) denotes the Euclidean distance between x and the set S, f (x) := maxi=1,...,p fi (x)

and [f (x)]+ := max{f (x), 0}. An expression of this kind is called a global error bound for the
set S. We say that a H¨older-type global error bound holds for the set S if the inequality (2)
holds with the exponent β = 1.
The study of error bounds has grown significantly and has found many important applications. For a summary of the theory and applications of error bounds, we refer the readers
to the survey of Pang [54] and the references cited therein.
The first error bound result is due to Hoffman [27]. His result deals with the case where
the polynomials f1 , . . . , fp are affine and states that the global error bound (2) holds with
the exponents α = β = 1. After the work of Hoffman, a lot of researchers have devoted
themselves to the study of global error bound; see, for example, [3, 16, 30, 31, 35, 46, 51, 58].
Under the convexity assumption of the polynomials fi , global H¨older-type error bounds
have been shown in [36, 37, 38, 41, 42, 44, 45, 50, 61, 60].
In the absence of convexity, global H¨older-type error bounds (even global error bounds) are
highly unlikely to hold. When the constrained set S defined by some affine linear functions
and a single quadratic polynomial, Luo and Sturm [44] showed that the H¨older-type global
1
error bound (2) holds with the exponents α = and β = 1. In particular, a global error
2
bound was obtained by H. V. H`a [26] for a nonlinear inequality defined by a single convenient
2


polynomial, which is (Newton) non-degenerate at infinity (see [32] and Section 2 for precise
definitions).
In this paper, we consider the class of polynomial maps which are (Newton) non-degenerate
at infinity. This notion extends the definitions of non-degenerate for analytic functions, in
the (local and at infinity) complex setting [29, 32]. It is worth paying attention to the fact
that Non-degenerate at infinity polynomial maps have a number of remarkable properties
which make them an attractive domain for various applications.
The main contributions of this paper are as follows:
(i) Suppose that the map (f0 , f1 , . . . , fp ) : Rn → Rp+1 is non-degenerate at infinity, and all

fi are convenient, i = 0, 1, . . . , p. If the objective function f0 is bounded from below on the
constrained set S, then f0 attains its infimum on S;
(ii) Suppose that the map (f1 , . . . , fp ) : Rn → Rp is non-degenerate at infinity, and all
fi are convenient, i = 1, . . . , p. Then there exists a constant c > 0 such that the following
H¨older-type global error bound holds
1

cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+

for all

x ∈ Rn ,

where d := maxi=1,...,p deg fi and H(d, n, p) := d(6d − 3)n+p−1 .
(iii) The class of polynomial maps (with fixed Newton polyhedra), which are non-degenerate
at infinity, is generic in the sense that it is an open and dense semi-algebraic set.
It should be emphasized that we do not require the polynomials fi to be convex, and their
degrees can be arbitrary. Moreover, our method is actually different from the argument in
[26]: the proofs use only the Curve Selection Lemma (see Lemma 2.1) as a tool.
The results presented in the paper suggest that the class of polynomial maps, which nondegenerate at infinity, may offer an appropriate domain on which the machinery of polynomial
optimization works with full efficiency.
The paper is structured as follows. Section 2 presents some backgrounds in the field. In
Section 3, we establish a Frank-Wolfe type theorem. Some H¨older-type global error bound
results will be given in Section 4. Finally, in Section 5, we show that the property of being
nondegenerate at infinity is generic.

2. Preliminaries
In this section, we give the notations, definitions, and preliminary results which will used
throughout the paper.
Throughout this paper, Rn denotes Euclidean space with dimension n. The corresponding

x, x
inner product (resp., norm) in Rn is defined by x, y for any x, y ∈ Rn (resp., x :=
n
for any x ∈ R ).
3


2.1. Semi-algebraic geometry. In this subsection, we recall some notions and results of
semi-algebraic geometry, which can be found in [4, 7, 8, 10, 18].
Definition 2.1.
of the form

(i) A subset of Rn is called semi-algebraic if it is a finite union of sets

{x ∈ Rn | fi (x) = 0, i = 1, . . . , k; fi (x) > 0, i = k + 1, . . . , p}
where all fi are polynomials.
(ii) Let A ⊂ Rn and B ⊂ Rp be semi-algebraic sets. A map F : A → B is said to be
semi-algebraic if its graph
{(x, y) ∈ A × B | y = F (x)}
is a semi-algebraic subset in Rn × Rp .
Semi-algebraic sets and functions enjoy a number of remarkable properties:
(i) The class of semi-algebraic sets is closed with respect to Boolean operators; a Cartesian product of semi-algebraic sets is a semi-algebraic set;
(ii) The closure and the interior of a semi-algebraic set is a semi-algebraic set;
(iii) A composition of semi-algebraic maps is a semi-algebraic map.
(iv) The image and inverse image of a semi-algebraic set under a semi-algebraic map are
semi-algebraic sets.
(v) If S is a semi-algebraic set, then the distance function
d(·, S) : Rn → R,

x → d(x, S) := inf{ x − a | a ∈ S},

is also semi-algebraic.
A major fact concerning the class of semi-algebraic sets is its stability under linear projections (see, for example, [7, 10]).
Theorem 2.1 (Tarski-Seidenberg Theorem). Let π(x1 , . . . , xn ) = (x1 , . . . , xn−1 ) be the canonical projection from Rn onto Rn−1 . If S is a semi-algebraic subset of Rn , then so is π(S) in
Rn−1 .
Remark 2.1. As an immediate consequence of Tarski-Seidenberg Theorem, we get semialgebraicity of any set {x ∈ A | ∃y ∈ B, (x, y) ∈ S}, provided that A, B, and S are semialgebraic sets in the corresponding spaces. It follows that also {x ∈ A | ∀y ∈ B, (x, y) ∈ S}
is a semi-algebraic set as its complement is the union of the complement of A and the set
{x ∈ A | ∃y ∈ B, (x, y) ∈ S}. Thus, if we have a finite collection of semi-algebraic sets, then
any set obtained from them with the help of a finite chain of quantifiers is also semi-algebraic.
4


We will need a version of the Curve Selection Lemma. Milnor [48] has proved this lemma
at points of the closure of a semi-algebraic set. N´emethi and Zaharia [49] showed how to
extend the result at infinity at some fibre of a polynomial maps. We give here a more general
statement, and for the sake of completeness we include a proof of this fact.
Lemma 2.1 (Curve Selection Lemma at infinity). Let A ⊂ Rn be a semi-algebraic set, and
let F := (f1 , . . . , fp ) : Rn → Rp be a semi-algebraic map. Assume that there exists a sequence
xk ∈ A such that limk→∞ xk = ∞ and limk→∞ F (xk ) = y ∈ (R)p , where R := R ∪ {±∞}.
Then there exists an analytic curve ϕ : (0, ) → A of the form
ϕ(t) = a0 tq + a1 tq+1 + · · ·
such that a0 ∈ Rn \ {0}, q < 0, q ∈ Z, and that limt→0 F (ϕ(t)) = y.
Proof. Replacing if necessary fi by

±1
, there is no
1+(fi (x))2
n
n+1

× Rp given by

We consider the semi-algebraic map Φ : R → R
Φ(x) :=

x1
1+ x

2

,...,

xn
1+ x

loss of generality to assume y ∈ Rp .

2

,

1
1+ x

2

, F (x) .

It follows that we can suppose that the sequence Φ(xk ) is convergent to some point (u, y) ∈
Sn × Rp . By Tarski-Seidenberg theorem, B := Φ(A) is a semi-algebraic set. Thus we can

apply the Curve Selection Lemma from [8, 10] for the point (u, y) ∈ B. We obtain an analytic
curve ψ(t) in B, which tends to (u, y) when t → +0. The desired curve ϕ(t) could be easily
obtained from ψ(t).
The following useful result is well-known (see, e.g., [18, 47]); for completeness we provide
a short proof below.
Lemma 2.2 (Growth Dichotomy Lemma). Let f : (0, ) → R be a semi-algebraic function
with f (t) = 0 for all t ∈ (0, ). Then there exist constants c = 0 and q ∈ Q such that
f (t) = ctq + o(tq ) as t → 0+ .
Proof. The set {(t, f (t)) ∈ R2 | 0 < t < } is semi-algebraic. By the Curve Selection Lemma
[8, 10], there exist δ > 0 and a parametrized analytic curve (x(s), y(s)), s ∈ (−δ, δ), such
that x(0) = 0, x(s) > 0 and f (x(s)) = y(s) for s ∈ (0, δ). By a change of the parameter s we
can assume that x(s) = sk , for some positive integer k. Then f (t) = y(t1/k ) has the desired
form.
2.2. The transversality theorem with parameters. Let P, X and Y be some C ∞ manifolds of finite dimension. Let S be a C ∞ sub-manifold of Y . Let F : X → Y be a C ∞ map.
Denote dx F : Tx X → TF (x) Y , the derivative map of F at x, where Tx X and TF (x) Y are,
respectively, the tangent space of X at x and Y at F (x).
5


Definition 2.2. The map F is said to be transverse to the sub-manifold S, abbreviated
F S, if either F (X) ∩ S = ∅ or we have for each x ∈ F −1 (S),
dx F (Tx X) + TF (x) S = TF (x) Y.
Remark 2.2. If dim X ≥ dim Y and S = {s}, then F S if and only if either F −1 (s) = ∅
or rankdx F = dim Y for all x ∈ F −1 (s). Moreover, if dim X < dim Y , then F
S if and
−1
only if F (S) = ∅.
The following result is useful in the sequel (see [23, 24]).
Theorem 2.2 (Transversality Theorem). Let F : P ×X → Y be a C ∞ map. For each p ∈ P,
consider the map Fp : X → Y defined by Fp (x) := F (p, x). If F transversal to S, then the set

D = {p ∈ P | Fp

S}

is open and dense in P. Moreover, if P, X, Y, and S are semi-algebraic sets and if F is a
semi-algebraic map, then D is also semi-algebraic.
Proof. The proof of openness and density of D is done in [23, 24]. The method used also
permits to prove that D is semi-algebraic if P, X, Y, S and F are semi-algebraic.
2.3. The set of asymptotic critical values. Let F = (f1 , . . . , fp ) : Rn → Rp be a C 1 -map,
and define the Rabier function νF : Rn → R by
p

νF (x) :=

λi ∇fi (x) .

min

p
i=1

|λi |=1

i=1

Remark 2.3. (i) By definition, νF (x) = 0 if and only if the gradient vectors ∇f1 (x), . . . , ∇fp (x)
are linearly dependent.
(ii) If the map F is semi-algebraic then so is νF .
Definition 2.3. [33, 57] We define the set of asymptotic critical values of F as
˜ ∞ (F ) := {y ∈ Rp | ∃{xk }k∈N ⊂ Rn such that

K
lim xk = ∞,

k→∞

lim F (xk ) = y,

k→∞

and

lim νF (xk ) = 0}.

k→∞

˜ ∞ (F ) is closed, and K
˜ ∞ (F ) = ∅ if F is a proper map in the sense that
Clearly the set K
lim

x →∞

F (x) = ∞.

6


2.4. Newton polyhedra. Throughout the text, we consider a fixed coordinate system
x1 , . . . , xn ∈ Rn . Let J ⊂ {1, . . . , n}, then we define
RJ := {x ∈ Rn | xj = 0, for all j ∈ J}.

We denote by R+ the set of non-negative real numbers. We also set Z+ := R+ ∩ Z.
If κ = (κ1 , . . . , κn ) ∈ Zn+ , we denote by xκ the monomial xκ1 1 · · · xκnn and by |κ| the sum
κ1 + · · · + κn .
Definition 2.4. A subset Γ ⊂ Rn+ is said to be a Newton polyhedron at infinity, if there
exists some finite subset A ⊂ Zn+ such that Γ is equal to the convex hull in Rn of A ∪ {0}.
Hence we say that Γ is the Newton polyhedron at infinity determined by A and we write
Γ = Γ(A). We say that a Newton polyhedron at infinity Γ ⊂ Rn+ is convenient if it intersects
each coordinate axis in a point different from the origin, that is, if for any i ∈ {1, . . . , n} there
exists some integer mj > 0 such that mj ej ∈ Γ, where {e1 , . . . , en } denotes the canonical
basis in Rn .
Given a Newton polyhedron at infinity Γ ⊂ Rn+ and a vector q ∈ Rn , we define
d(q, Γ) := min{ q, κ | κ ∈ Γ},
∆(q, Γ) := min{κ ∈ Γ | q, κ = d(q, Γ)}.
We say that a subset ∆ of Γ is a face of Γ if there exists a vector q ∈ Rn such that
∆ = ∆(q, Γ). The dimension of a face ∆ is defined as the minimum of the dimensions of the
affine subspaces containing ∆. The faces of Γ of dimension 0 are called the vertices of Γ. We
denote by Γ∞ the union of the faces of Γ which do not contain the origin 0 in Rn .
Let Γ1 , . . . , Γp be a collection of p Newton polyhedra at infinity in Rn+ , for some p ≥ 1.
The Minkowski sum of Γ1 , . . . , Γp is defined as the set
Γ1 + · · · + Γp = {κ1 + · · · + κp | κi ∈ Γi , for all i = 1, . . . , p}.
By definition, Γ1 + · · · + Γp is again a Newton polyhedron at infinity. Moreover, by applying
the definitions given above, it is easy to check that
d(q, Γ1 + · · · + Γp ) = d(q, Γ1 ) + · · · + d(q, Γp ),
∆(q, Γ1 + · · · + Γp ) = ∆(q, Γ1 ) + · · · + ∆(q, Γp ),
for all q ∈ Rn . As an application of these relations, we obtain
Lemma 2.3. Let ∆ be a face of Γ1 + · · · + Γp . Then there exists a unique collection of faces
∆1 , . . . , ∆p of Γ1 , . . . , Γp , respectively, such that
∆ = ∆ 1 + · · · + ∆p .
In particular, Γ∞ ⊂ Γ1,∞ + · · · + Γp,∞ .
7

Let f : Rn → R be a polynomial function. Suppose that f is written as f = κ aκ xκ .
Then the support of f, denoted by supp(f ), is defined as the set of those κ ∈ Zn+ such that
aκ = 0. We denote the set Γ(supp(f )) by Γ(f ). This set will be called the Newton polyhedron
at infinity of f. The polynomial f is said to be convenient when Γ(f ) is convenient. If f ≡ 0,
then we set Γ(f ) = ∅. Note that, if f is convenient, then for each nonempty subset J of
{1, . . . , n}, we have Γ(f ) ∩ RJ = Γ(f |RJ ). The Newton boundary at infinity of f , denoted by
Γ∞ (f ), is defined as the union of the faces of Γ(f ) which do not contain the origin 0 in Rn .
Let us fix a face ∆ of Γ∞ (f ). We define the principal part of f at infinity with respect to
∆, denoted by f∆ , as the sum of those terms aκ xκ such that κ ∈ ∆.
Remark 2.4. By definition, for each face ∆ of Γ∞ there exists a vector q = (q1 , . . . , qn ) ∈ Rn
with minj=1,...,n qj < 0 such that ∆ = ∆(q, Γ).
2.5. Non-degeneracy at infinity. In [29] (see also [32]), Khovanskii introduced a condition
of non-degeneracy of complex analytic maps F : (Cn , 0) → (Cp , 0) in terms of the Newton
polyhedra of the component functions of F. This notion has been applied extensively to
the study of several questions concerning isolated complete intersection singularities (see for
instance [9, 15, 22, 53]). We will apply this condition for real polynomial maps. First we
need to introduce some notation.
Let F := (f1 , . . . , fp ) : Rn → Rp , 1 ≤ p ≤ n, be a polynomial map. Let Γ(F ) denote
the Minkowski sum Γ(f1 ) + · · · + Γ(fp ), and we denote by Γ∞ (F ) the union of the faces of
Γ(F ) which do not contain the origin 0 in Rn . Let ∆ be a face of the Γ(F ). According to
Lemma 2.3, let us consider the decomposition ∆ = ∆1 + · · · + ∆p , where ∆i is a face of Γ(fi ),
for all i = 1, . . . , p. We denote by F∆ the polynomial map (f1,∆1 , . . . , fp,∆p ) : Rn → Rp , and
the Jacobian matrix of F∆ at x is denoted by DF∆ (x).
Definition 2.5. We say that F is Khovanskii non-degenerate at infinity if and only if for
any face ∆ of Γ∞ (F ), we have
F∆−1 (0) ∩ {x ∈ Rn | rank(DF∆ (x)) < p} ⊂ {x ∈ Rn | x1 · · · xn = 0}.
The following result will be useful for our later analysis.
Theorem 2.3. Let F = (f1 , . . . , fp ) : Rn → Rp be a polynomial map such that fi is convenient, for all i = 1, . . . , p. Suppose that F is Khovanskii non-degenerate at infinity. Then

˜ ∞ (F ) = ∅.
K
˜ ∞ (F ) = ∅; i.e., there exist a point y ∈ Rp and a
Proof. By contradiction, suppose that K
sequence {xk }k∈N ⊂ Rn such that
lim xk = ∞,

k→∞

lim F (xk ) = y,

k→∞

8

and

lim νF (xk ) = 0.

k→∞


By definition, there exists a sequence λk := (λk1 , . . . , λkp ) ∈ Rp , with
we have for all k ≥ 1,

p
i=1

|λki | = 1, such that

p

λki ∇fi (xk ) .

k

νF (x ) =
i=1

By the Curve Selection Lemma at infinity (Lemma 2.1), there exist analytic curves ϕ(t) :=
(ϕ1 (t), . . . , ϕn (t)) and λ(t) := (λ1 (t), . . . , λp (t)), 0 < t
1, such that
(a1) limt→0 ϕ(t) = ∞;
(a2) limt→0 F (ϕ(t)) = y ∈ Rp ;
(a3) pi=1 |λi (t)| = 1; and
p
(a4) limt→0
i=1 λi (t)∇fi (ϕ(t)) = 0.
Let J := {j | ϕj ≡ 0}. By Condition (a1), J = ∅. Thanks to Growth Dichotomy Lemma
(Lemma 2.2), for each j ∈ J, we can expand the coordinate ϕj in terms of the parameter:
say
ϕj (t) = x0j tqj + higher order terms in t,
where x0j = 0. From Condition (a1), we get minj∈J qj < 0.
Recall that RJ := {κ := (κ1 , κ2 , . . . , κn ) ∈ Rn | κj = 0 for j ∈ J}. Since fi is convenient,
Γ(fi ) ∩ RJ = ∅. Let di be the minimal value of the linear function j∈J qj κj on Γ(fi ) ∩ RJ ,
and let ∆i be the (unique) maximal face of Γ(fi ) ∩ RJ where the linear function takes this
value. Since fi is convenient, di < 0 and ∆i is a face of Γ∞ (fi ). Note that fi,∆i does not
dependent on xj for all j ∈ J. By a direct calculation, then
fi (ϕ(t)) = fi,∆i (x0 )tdi + higher order terms in t,
where x0 := (x01 , . . . , x0n ) with x0j = 1 for j ∈ J. By Condition (a2) and di < 0, we have

(3)

fi,∆i (x0 ) = 0,

for all i = 1, . . . , p.

Let I := {i | λi ≡ 0}. It follows from Condition (a3) that I = ∅. For i ∈ I, expand the
coordinate λi in terms of the parameter: say
λi (t) = λ0i tθi + higher order terms in t,
where λ0i = 0.
For i ∈ I and j ∈ J we have
∂fi
∂fi,∆i 0 di −qj
(ϕ(t)) =
(x )t
+ higher order terms in t.
∂xj
∂xj
9


It implies that
λi (t)
i∈I

∂fi
(ϕ(t)) =
∂xj

λ0i

∂fi,∆i 0 di +θi −qj
(x )t
+ higher order terms in t
∂xj

λ0i

∂fi,∆i 0
(x ) t −qj + higher order terms in t,
∂xj

i∈I

=
i∈I

where := mini∈I (di + θi ) and I := {i ∈ I | di + θi = } = ∅. Then by Condition (a4), we
have for all j ∈ J,

i∈I

p

∂fi
λi (t)
(ϕ(t)) =
∂xj

λi (t)

i=1

∂fi
(ϕ(t)) → 0,
∂xj

as t → 0.

There are two cases to be considered.
Case 1:

≤ qj∗ := minj∈J qj . We have for all j ∈ J,
λ0i
i∈I

∂fi,∆i 0
(x ) = 0,
∂xj

which implies easily that


rank 


x01
x01

∂f1,∆1
(x0 )

∂x1

..
.

∂fp,∆p
(x0 )
∂x1

···

x0n

···
···

x0n

∂f1,∆1
(x0 )
∂xn

..
.

∂fp,∆p
(x0 )
∂xn




 < p.


This, together with (3), contradicts with the assumption that the polynomial map F =
(f1 , . . . , fp ) is Khovanskii non-degenerate at infinity.
Case 2: > qj∗ := minj∈J qj . It follows from Condition (a3) that θi ≥ 0 for all i ∈ I and
θi = 0 for some i ∈ I. Without lost of generality, we may assume that 1 ∈ I and θ1 = 0.
Since f1 is convenient, for any j = 1, . . . , n, there exists a natural number mj ≥ 1 such that
mj ej ∈ Γ∞ (f1 ). Then it is clear that
qj mj ≥ d1 ,

for all j ∈ J.

On the other hand, we have
d1 = d1 + θ1 ≥ min(di + θi ) = .
i∈I

Therefore
qj∗ mj∗ ≥ d1 ≥ > qj∗ .
Since qj∗ = minj∈J qj < 0, it implies that mj∗ < 1, which is a contradiction.
10


Definition 2.6. Let F := (f1 , . . . , fp ) : Rn → Rp , 1 ≤ p
say that F is non-degenerate at infinity if and only if for
x ∈ (R \ {0})n , we have
 ∂f
∂f1,∆1
1,∆1

(x) · · · xn ∂x
(x) f1
x1 ∂x
n
1

.
.
..
..
rank 
···

∂fp,∆p
∂fp,∆p
(x) · · · xn ∂x
(x) 0
x1 ∂x
n
1

≤ n, be a polynomial map. We
any face ∆ of Γ∞ (F ) and for all

···
...

0

···

fp



 = p.


For each a subset I := {i1 , . . . , iq } ⊂ {1, . . . , p}, we define the polynomial map FI : Rn →
Rq by FI (x) = (fi1 (x), . . . , fiq (x)).
The connection between non-degeneracy conditions is given by the following result:
Lemma 2.4. Let F := (f1 , . . . , fp ) : Rn → Rp , 1 ≤ p ≤ n, be a polynomial map. Then the
following statements hold
(i) F is non-degenerate at infinity if and only if FI is Khovanskii non-degenerate at
infinity, for all subset I ⊂ {1, . . . , p}.
˜ I ) = ∅ for all nonempty subset I ⊂
(ii) If F is non-degenerate at infinity then K(F
{1, . . . , p}.
Proof. The first statement is straightforward from the definition, and so the second statement
follows from Theorem 2.3.
The above lemma implies that if F is non-degenerate at infinity then F is Khovanskii
non-degenerate at infinity. The converse does not hold. However, both conditions constitute
generic conditions in a sense that we will explain in Section 5.
3. A Frank-Wolfe type Theorem
In this section we prove a Frank-Wolfe type theorem for polynomial maps that are nondegenerate at infinity with respect to their Newton polyhedron.
Let f0 , f1 , . . . , fp : Rn → R be polynomial functions, and let
S := {x ∈ Rn | fi (x) ≤ 0, i = 1, . . . , p} = ∅.
The main result of this section is as follows.
Theorem 3.1. Assume that the polynomial functions f0 , f1 , . . . , fp are convenient and the
polynomial map (f0 , f1 , . . . , fp ) : Rn → Rp+1 is non-degenerate at infinity. If f0 is bounded

from below on S, then f0 attains its infimum on S.
Before proving the theorem, we need the following definition.
11


Definition 3.1. For each x ∈ S, let I(x) be the set of indices i for which fi vanishes at
x. The closed semi-algebraic set S is called regular at infinity if there exists a real number
R0 > 0 such that for each x ∈ S, x ≥ R0 , the gradient vectors ∇fi (x), i ∈ I(x), are linearly
independent.
The following lemma follows easily from the Curve Selection Lemma at infinity.
Lemma 3.1. Suppose that the closed semi-algebraic set S is unbounded and regular at infinity. Then there exists a real number R0 > 0 such that for all R ≥ R0 , the set
SR := {x ∈ S | x

2

= R2 }

is a nonempty compact set, and it is regular, i.e., for each x ∈ SR , the vectors x and ∇fi (x),
i ∈ I(x), are linearly independent.
Proof. See [25, Lemma 3.1].
Lemma 3.2. Assume that the polynomial functions f1 , . . . , fp are convenient and the polynomial map F := (f1 , . . . , fp ) : Rn → Rp is non-degenerate at infinity. Then the set S is
regular at infinity.
Proof. Suppose that the lemma does not hold. Then, by the Curve Selection Lemma at
infinity, there exist a nonempty subset I := {i1 , . . . , iq } ⊂ {1, . . . , p} and an analytic curve
ϕ(t) := (ϕ1 (t), . . . , ϕn (t)), 0 < t
1, such that
(b1) limt→0 ϕ(t) = ∞;
(b2) fi (ϕ(t)) ≡ 0 for i ∈ I and fi (ϕ(t)) < 0 for i ∈ I;
(b3) The gradient vectors ∇fi (ϕ(t)), i ∈ I, are linearly dependent.
By definition, νFI (ϕ(t)) ≡ 0 for 0 < t

1, where FI is the map x → (fi1 (x), . . . , fiq (x)).
˜ I ), which contradicts Lemma 2.4.
Consequently, we have 0 ∈ K(F
Now we are ready to prove the Frank-Wolfe type theorem 3.1.
Proof of Theorem 3.1. We will prove a stronger statement; namely that f0 is coercive on S
in the sense that
lim

k→∞

min2

x∈S, x

=k2

f0 (x) = +∞.

Suppose that it is not so; i.e., there exists a sequence {xk }k∈N ⊂ S such that
(c1) limk→∞ xk = ∞, limk→∞ f0 (xk ) = y ∈ R; and
(c2) xk is a solution of the following problem
min2

x∈S, x

=k2

12

f0 (x).

By Lemma 3.2, the set S is regular at infinity, and so the set Sk := S ∩{ x 2 = k 2 }, k
1,
is regular, in view of Lemma 3.1. It follows from Lagrange’s multipliers theorem that there
exist real numbers λki , i = 1, . . . , p, and µk such that
(c3) λki fi (xk ) = 0 for i = 1, . . . , p; and
(c4) ∇f0 (xk ) + pi=1 λki ∇fi (xk ) + µk xk = 0.
By the Curve Selection Lemma at infinity, there exist a nonempty subset I := {i1 , . . . , iq } ⊂
{1, . . . , p}, an analytic curve ϕ(t), and some analytic functions λi (t), i ∈ I, and µ(t), for
01, such that
(c5) limt→0 ϕ(t) = ∞ and limt→0 f0 (ϕ(t)) = y;
(c6) fi (ϕ(t)) ≡ 0 for i ∈ I and fi (ϕ(t)) < 0 for i ∈ I;
(c7) ∇f0 (ϕ(t)) + i∈I λi (t)∇fi (ϕ(t)) + µ(t)ϕ(t) ≡ 0.
It implies that
µ(t) d ϕ(t)
2
dt

2

= µ(t) ϕ(t),

dϕ
dt

= − ∇f0 (ϕ(t)),
= −

dϕ
dt

d
(f0 ◦ ϕ)(t) −
dt

λi (t) ∇fi (ϕ(t)),

−
i∈I

λi (t)
i∈I

dϕ
dt

d
d
(fi ◦ ϕ)(t) = − (f0 ◦ ϕ)(t).
dt
dt

Therefore
d
(f0 ◦ ϕ)(t)
dt

=

=

µ(t) d ϕ(t) 2
2
dt
∇f0 (ϕ(t)) + i∈I λi (t)∇fi (ϕ(t))
2 ϕ(t)

d ϕ(t)
dt

2

.

On the other hand, we may write
ϕ(t)

= c1 tα + higher order terms in t,

f0 (ϕ(t)) = c2 tβ + higher order terms in t,
where c1 = 0, c2 = 0 and α < 0, β ≥ 0 (by Condition (c5)). By a direct computation, then
∇f0 (ϕ(t)) +

λi (t)∇fi (ϕ(t))

= ctβ−α + higher order terms in t,

i∈I

for some constant c = 0. Consequently, we get
lim ∇f0 (ϕ(t)) +
t→0

λi (t)∇fi (ϕ(t))

= 0.

i∈I

˜ 0 , fi1 , . . . , fiq ), which contradicts
Hence, limt→0 νFI (ϕ(t)) = 0. Therefore, (y, 0, . . . , 0) ∈ K(f
Lemma 2.4. The theorem is proved.
13


Remark 3.1. The assumption in Theorem 4.1 that the polynomials fi are convenient cannot
be removed. A counterexample is f0 (x1 , x2 ) := x21 + (x1 x2 − 1)2 and S := R2 . It is easy to
check that f0 is non-degenerate at infinity. However, f0 has infimum 0 but it is not achievable.
4. Global error bound results
4.1. The existence of a global error bound. The purpose of this subsection is to extend
the H`a’s error bound result [26, Theorem A] from polynomial functions to semi-algebraic
functions.
Let f : Rn → R be a continuous semi-algebraic function. Assume that S := {x ∈
Rn | f (x) ≤ 0} = ∅. Let [f (x)]+ := max{f (x), 0}.
Theorem 4.1. With the notations above, the following two statements are equivalent.
(i) For any sequence xk ∈ Rn \ S, xk → ∞, we have
(i1) if f (xk ) → 0 then d(xk , S) → 0;
(i2) if d(xk , S) → ∞ then f (xk ) → ∞.
(ii) There exist some constants c > 0, α > 0, and β > 0 such that

cd(x, S) ≤ [f (x)]α+ + [f (x)]β+

for all

x ∈ Rn .

The proof follows the steps in the proof of [26, Theorem A] (see also [17]). The case
S = Rn is trivial so assume that S = Rn . Let
µ(t) :=

for t ≥ 0.

sup d(x, S),
x∈f −1 (t)

Then the theorem follows from the next two claims.
Claim 4.1 (H¨older-type error bound “near from S”). The following two statements are
equivalent.
(i) For any sequence xk ∈ Rn \ S, with xk → ∞, it holds that
f (xk ) → 0

=⇒

d(xk , S) → 0;

(ii) There exist some constants c > 0, δ > 0, and α > 0 such that
cd(x, S) ≤ [f (x)]α+

for all

x ∈ f −1 ((−∞, δ]).

Proof. The conclusion will follow if we show that [(i) ⇒ (ii)] as [(ii) ⇒ (i)] follows easily.
Since S = Rn , there exists δ1 > 0 such that f −1 (t) = ∅ for all t ∈ [0, δ1 ]. Then the
condition (i) implies that there exists 0 < δ2 ≤ δ1 such that µ(t) < +∞ for all t ∈ [0, δ2 ]
and µ(t) → 0 when t → 0. In view of Tarski-Seidenberg Theorem, the function µ is semialgebraic on [0, δ1 ]. By Growth Dichotomy Lemma (Lemma 2.2), we can expand the function
µ at t = 0 to get
µ(t) = atα + higher order term in t,
14

0
1,


where a and α are some positive constants. We deduce that there exist c > 0 and 0 < δ ≤ δ2
such that
cd(x, S) ≤ [f (x)]α+

for all

x ∈ f −1 ((−∞, δ]).

Claim 4.2 (H¨older-type error bound “far from S”). Suppose that for any sequence xk ∈
Rn \ S, with xk → ∞, it holds that
d(xk , S) → ∞

=⇒

f (xk ) → ∞;

Then there exist some constants c > 0, r > 0, and β > 0 such that
cd(x, S) ≤ [f (x)]β+

for all

x ∈ f −1 ([r, +∞)).

Proof. We firstly assume that f is bounded from above, say by r > 0. It follows from the
assumption that there exists M > 0 such that d(x, S) ≤ M for all x ∈ Rn . Then we have for
all x ∈ f −1 ([r, +∞)),
r
r
f (x) ≥ r =
M≥
d(x, S),
M
M
which implies the claim.
We now suppose that f is not bounded from above. Since S = ∅ and by the continuity
of f, we get f −1 (t) = ∅ for any t ≥ 0. The assumption implies that there exists r1 ≥ 0 such
that µ(t) < +∞ for all t ∈ [r1 , +∞). Thanks to Tarski-Seidenberg Theorem, the function
µ(t) is semi-algebraic on [r1 , +∞). By Growth Dichotomy Lemma, we can write
µ(t) = btβ + lower order term in t,

t

1,

for some b > 0 and β ∈ Q. Let

M :=

sup

µ(t).

t∈[r1 ,+∞)

We distinguish two cases.
Case 1: M = +∞.
Then limt→∞ µ(t) = ∞. Therefore β > 0, and so we find that there exist c > 0 and r > r1
such that
cd(x, S) ≤ [f (x)]β+

for all

which proves Item (ii).
15

x ∈ f −1 ([r, +∞)),


Case 2: M < +∞.
Take any r ≥ r1 . Then we have for all x ∈ f −1 ([r, +∞)),
f (x) ≥ r =

r
r
M≥
d(x, S).

M
M

This implies Item (ii).
Now, we are in position to finish the proof of Theorem 4.1.
Proof of Theorem 4.1. The implication (ii) ⇒ (i) is straightforward, so we will prove the
implication (i) ⇒ (ii).
Indeed, by Claims 4.1 and 4.2, there exist positive constants c1 , c2 , δ, and r, and exponents
α > 0, β > 0 such that
c1 d(x, S) ≤ [f (x)]α+

for all

x ∈ f −1 ((−∞, δ])

c2 d(x, S) ≤ [f (x)]β+

for all

x ∈ f −1 ([r, +∞)).

and

On the other hand, it follows easily from the condition (i2) that there is a constant M > 0
such that d(x, S) ≤ M for all x ∈ f −1 ([δ, r]); and so we have
[f (x)]α+

+

[f (x)]β+

δα + δβ
≥δ +δ ≥
d(x, Z)
M
α

β

for all

x ∈ f −1 ([δ, r]).

This inequality, together with the above two inequalities, implies Item (ii).
4.2. The Palais-Smale condition and global error bound results. The notion of
subdifferential-that is, an appropriate multivalued operator playing the role of the usual
gradient map-is crucial for our considerations. For nonsmooth analysis we refer to the comprehensive texts [13, 14, 59].
ˆ (x) of a continuous function f : Rn →
Definition 4.1.
(i) The Fr´echet subdifferential ∂f
R at x ∈ Rn is given by
ˆ (x) :=
∂f

v ∈ Rn |

f (x + h) − f (x) − v, h
≥0 .
h=0
h

lim inf

h →0,

(ii) The limiting subdifferential at x ∈ Rn , denoted by ∂f (x), is the set of all cluster
ˆ (xk ) and (xk , f (xk )) → (x, f (x)) as
points of sequences {v k }k≥1 such that v k ∈ ∂f
k → ∞.
ˆ (x) (and a fortiori
Remark 4.1. It is a well-known result of variational analysis that ∂f
∂f (x)) is not empty in a dense subset of the domain of f (see [59], for example).

16


Definition 4.2. Using the limiting subdifferential ∂f, we define the nonsmooth slope of f
by
mf (x) := inf{ v | v ∈ ∂f (x)}.
By definition, mf (x) = +∞ whenever ∂f (x) = ∅.
Remark 4.2. (i) If the function f is of class C 1 , the above notion coincides with the usual
ˆ (x) = {∇f (x)}, and hence mf (x) = ∇f (x) .
concept of gradient; that is, ∂f (x) = ∂f
(ii) It is not hard to show that if the function f is semi-algebraic then so is mf .
Definition 4.3. Given a continuous function f : Rn → R and a real number t, we say that f
satisfies the Palais-Smale condition at the level t, if every sequence {xk }k∈N ⊂ Rn such that
f (xk ) → t and mf (xk ) → 0 as k → ∞ possesses a convergence subsequence.
The following result extends [26, Theorem B] from polynomial functions to semi-algebraic
functions.
Theorem 4.2. Let f : Rn → R be a continuous semi-algebraic function. Assume that S :=

{x ∈ Rn | f (x) ≤ 0} = ∅. If f satisfies the Palais-Smale condition at each level t ≥ 0, then
there exist some constants c > 0, α > 0, and β > 0 such that
cd(x, S) ≤ [f (x)]α+ + [f (x)]β+

x ∈ Rn .

for all

Proof. The proof is similar to that of [26, Theorem B]. However, instead of using the Ekeland
Variational Principle [19], we use a version of the variational principle of Borwein and Preiss
(see [12], [14, Theorem 4.2]).
It is sufficient to show that the condition (i) in Theorem 4.1 holds. We proceed by the
method of contradiction.
We first assume that there exist a number δ > 0 and a sequence xk ∈ Rn \S, with xk → ∞,
such that
f (xk ) → 0

and

d(xk , S) ≥ δ.

Let us consider the continuous semi-algebraic function
f+ : Rn → R,

x → max{f (x), 0}.

Clearly, infn f+ (x) = 0. Applying the Minimization Principle [14, Theorem 4.2] to the funcx∈R

tion f+ with data
such that

:= f+ (xk ) = f (xk ) > 0 and λ :=
z k − xk < λ,

δ
4

y k − z k < λ,

> 0, we find points y k and z k in Rn
f+ (y k ) ≤ f+ (xk ),

and such that the function
x → f+ (x) +

17

λ2

x − zk

2


is minimized over Rn at y k . We deduce from the above inequalities that
y k − xk

≤

z k − xk + y k − z k

< 2λ =

δ
,
2

which yields that limk→∞ y k = ∞ and
d(y k , S) ≥ d(xk , S) − d(xk , y k ) > d(xk , S) −

δ
δ
≥ .
2
2

Hence,
δ
δ
⊂ Rn \ S.
B(y k , ) := x ∈ Rn | x − y k <
2
2
In particular, we have f+ (x) = f (x) for all x ∈ B(y k , 2δ ). Consequently, the function
x → f (x) +

λ2

x − zk

2

attains its minimum on the open ball B(y k , 2δ ) at y k . Then, by the Fermat’s rule generalized
[59, Theorem 10.1], we get
−2 2 (y k − z k ) ∈ ∂f (y k ).
λ
Therefore
8f (xk )
mf (y k ) ≤ 2 2 y k − z k ≤ 2 ≤
.
λ
λ
δ
By letting k tend to infinity, we obtain
lim y k = ∞,

k→∞

lim f (y k ) = 0,

k→∞

lim mf (y k ) = 0.

and

k→∞

So, f does not satisfy the Palais-Smale condition at the value t = 0, and a contradiction
follows.

We next suppose that there exist a number M > 0 and a sequence xk ∈ Rn \ S, with
xk → ∞, such that
d(xk , S) → ∞ and 0 < f (xk ) ≤ M.
Again, we see that inf x∈Rn f+ (x) = 0. We now apply the Minimization Principle [14, Theorem
k
4.2] to the function f+ with data := f+ (xk ) = f (xk ) > 0 and λ := d(x4 ,S) > 0; there exist
points y k and z k in Rn with
z k − xk < λ,

y k − z k < λ,

f+ (y k ) ≤ f+ (xk ),

and having the property that the function
x − zk 2
λ2
has a unique minimum at y k . We deduce from the above inequalities that
x → f+ (x) +

d(y k , S) ≥ d(xk , S) − d(xk , y k )
d(xk , S)
,
2
which yields limk→∞ d(y k , S) = ∞. In particular, we get y k ∈ Rn \ S and y k → ∞.
≥ d(xk , S) − 2λ =

18


By an argument as above, we can easily deduce again that

mf (y k ) ≤ 2

λ2

yk − zk

≤ 2

8M
.
d(xk , S)

≤

λ

Hence,
lim mf (y k ) = 0.

k→∞

Note that 0 < f (y k ) ≤ f (xk ) ≤ M for all k ≥ 1. Hence, by passing to subsequences if
necessary, we may assume that there exists the limit t := limk→∞ f (y k ). Therefore f does not
satisfy the Palais-Smale condition at t, which is a contradiction. The proof of Theorem 4.2
is complete.
The following lemma is useful in the sequel.
Lemma 4.1. Let F = (f1 , . . . , fp ) : Rn → Rp be a map of class C 1 and let f (x) :=
maxi=1,...,p fi (x). Then f is a continuous function and
mf (x) =

λi ∇fi (x) ,

min
λi ≥ 0,

i∈I λi = 1

i∈I

where I := {i | fi (x) = f (x)}. In particular, we have for all x ∈ Rn ,
0 ≤ νF (x) ≤ mf (x).
Proof. The first statement follows from [59, Exercise 8.31], and so the second statement is
clear.
Corollary 4.1. Let F = (f1 , . . . , fp ) : Rn → Rp be a map of class C 1 , and let f (x) :=
˜ ) = ∅, then there exist
maxi=1,...,p fi (x). Suppose that S := {x ∈ Rn | f (x) ≤ 0} = ∅. If K(F
some constants c > 0, α > 0, and β > 0 such that
cd(x, S) ≤ [f (x)]α+ + [f (x)]β+

for all

x ∈ Rn .

˜ ) = ∅. By Lemma 4.1, the function f satisfies the Palais-Smale
Proof. Suppose that K(F
condition at each level t ∈ R. Then the desired result follows from Theorem 4.2.
4.3. Non-degeneracy at infinity and H¨
older-type global error bound results. In
this part, we establish the H¨older-type global error bound for polynomial maps which are
non-degenerate at infinity, where the corresponding exponents α and β can be explicitly

determined. To begin with, for any positive integers d, n, and p, we let
H(d, n, p) := d(6d − 3)n+p−1 .

19


Theorem 4.3. Let F = (f1 , . . . , fp ) : Rn → Rp , 1 ≤ p ≤ n, be a polynomial map such
that fi is convenient, for all i = 1, . . . , p. Suppose that F is non-degenerate at infinity. Let
f (x) := maxi=1,...,p fi (x) and S := {x ∈ Rn | f (x) ≤ 0} = ∅. Then there exists a constant
c > 0 such that
1

cd(x, S) ≤ [f (x)]+H(d,n,p) + [f (x)]+

for all

x ∈ Rn ,

where d := maxi=1,...,p deg fi .
The following lemma is crucially used in the proof of Theorem 4.3.
Lemma 4.2. Under the assumptions of Theorem 4.3, there exist some constants c > 0 and
R > 0 such that
mf (x) ≥ c for all
x ≥ R.
Proof. Suppose that by contradiction there exists a sequence {xk }k∈N ⊂ Rn such that
lim xk = ∞,

k→∞

and

lim mf (xk ) = 0.

k→∞

We remark that the function mf (x) is semi-algebraic. By Lemma 4.1 and by the Curve
Selection Lemma at infinity, there exist a nonempty subset I˜ ⊂ {1, . . . , p}, an analytic curve
˜ for 0 < t
ϕ(t) := (ϕ1 (t), . . . , ϕn (t)) and some analytic functions λi (t), i ∈ I,
1, such that
(d1)
(d2)
(d3)
(d4)

limt→0 ϕ(t) = ∞;
˜ and fi (ϕ(t)) < f (ϕ(t)) for i ∈ I;
˜
fi (ϕ(t)) = f (ϕ(t)) for i ∈ I,
˜ and
λi (t) ≥ 0 for all i ∈ I,
i∈I˜ λi (t) = 1.
limt→0 mf (ϕ(t)) = limt→0
i∈I˜ λi (t)∇fi (ϕ(t)) = 0.

Let J := {j | ϕj ≡ 0}. By Condition (d1), J = ∅. In view of Growth Dichotomy Lemma,
for j ∈ J, we can expand the coordinate ϕj in terms of the parameter: say
ϕj (t) = x0j tqj + higher order terms in t,
where x0j = 0. From Condition (d1), we get qj∗ := minj∈J qj < 0. Note that ϕ(t)
ctqj∗ + o(tqj∗ ) as t → 0, for some c > 0.

=

Since fi is convenient, Γ(fi ) ∩ RJ = ∅. Let di be the minimal value of the linear function
J
J
j∈J qj κj on Γ(fi ) ∩ R , and let ∆i be the (unique) maximal face of Γ(fi ) ∩ R where the
linear function takes this value. Since fi is convenient, di < 0 and ∆i is a face of Γ∞ (fi ).
Note that fi,∆i does not dependent on xj for all j ∈ J. By a direct calculation, then
fi (ϕ(t)) = fi,∆i (x0 )tdi + higher order terms in t,
where x0 := (x01 , . . . , x0n ) with x0j = 1 for j ∈ J.
Let I := {i ∈ I˜ | λi ≡ 0}. It follows from Condition (d3) that I = ∅. For i ∈ I, expand
the coordinate λi in terms of the parameter: say
λi (t) = λ0i tθi + higher order terms in t,
20


where λ0i = 0.
For i ∈ I and j ∈ J we have
∂fi
∂fi,∆i 0 di −qj
(ϕ(t)) =
(x )t
+ higher order terms in t.
∂xj
∂xj
It implies that
λi (t)
i∈I

∂fi
(ϕ(t)) =
∂xj

λ0i

∂fi,∆i 0 di +θi −qj
(x )t
+ higher order terms in t
∂xj

λ0i

∂fi,∆i 0
(x ) t −qj + higher order terms in t,
∂xj

i∈I

=
i∈I

where

:= mini∈I (di + θi ) and I := {i ∈ I | di + θi = } = ∅.

There are two cases to be considered.
Case 1:

≤ qj∗ := minj∈J qj .

We deduce from Condition (d4) that
λ0i
i∈I

∂fi,∆i 0
(x ) = 0,
∂xj

j ∈ J.

for all

Hence
rank x0j

∂fi,∆i
(x0 )
∂xj

< #I .
i∈I ,1≤j≤n

On the other hand, it follows from Lemma 2.4 that the map FI is Khovanskii non-degenerate
at infinity. Therefore there exists an index i0 ∈ I such that fi0 ,∆i0 (x0 ) = 0. Then, by
˜
Condition (d2), we have for all i ∈ I,
f (ϕ(t)) = fi (ϕ(t)) = fi0 (ϕ(t)) = fi0 ,∆i0 (x0 )tdi0 + higher order terms in t.
By taking the derivative in t of the function (f ◦ ϕ)(t), we deduce that
d(f ◦ ϕ)(t)

d(fi ◦ ϕ)(t)
=
=
dt
dt

∇fi (ϕ(t)),

dϕ(t)
dt

,

for all

˜
i ∈ I.

By Condition (d3), then
d(f ◦ ϕ)(t)
=
dt

λi (t)
i∈I˜

d(f ◦ ϕ)(t)
=
dt

λi (t)∇fi (ϕ(t)),
i∈I˜

Thus
d(f ◦ ϕ)(t)
dt

≤ mf (ϕ(t))

dϕ(t)
,
dt

which implies that
mf (ϕ(t)) ≥ c tdi0 −qj∗ + higher order terms in t,
21

dϕ(t)
dt

.


for some c > 0. But this inequality contradicts Condition (d4) since
di0 ≤ di0 + θi0 = ≤ qj∗ .
Case 2:

> qj∗ := minj∈J qj .

With a similar argument as in the proof of Theorem 2.3, it is not hard to get a contradiction.

By Lemma 4.2, the semi-algebraic function f (x) := maxi=1,...,p fi (x) satisfies the PalaisSmale condition at each t ∈ R. Then, it follows from Theorem 4.2 that a global error bound
result holds for f.
Before proving Theorem 4.3 which establishes that a H¨older-type global error bound holds,
we give an error bound result on a bounded region.
Lemma 4.3. Let S denote the set of x in Rn satisfying f1 (x) ≤ 0, . . . , fp (x) ≤ 0, where each
fi is a real polynomial. Let R be a positive number such that S contains an element x with
x ≤ R. Then, there exists a constant c > 0 such that
1

cd(x, S) ≤ [f (x)]+H(d,n,p)

for all x with x ≤ R.

Here f (x) := maxi=1,...,p fi (x) and d := maxi=1,...,p deg fi .
Proof. Consider the polynomial h : Rn × Rp given by
h(x, z) := (f1 (x) + z12 )2 + · · · + (fp (x) + zp2 )2 ,
where z := (z1 , . . . , zp ). Let
S¯ := {(x, z) ∈ Rn+p | h(x, z) = 0}.
It is easy seen that
x∈S

⇔

¯
(x, z) ∈ S,

with zi :=

[−fi (x)]+ , i = 1, . . . , p.

Further, since S = ∅ is nonempty, it follows that S¯ = ∅.
Let
R := max{ z | zi :=

[−fi (x)]+ , i = 1, . . . , p, x ≤ R} > 0.

By [56, Theorem 2.2] (see also [34]), it is not hard to check that
1

¯ ≤ h(x, z) 2H(d,n,p)
c1 d((x, z), S)

for all

x ≤ R and z ≤ R ,

for some c1 > 0.
Now by a similar argument as in the proof of [41, Theorem 2.2], one can get the conclusion
easily. In fact, given any x ∈ Rn with x ≤ R, let z ∈ Rp be given by zi := [−fi (x)]+ , i =
1, . . . , p. Then, by definition, we can see that z ≤ R and
fi (x) + zi2 = fi (x) + [−fi (x)]+ = [fi (x)]+ ≤ [f (x)]+ ,
22

for i = 1, . . . , p.


It implies that
p

(fi (x) + zi2 )2 ≤ p[f (x)]2+ .

h(x, z) =
i=1

Therefore
c1 x − x ∗

1

¯ ≤ h(x, z) 2H(d,n,p)
≤ c1 d((x, z), S)
1

≤ (p[f (x)]2+ ) 2H(d,n,p) ,
from which the desired result follows immediately.
Now, we are in position to finish the proof of Theomrem 4.3.
Proof of Theorem 4.3. Let us again consider the continuous semi-algebraic function
f+ : Rn → R,

x → max{f (x), 0}.

By definition, if f (x) > 0 then f+ (x) = f (x), ∂f+ (x) = ∂f (x) and mf+ (x) = mf (x). In view
of Lemma 4.2, there exist c1 > 0, δ > 0 and R > 0 such that
mf+ (x) ≥ c1

(4)

x ∈ f+−1 ((0, +∞)) and

for all

x ≥ R.

Thanks to Lemma 4.3, there is a constant c2 > 0 such that
1

c2 d(x, S) ≤ f+ (x) H(d,n,p)

(5)

for all

x ≤ R.

Let x ∈ Rn be such that x ∈ f+ −1 ((0, ∞)) and x > R. By [11, Corollary 4.1], there
exists a maximal absolutely continuous curve u : [0, ∞) → Rn of the dynamical system
0 ∈ u(s)
˙
+ ∂[f+ (u(s))]
satisfying u(0) = x. In addition, the function s → f+ ◦ u(s) is absolutely continuous and
strictly decreasing on [0, ∞). By [11, Corollary 4.2], we have for almost all s ∈ [0, ∞),
u(s)
˙
= mf+ (u(s))

and

d
(f+ ◦ u)(s) = −[mf+ (u(s))]2 .
ds

We have the following remark. Suppose that f+ (u(s)) > 0 and u(s) ≥ R for all s ∈
[t1 , t2 ], for some 0 ≤ t1 < t2 . It follows from the inequality (4) that
t2

f+ (u(t1 )) − f+ (u(t2 )) = −
t1
t2

≥

d
(f+ ◦ u)(s)ds =
ds

t2

[mf+ (u(s))]2 ds
t1

t2

c1 mf+ (u(s))ds =
t1

c1 u(s)
˙
ds,
t1

which yields
(6)

f+ (u(t1 )) − f+ (u(t2 )) ≥ c1 | u(t1 ) − u(t2 ) |.
23


Hence the curve u has finite length, and so it is bounded. In view of [11, Theomrem 4.5],
there exists the limit a := lims→∞ f (u(s)). In addition, we have mf (a) = 0. Let
t := inf{s | u(s) > R}.
There are two cases to be considered.
Case 1: t = ∞; i.e., u(s) > R for all s ≥ 0.
Since mf (a) = 0, it follows from the inequality (4) that f+ (a) = 0. Therefore, by (6), we
obtain
f+ (x) = f+ (x) − f+ (a) ≥ c1 u(0) − a

= c1 x − a

≥ c1 d(x, S).

Case 2: t < ∞.
We have u(t) = R. Then it follows from (4), (5) and (6) that

d(x, S) ≤ d(x, u(t)) + d(u(t), S)
1

f+ (x) − f+ (u(t)) (f+ (u(t))) H(d,n,p)
+
≤
c1

c2
1

f+ (x) (f+ (x)) H(d,n,p)
≤
+
.
c1
c2
In summary, in both cases, we have
1

cd(x, S) ≤ f+ (x) + (f+ (x)) H(d,n,p) ,
where c := min{c1 , c2 }. This, together with (5), completes the proof of Theorem 4.3.
5. Openness and density
Let Γ1 , . . . , Γp be a collection of p Newton polyhedra in Rn+ for some 1 ≤ p ≤ n. Let Γ
denote the Minkowski sum Γ1 + · · · + Γp . Then we define
D(Γ) := {c := (c1 , . . . , cp ) ∈ Rm1 × · · · × Rmp | ci = (ci,κ ),
ci,κ xκ ∈ R[x], Γ(fi ) = Γi , and

fi,ci (x) := fi (x, ci ) =
κ

Fc (x) := (f1,c1 (x), . . . , fp,cp (x)) is non-degenerate at infinity}.
For each a subset I = {i1 , . . . , iq } ⊂ {1, . . . , p}, we denote
DI (Γ) := {c := (c1 , . . . , cp ) ∈ Rm1 × · · · × Rmp | ci = (ci,κ ),
ci,κ xκ ∈ R[x], Γ(fi ) = Γi , and

fi,ci (x) := fi (x, ci ) =
κ

FI,c (x) := (fi1 ,ci1 (x), . . . , fiq ,ciq (x)) is Khovanskii non-degenerate at infinity}.
It follows from Lemma 2.4 that D(Γ) = ∩I⊂{1,...,p} DI (Γ).
24


The main result of this section is the following.
Theorem 5.1. The set D(Γ) ⊂ Rm1 × · · · × Rmp is an open and dense semi-algebraic set.
Proof. Since the number of subsets of {1, . . . , p} is finite and a finite intersection of open
and dense semi-algebraic sets is open and dense semi-algebraic, it is sufficient to prove that
for every I ⊂ {1, . . . , p}, the set DI (Γ) is open dense semi-algebraic. For simplifying the
notation, we will only consider the case I = {1, . . . , p}, the other cases are completely
similar. The proof follows immediately from Claims 5.1 and 5.2 below.
Claim 5.1. The set D{1,...,p} (Γ) is open and semi-algebraic.
Proof. The idea of the proof took from [52, Appendix]. For every face ∆ = ∆1 + · · · + ∆p ⊂
Γ∞ , we define:
V (∆) := {(x, c1 , . . . , cp ) ∈ Rn × Rm1 × · · · × Rmp | f1,∆1 (x, c1 ) = · · · = fp,∆p (x, cp ) = 0,
 ∂f

1,∆1
0 ∂f1,∆1
x01 ∂x
(x,
c
)
(x,
c
)
·
·

·
x
1
1
n ∂xn
1


..
..

 = p,
rank 
.
···
.

0 ∂fp,∆p
0 ∂fp,∆p
x1 ∂x1 (x, cp ) · · · xn ∂xn (x, cp )
V (∆)∗ := V (∆) ∩ {(x, c1 , . . . , cp ) ∈ Rn × Rm1 × · · · × Rmp | x1 · · · xn = 0}.
Note that V (∆) is closed and that V (∆)∗ = V (∆). Let us consider the union V ∗ :=
V (∆)∗ and the projection π : Rn × Rm1 × · · · × Rmp → Rm1 × · · · × Rmp . Showing that
∆⊂Γ∞

D{1,...,p} (Γ) is an open set means to prove that its complement W := π(V ∗ ) is a closed set.
One observes that W is a semi-algebraic set, since it is the projection of a semi-algebraic set
(Theorem 2.1).
Let (c01 , . . . , c0p ) ∈ W . By the Curve Selection Lemma at infinity, there exists a face ∆
of Γ∞ and a real analytic path (ϕ(t), c1 (t), . . . , cp (t))) ∈ V (∆)∗ defined on a small enough

interval (0, ) such that
(f1) limt→0 ci (t) = c0i , i = 1, . . . , p; and
(f2) either limt→0 ϕ(t) = ∞ or limt→0 ϕ(t) = a ∈ V (∆).
Let us expand the coordinate ϕj , j = 1, . . . , n, in terms of the parameter, say
ϕj (t) = x0j tqj + higher order terms in t,
where x0j = 0 and qj ∈ Z.
According to Lemma 2.3, let us consider the decomposition ∆ = ∆1 + · · · + ∆p , where
∆i is a face of Γ(fi ), for all i = 1, . . . , p. Let di be the minimal value of the linear function
n

qj κj on ∆i , and let ∆i be the (unique) maximal face of ∆i where the linear function
j=1

25