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

Error bounds, Frank-Wolfe type theorem, Newton polyhedron, nondegenerate polynomial maps.. polynomial, which is Newton non-degenerate at infinity see [32] and Section 2 for precisedefini

A FRANK-WOLFE TYPE THEOREM AND H ¨OLDER-TYPE GLOBALERROR BOUNDS FOR GENERIC POLYNOMIAL SYSTEMS

S˜I TI ˆ E P D- INH † , HUY VUI H ` A†, AND TI ˆ E ´N SO.N PHA.M ‡

Abstract This paper studies generic polynomial systems More precisely, let f 0 and

f 1 , , f p : R n → R be convenient polynomial functions, and let S := {x ∈ R n | f i (x) ≤

0, i = 1, , p} 6= ∅ The following results are shown:

(i) A Frank-Wolfe type Theorem: Suppose that the map (f 0 , f 1 , , f p ) : R n → R p+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 (f 1 , , f p ) : R n → R p is

non-degenerate at infinity Let d := max i=1, ,p deg f i and H(d, n, p) := d(6d − 3)n+p−1.

Then there exists a constant c > 0 such that

cd(x, S) ≤ [f (x)]

1 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, ,pfi(x) and [f (x)] + := max{f (x), 0}; and

(iii) For polynomial maps with fixed Newton polyhedra, the property of being

nondegen-erate at infinity is generic.

1 IntroductionLet 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 optimizationproblem

The purpose of this paper is twofold Firstly, we are concerned with the question ofexistence 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 nology Development (NAFOSTED) grant 101.01-2010.08.

Tech-are linear, it is well known that the set of optimal solutions is nonempty provided theproblem is bounded below In 1956, Frank and Wolfe [20] proved that if fi’s remain affinelinear functions for i = 1, , p, and f0 is an arbitrary quadratic polynomial, then f0 beingbounded from below over S implies that an optimal solution exists If the statement holdswith respect to other classes of polynomial functions f0, , fp we will speak of a Frank-Wolfetype 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-quadraticobjective functions and linear constraints Andronov et al [2] extended the Frank-Wolfetheorem 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 generalconvex/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 ofarbitrary degree

Secondly, we are interested in the question of whether one can use the residual (constraintviolation) at a point x ∈ Rn to bound the distance from x to the set S More precisely, westudy 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, ,pfi(x)and [f (x)]+ := max{f (x), 0} An expression of this kind is called a global error bound for theset 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 cations For a summary of the theory and applications of error bounds, we refer the readers

appli-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 wherethe polynomials f1, , fp are affine and states that the global error bound (2) holds withthe exponents α = β = 1 After the work of Hoffman, a lot of researchers have devotedthemselves 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 boundshave 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) arehighly unlikely to hold When the constrained set S defined by some affine linear functionsand a single quadratic polynomial, Luo and Sturm [44] showed that the H¨older-type globalerror bound (2) holds with the exponents α = 1

2 and β = 1 In particular, a global errorbound was obtained by H V H`a [26] for a nonlinear inequality defined by a single convenient

polynomial, which is (Newton) non-degenerate at infinity (see [32] and Section 2 for precisedefinitions).

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, inthe (local and at infinity) complex setting [29, 32] It is worth paying attention to the factthat Non-degenerate at infinity polynomial maps have a number of remarkable propertieswhich 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 theconstrained 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 followingH¨older-type global error bound holds

cd(x, S) ≤ [f (x)]

1 H(d,n,p)

+ + [f (x)]+ for all x ∈ Rn,where d := maxi=1, ,pdeg 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 theirdegrees 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 degenerate at infinity, may offer an appropriate domain on which the machinery of polynomialoptimization works with full efficiency

non-The paper is structured as follows Section 2 presents some backgrounds in the field InSection 3, we establish a Frank-Wolfe type theorem Some H¨older-type global error boundresults will be given in Section 4 Finally, in Section 5, we show that the property of beingnondegenerate at infinity is generic

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

Definition 2.1 (i) A subset of Rn is called semi-algebraic if it is a finite union of sets

of the form

{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 besemi-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 sian product of semi-algebraic sets is a semi-algebraic set;

Carte-(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 aresemi-algebraic sets

(v) If S is a semi-algebraic set, then the distance function

semi-is a semi-algebraic set as its complement semi-is the union of the complement of A and the set{x ∈ A | ∃y ∈ B, (x, y) 6∈ S} Thus, if we have a finite collection of semi-algebraic sets, thenany set obtained from them with the help of a finite chain of quantifiers is also semi-algebraic

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 toextend the result at infinity at some fibre of a polynomial maps We give here a more generalstatement, 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, andlet F := (f1, , fp) : Rn → Rp be a semi-algebraic map Assume that there exists a sequence

xk ∈ A such that limk→∞kxkk = ∞ and limk→∞F (xk) = y ∈ (R)p, where R := R ∪ {±∞}.Then there exists an analytic curve ϕ : (0, ) → A of the form

ϕ(t) = a0tq+ a1tq+1+ · · ·such that a0 ∈ Rn

\ {0}, q < 0, q ∈ Z, and that limt→0F (ϕ(t)) = y

Proof Replacing if necessary fi by 1+(f±1

i (x)) 2, there is no loss of generality to assume y ∈ Rp

We consider the semi-algebraic map Φ : Rn → Rn+1× Rp given by

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 canapply the Curve Selection Lemma from [8, 10] for the point (u, y) ∈ B We obtain an analyticcurve ψ(t) in B, which tends to (u, y) when t → +0 The desired curve ϕ(t) could be easily

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 functionwith f (t) 6= 0 for all t ∈ (0, ) Then there exist constants c 6= 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), suchthat x(0) = 0, x(s) > 0 and f (x(s)) = y(s) for s ∈ (0, δ) By a change of the parameter s wecan assume that x(s) = sk, for some positive integer k Then f (t) = y(t1/k) has the desired

2.2 The transversality theorem with parameters Let P, X and Y be some C∞ ifolds of finite dimension Let S be a C∞ sub-manifold of Y Let F : X → Y be a C∞ map.Denote dxF : TxX → TF (x)Y , the derivative map of F at x, where TxX and TF (x)Y are,respectively, the tangent space of X at x and Y at F (x)

man-Definition 2.2 The map F is said to be transverse to the sub-manifold S, abbreviated

F t S, if either F (X) ∩ S = ∅ or we have for each x ∈ F−1(S),

dxF (TxX) + TF (x)S = TF (x)Y

Remark 2.2 If dim X ≥ dim Y and S = {s}, then F t S if and only if either F−1(s) = ∅

or rankdxF = dim Y for all x ∈ F−1(s) Moreover, if dim X < dim Y , then F t S if andonly if F−1(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

(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

lim

kxk→∞kF (x)k = ∞

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 6∈ 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

Γ = Γ(A) We say that a Newton polyhedron at infinity Γ ⊂ Rn

+ is convenient if it intersectseach coordinate axis in a point different from the origin, that is, if for any i ∈ {1, , n} thereexists some integer mj > 0 such that mjej ∈ Γ, where {e1, , en} denotes the canonicalbasis in Rn

Given a Newton polyhedron at infinity Γ ⊂ Rn+ and a vector q ∈ Rn, we define

d(q, Γ) := min{hq, κi | κ ∈ Γ},

∆(q, Γ) := min{κ ∈ Γ | hq, κi = 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 theaffine subspaces containing ∆ The faces of Γ of dimension 0 are called the vertices of Γ Wedenote 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

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,∞

Let f : Rn → R be a polynomial function Suppose that f is written as f = κaκxκ.Then the support of f, denoted by supp(f ), is defined as the set of those κ ∈ Zn

+ such that

aκ 6= 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 aκxκ such that κ ∈ ∆

Remark 2.4 By definition, for each face ∆ of Γ∞there exists a vector q = (q1, , qn) ∈ Rn

with minj=1, ,nqj < 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 Newtonpolyhedra of the component functions of F This notion has been applied extensively tothe study of several questions concerning isolated complete intersection singularities (see forinstance [9, 15, 22, 53]) We will apply this condition for real polynomial maps First weneed to introduce some notation

Let F := (f1, , fp) : Rn → Rp, 1 ≤ p ≤ n, be a polynomial map Let Γ(F ) denotethe Minkowski sum Γ(f1) + · · · + Γ(fp), and we denote by Γ∞(F ) the union of the faces ofΓ(F ) which do not contain the origin 0 in Rn Let ∆ be a face of the Γ(F ) According toLemma 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, andthe 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 forany 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 nient, for all i = 1, , p Suppose that F is Khovanskii non-degenerate at infinity Then

By definition, there exists a sequence λk := (λk1, , λkp) ∈ Rp, withPp

ϕj(t) = x0jtqj + higher order terms in t,where x0j 6= 0 From Condition (a1), we get minj∈Jqj < 0

Recall that RJ := {κ := (κ1, κ2, , κn) ∈ Rn | κj = 0 for j 6∈ J } Since fi is convenient,Γ(fi) ∩ RJ 6= ∅ Let di be the minimal value of the linear functionP

j∈Jqjκj on Γ(fi) ∩ RJ,and let ∆i be the (unique) maximal face of Γ(fi) ∩ RJ where the linear function takes thisvalue Since fi is convenient, di < 0 and ∆i is a face of Γ∞(fi) Note that fi,∆i does notdependent on xj for all j 6∈ 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 6∈ J By Condition (a2) and di < 0, we have(3) fi,∆i(x0) = 0, for all i = 1, , p

Let I := {i | λi 6≡ 0} It follows from Condition (a3) that I 6= ∅ For i ∈ I, expand thecoordinate λi in terms of the parameter: say

λi(t) = λ0itθi + higher order terms in t,where λ0

t`−qj + higher order terms in t,

where ` := mini∈I(di + θi) and I0 := {i ∈ I | di+ θi = `} 6= ∅ Then by Condition (a4), wehave for all j ∈ J,

There are two cases to be considered

Case 1: ` ≤ qj∗ := minj∈Jqj We have for all j ∈ J,

x0 1

Case 2: ` > qj ∗ := minj∈Jqj 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

mjej ∈ Γ∞(f1) Then it is clear that

Definition 2.6 Let F := (f1, , fp) : Rn → Rp, 1 ≤ p ≤ n, be a polynomial map Wesay that F is non-degenerate at infinity if and only if for any face ∆ of Γ∞(F ) and for all

(i) F is non-degenerate at infinity if and only if FI is Khovanskii non-degenerate atinfinity, for all subset I ⊂ {1, , p}

(ii) If F is non-degenerate at infinity then ˜K(FI) = ∅ for all nonempty subset I ⊂{1, , p}

Proof The first statement is straightforward from the definition, and so the second statement

The above lemma implies that if F is non-degenerate at infinity then F is Khovanskiinon-degenerate at infinity The converse does not hold However, both conditions constitutegeneric 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 degenerate at infinity with respect to their Newton polyhedron

non-Let f0, f1, , fp: Rn → R be polynomial functions, and let

S := {x ∈ Rn | fi(x) ≤ 0, i = 1, , p} 6= ∅

The main result of this section is as follows

Theorem 3.1 Assume that the polynomial functions f0, f1, , fp are convenient and thepolynomial map (f0, f1, , fp) : Rn → Rp+1 is non-degenerate at infinity If f0 is boundedfrom below on S, then f0 attains its infimum on S

Before proving the theorem, we need the following definition

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, kxk ≥ R0, the gradient vectors ∇fi(x), i ∈ I(x), are linearlyindependent

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 finity Then there exists a real number R0 > 0 such that for all R ≥ R0, the set

in-SR := {x ∈ S | kxk2 = 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

Lemma 3.2 Assume that the polynomial functions f1, , fp are convenient and the nomial map F := (f1, , fp) : Rn → Rp is non-degenerate at infinity Then the set S isregular at infinity

poly-Proof Suppose that the lemma does not hold Then, by the Curve Selection Lemma atinfinity, there exist a nonempty subset I := {i1, , iq} ⊂ {1, , p} and an analytic curveϕ(t) := (ϕ1(t), , ϕn(t)), 0 < t  1, such that

(b1) limt→0kϕ(t)k = ∞;

(b2) fi(ϕ(t)) ≡ 0 for i ∈ I and fi(ϕ(t)) < 0 for i 6∈ 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 7→ (fi1(x), , fiq(x)).Consequently, we have 0 ∈ ˜K(FI), which contradicts Lemma 2.4 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

x∈S,kxk 2 =k 2f0(x) = +∞

Suppose that it is not so; i.e., there exists a sequence {xk}k∈N ⊂ S such that

(c1) limk→∞kxkk = ∞, limk→∞f0(xk) = y ∈ R; and

(c2) xk is a solution of the following problem

min

x∈S,kxk 2 =k 2f0(x)

By Lemma 3.2, the set S is regular at infinity, and so the set Sk := S ∩{kxk2 = k2}, k  1,

is regular, in view of Lemma 3.1 It follows from Lagrange’s multipliers theorem that thereexist real numbers λk

i, i = 1, , p, and µk such that(c3) λkifi(xk) = 0 for i = 1, , p; and

(c5) limt→0kϕ(t)k = ∞ and limt→0f0(ϕ(t)) = y;

(c6) fi(ϕ(t)) ≡ 0 for i ∈ I and fi(ϕ(t)) < 0 for i 6∈ I;

=

µ(t)2

dkϕ(t)k2dt

= k∇f0(ϕ(t)) +P

i∈Iλi(t)∇fi(ϕ(t))k2kϕ(t)k