DOI 10.1007/s10107-014-0806-9 F U L L L E N G T H PA P E R
New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors
G. Li · B. S. Mordukhovich · T. S. Pha.m
Received: 7 January 2013 / Accepted: 2 August 2014 / Published online: 20 August 2014
© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014
Abstract In this paper we derive new fractional error bounds for polynomial sys- tems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a gen- eral polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed tech- niques are largely based on variational analysis and generalized differentiation, which
This research was partially supported by the Australian Research Council under Grant DP-12092508.
Research of G. Li was also partially supported by the Australian Research Council Future Fellowship FT130100038. Research of B. S. Mordukhovich was also partially supported by the USA National Science Foundation under Grant DMS-1007132 and by the Portuguese Foundation of Science and Technologies under Grant MAT/11109. Research of T. S. Pha.m was also partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 101.04–2013.07.
G. Li
Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia e-mail: [email protected]
B. S. Mordukhovich (
B
)Department of Mathematics, Wayne State University, Detroit, MI 48202, USA e-mail: [email protected]
B. S. Mordukhovich
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
T. S. Pha.m
Center of Research and Development, Duy Tan University, K7/25, Danang, Quang Trung, Vietnam T. S. Pha.m
Department of Mathematics, University of Dalat, 1, Phu Dong Thien Vuong, Dalat, Vietnam e-mail: [email protected]
allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradi- ent inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for para- meterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.
Keywords Error bounds·Polynomials·Variational analysis·
Generalized differentiation · Łojasiewicz’s inequality · Hölderian stability · Polynomial optimization and complementarity
Mathematics Subject Classification 90C26·90C31·49J52·49J53·26D10
1 Introduction
Constraint sets in many optimization problems can be described by systems of inequal- ities and equalities
gi(x)≤0, i =1, . . . ,r, and hj(x)=0, j =1, . . . ,s, (1.1) wheregi,hj :Rn→Rfori =1, . . . ,randj =1, . . . ,sare real-valued functions on Rn. One of the most important issues for (1.1) is the so-callederror bounds. Denoting bySthe set of solutions to (1.1), recall that this system has a (local)error boundwith exponentτ >0 atx∈Rnif there exist a constantc>0 and an neighborhoodUofx such that
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
τ
for all x∈U, (1.2)
whered(x,S)signifies the Euclidean distance betweenx and the setS, and where [α]+ := max{α,0}. This estimates bounds the distance from an arbitrary point x around the reference one x to the solution set S via a constant multiple of a com- putableresidual function, which measures the violation of the constraintS := {x ∈ Rngi(x)≤ 0, hj(x)=0}. The study of error bounds has attracted a lot of atten- tion of many researchers over the years and has found numerous applications to, in particular, sensitivity analysis for various problems of mathematical programming, termination criteria for descent algorithms, etc. We refer the reader to [25,48,60]
for excellent surveys in these directions. It is worth noting relationships between error bounds andmetric regularity/subregularityissues in basic variational analysis [41,56], where the main attention has been paid to the case of “linear rate”(τ =1); see also [1,12,16,29] and their references for certain “fractional/root” versions.
One of the most important and celebrated error bound/metric regularity result is due to Hoffman [18] who proved, in the case of linear functionsgiandhjand solvability of system (1.1), the existence of c > 0 such that the error bound (1.2) holds with
U =Rnandτ =1.Extensions of Hoffman’s error bound result to convex inequali- ties have been well established in the literature; see, e.g. [10,19,20,22,27,30,54,59]
and the references therein. Quite recently [26] various extensions of these results have been obtained for convex inequality systems on finite-dimensional Riemannian and Hadamard manifolds. For nonconvex inequality and equality systems some local error bound results have been established in [13,44–46,58] under certain regularity condi- tions, which bound the size of a suitable subdifferential of the function in question via its values around the reference point. On the other hand, it is proved in [35,36] by using the cerebrated Łojasiewicz’s inequality [33] that (1.2) holds with someunknown fractional exponentτ when allgi andhj are polynomials or analytic functions. Fur- thermore, it is stated by the authors of [36] in their concluding remarks that “we have not been able to obtain explicit formulas for the multiplier or the exponent in the error bound. We feel that such formulas would be useful for computational and other purposes.” Note to this end that local error bound results withexplicitexponents are indeed important for both theory and applications since they can be used, e.g., to establish explicitconvergence ratesof the proximal point algorithm as demonstrated in [5,29,32]. We also refer the reader to [37] for relevant discussions on other algo- rithms and to Sect.5 below for new applications to quantitative Hölderian stability of polynomial optimization problems and nonlinear complementarity systems with polynomial data. There are some important progress along this direction for special polynomial systems. For example, as shown in [38], regularity assumptions are not needed to obtain (1.2) withτ =12if system (1.1) involves only one quadratic function;
see also [45,46] for infinite-dimensional extensions. Moreover, error bound results for system (1.1) that involves only one single polynomial has also been established in [8]
without regularity assumptions.
Among major goals of this paper are extending the results in [8] from a single poly- nomial to general polynomial systems and establishingerror boundresults (1.1) with explicit exponentsτ in (1.2). Employing advanced techniques of variational analy- sis and generalized differentiation allows us to derive error bounds for such systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials without any regularity conditions.
Besides meeting the aforementioned general goals formulated in [36], in this way we resolve, in particular, a long-standingopen questionraised in [38] about Hölderian error bounds with explicit exponents fornonconvex quadratic systems. Furthermore, we apply our error bound results to deriving verifiable conditions forHölderian stabil- ityof general polynomial optimization problems as well as nonlinear complementarity problems with polynomial data. As a by-product of our analysis, we give a positive answer to anotheropen questionraised in [31] about theρth-order semismoothnessof the maximum eigenvalue for asymmetric tensorwith explicit estimating the exponent ρ. Since the concept of symmetric tensors has been well recognized as a high-order extension of symmetric matrices with various applications to automatic control and image science [47,51,53], the result obtained is of undoubted importance for further applications to these areas. Note that much of our study on error bounds is in the spirit of [27,28,35,36] being largely motivated by the recent work on nonsmooth extensions of Łojasiewicz’s inequality initiated in [3]. It is worth emphasizing that, as demonstrated in this paper,generalized differentialtechniques can be very instrumen-
tal for revolving applied quantitative issues even forsmooth/polynomial systems. We also refer the reader to [6,7,22,23,25,30,48,49,58] and the bibliographies therein for other approaches to error bounds and their numerous applications.
The rest of the paper is organized as follows. In Sect. 2 we present some con- structions and statements from generalized differentiation of variational analysis and polynomial theory, which are widely used in the formulations and proofs of the main results below. Section3is devoted to establishing major error bounds for polynomial systems with explicitly calculated exponents. In Sect.4we consider some special set- tings for which the exponents in error bounds obtained in Sect.3can be significantly sharpen. Section5concerns applications of the error bounds established in the previ- ous sections to deriving new results on quantitative Hölderian stability for polynomial optimization problems and as well as for nonlinear complementarity problems with polynomial data. Finally, in Sect.6we present concluding remarks and discuss some directions of the future research.
Our notation is basically standard in variational analysis and optimization the- ory. All the spaces under consideration are finite-dimensional with the inner product x,y :=xTyand the Euclidean normx :=(xTx)1/2for anyx,y∈Rn, wherexT signifies the vector (as well as matrix) transposition. We use the symbolsB(x, )and B(x, )to denote the open and closed balls, respectively, of the space in question with centerxand radius >0. Given a set⊂Rn, its interior (resp. closure, boundary and convex hull) is denoted by int(resp. cl,bd,and co). Recall also that N:= {1,2, . . .}.
2 Preliminaries
This section contains the necessary preliminaries needed in the paper. We start with generalized differentiation of variational analysis referring the reader to the books [41,56] for more details and commentaries.
Given a proper extended-real-valued function f:Rn →R:=(−∞,∞], we use the symbolz →f x to indicate thatz → x and f(z)→ f(x). Our basicsubdiffer- entialof f atx ∈ dom f (known also as the general, or limiting, or Mordukhovich subdifferential) is defined by
∂f(x):=
v∈Rn∃xk
→f x, vk →v with lim inf
z→xk
f(z)− f(xk)− vk,z−xk z−xk ≥0
. (2.1) For convex functions f the subdifferential (2.1) reduces to the classical subdifferential of convex analysis
∂f(x)=
v∈Rnv,z−x ≤ f(z)− f(x) whenever z∈Rn , x∈ domf.
(2.2) In the general case the subdifferential set (2.1) is often nonconvex (e.g., for f(x)=
−|x| at 0 ∈ R) while ∂f enjoys comprehensive calculus rules based on varia-
tional/extremal principlesof variational analysis [41,56]. Note also that∂f(x)= ∅if f is locally Lipschitzian aroundx.
Definition 2.1 (Subdifferential slope) Given f:Rn→Rand using (2.1), thesubd- ifferential slopeof f atx∈ domf is defined by
mf(x):=inf
vv∈∂f(x) .
We can see directly from the definition thatmf(x) = ∞whenever∂f(¯x)= ∅.
Observe also that for f ∈C1aroundxwe have∂f(x)= {∇f(x)}and hencemf(x)=
∇f(x).
The following useful result is a consequence of [41, Theorem 3.46(ii)]; cf. also [56, Exercise 8.31].
Lemma 2.2 (Subdifferential slope for maximum functions)Let g1, . . . ,gl:Rn→R be functions of classC1, and let f(x):=maxi=1,...,lgi(x). Then f is a locally Lipschitz function, and we have
mf(x)=min
⎧⎨
⎩
i∈I(x)
λi∇gi(x)λi ≥0,
i∈I(x)
λi =1
⎫⎬
⎭,
where I(x)is the active index set at x defined by I(x):= {i |gi(x)= f(x)}.
Next let us recall some facts concerning real polynomials (or polynomials with real coefficients). As usual, we say that f: Rn →Ris apolynomialif there is a number r∈Nsuch that
f(x)=
0≤|α|≤r
λαxα,
where λα ∈ R, x = (x1, . . . ,xn), xα := xα11· · ·xnαn,αi ∈ N∪ {0}, and|α| :=
n
j=1αj. The corresponding constantris called thedegreeof f. Recall further that f: Rn →Ris (real)analyticif it can be locally represented onRnby a convergent infinite power series, i.e., for all vectorsx=(x1, . . . ,xn)∈Rn there is a neighbor- hoodU ofxsuch that for everyx=(x1, . . . ,xn)∈Uwe have
f(x)= ∞
|α|=0
λα(x−x)α.
A major property of analytic functions that is most important for this paper is given by the following classical result by Łojasiewicz [33]:
• (Łojasiewicz’s gradient inequality) If f is an analytic function with f(0)=0 and
∇f(0)=0, then there exist positive constantsc, τ,andsuch that
∇f(x) ≥c|f(x)|τ for all x ≤.
As pointed out in [37], it is often difficult to determine the corresponding exponents τin Łojasiewicz’s gradient inequality, and they are typically unknown. Some estimates of the exponentτ in the gradient inequality were derived in [8,15] in the case when f is a polynomial. To formulate these results, for eachn,d ∈Ndefine the following two constants:
κ(n,d):=(d−1)n+1 and R(n,d):=
1 if d=1,
d(3d−3)n−1 if d≥2. (2.3) It is not hard to verify that R(n,d)≥ κ(n,d)for any natural numbersnandd and that this inequality is strict whenn≥2 andd ≥2.
Lemma 2.3 (Exponent estimates in Łojasiewicz’s gradient inequality for polynomi- als)Let f be a real polynomial onRnwith degree d∈N. The following hold:
(i) (cf.[8, Theorem 4.2]) Suppose that f(0)=0and∇f(0)=0. Then there exist constants c, >0such that for allx ≤we have
∇f(x) ≥c|f(x)|τ with τ =1−R(n,d)−1.
(ii) (cf.[15,21]) Suppose that x¯ = 0 is an isolated zero of f in the sense that f(0)=0and there isδ >0with f(x) >0for all x ∈ B(0, δ)\{0}. Then there exist positive constants c, such that for allx ≤we have
∇f(x) ≥c|f(x)|τ with τ =1−κ(n,d)−1.
3 Error bounds for polynomial systems
In this section we establish new error bound results for polynomial systemwithout any regularity conditions. Let us begin with specifying the definition of local error bounds.
Definition 3.1 (Local error bounds) We say that system (1.1) has alocal Hö lderian (or Hö lder type) error bound with exponentτ >0 atx ∈Rnif there are positive constantscandsuch that
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
τ
for all x with x−x ≤, (3.1)
whereSis the solution set for the system (1.1) given by S:=
x ∈Rngi(x)≤0, i=1, . . . ,r, and hj(x)=0, j =0, . . . ,s . (3.2) Throughout this paper, to avoid triviality, we always assume that∅ =S=Rn.
Prior to deriving the main results of this section we present an example illustrating the dependence of error bounds for polynomial systems on the degree of the polyno- mials involved and on the dimension of the problem/space in question. Note that for d =2 this example is given in [36] (see also [21]).
Example 3.2 [Dependence of error bounds on polynomial degrees and space dimen- sions] Let d ∈ N, and let hj(x1, . . . ,xn) := xj+1 −xdj for j = 1, . . . ,n −1, hn(x1, . . . ,xn):=xnd, andgi(x)≡0 for alli =1, . . . ,r in (3.1). Then the solution setSfor (3.1) isS = {x ∈Rn|x =0}. Take furtherx =0 and consider the family of vectors x() := (, d, . . . , dn−1) ∈ Rn with ∈ (0,1]. It is easy to see that d(x(),S)=n
i=12di−1 =O(),n
j=1|hj(x)| =dn, and thus we have d
x(),S
=O n
j=1
|hj(x())|
1
dn
,
which shows that the exponentτ in (3.1) for this system atxdoes not exceedd−n. Our first goal in this section is employing Lemmas 2.2 and 2.3(i) to obtain a nonsmoothversion of Łojasiewicz’s gradient inequality formaximumfunctions over finitely many polynomials with anexplicit exponent. It is certainly of its own interest while being applied in what follows to deriving error bounds for polynomial systems with explicit fractional exponents.
Theorem 3.3 (Nonsmooth Łojasiewicz’s inequality with explicit exponent for max- imum functions)Let f(x):=max{g1(x), . . . ,gl(x)}, where gi for i =1, . . . ,l are real polynomials on Rn with their degrees not exceeded d, and let x ∈ Rn with
f(x)=0. Then there are numbers c, >0such that
mf(x)≥c|f(x)|1−R(n+l−11,d+1) for all x with x−x ≤,
wheremf(x)is the subdifferential slope from Definition2.1, and where the constant R is defined in(2.3).
Proof Without loss of generality, assume thatgi(x)=0 for alli =1, . . . ,l. Then for each subsetI := {i1, . . . ,iq} ⊂ {1, . . . ,l},we define the functionFI :Rn×Rq−1→ Rby
FI(x, λ):=
q−1
j=1λjgij(x)+
1−q−1 j=1λj
giq(x) ifq ≥2,
gi1(x) ifq =1,
which is clearly a polynomial onRn+q−1with degree at mostd+1 andF(x, λ)¯ =0 for allλ∈Rq−1.Define further the setP⊂Rq−1by
P:=
⎧⎨
⎩λ∈Rq−1λj ≥0,
q−1
j=1
λj ≤1
⎫⎬
⎭.
Then, there exist numberscI >0 andI >0 for which we have
∇FI(x, λ) ≥cI|FI(x, λ)|1−R(n+q−11,d+1) whenever x−x ≤I and λ∈P.
(3.3) To verify (3.3), by standard compactness arguments, we only need to check that for eachλ¯ ∈Pthere are numbersc(λ) >¯ 0 and(λ) >¯ 0 such that
∇F(x, λ) ≥c(λ)¯ |F(x, λ)|1−R(n+q−1,d+1)1 for allx−x ≤c(λ),¯ λ− ¯λ ≤(λ).¯ Indeed, sinceF(x, λ)¯ =0 for allλ∈P, it is obvious if∇F(x,λ) =¯ 0, while in the remaining case of∇F(x,λ) =¯ 0 this inequality follows from Lemma2.3(i).
Let c := min
cI| I ⊂ {1, . . . ,l}, I = ∅ > 0 and := min
I| I ⊂ {1, . . . ,l}, I = ∅ > 0. Pick an arbitrary pointx inRn withx −x ≤ and denoteI¯ = I(x):= {i |gi(x)= f(x)}. Lemma2.2tells us that there are numbers λi ≥0 fori ∈ ¯Isuch that
i∈ ¯Iλi =1 and mf(x)=
i∈ ¯I
λi∇gi(x) .
Let us renumerate the index setI¯asI¯= {i1, . . . ,iq0}, whereq0signifies its cardinality.
Then
FI¯(x, λi1, . . . , λiq0−1)=
q0
j=1
λijgij(x)=
i∈ ¯I
λigi(x)=
i∈I(x)
λigi(x)= f(x).
Furthermore, we have the representations
∇FI¯(x, λi1, . . . , λiq0−1) =
⎛
⎝q0
j=1
λij∇gij(x),gi1(x)−giq
0(x), . . . ,giq0−1(x)−giq
0(x)
⎞
⎠
=
q0
j=1
λij∇gij(x)=
i∈I(x)
λi∇gi(x)=mf(x),
which, being combined with inequality (3.3), allow us to conclude that mf(x)= ∇FI¯(x, λi1, . . . , λiq0−1)
≥cI¯|FI¯(x, λi1, . . . , λiq0−1)|1−R(n+q0−11,d+1)
=cI¯|f(x)|1−R(n+q0−11,d+1),
≥c|f(x)|1−R(n+l−11,d+1)
and thus to complete the proof of the theorem.
Employing further the nonsmooth Łojasiewicz’s inequality of Theorem3.3leads us to effective error bounds of polynomial systems with explicit exponents. To proceed, we need the following lemma on error bounds for locally Lipschitz functions taken from [46, Corollary 2].
Lemma 3.4 (Sufficient condition for error bounds of Lipschitz functions)Let f:Rn
→Rbe locally Lipschitzian around x ∈ bd Sf, where Sf = {x| f(x)≤0}. Assume that there are numbers c, > 0 such that τmf(x) ≥ c|f(x)|1−τ for all x with x−x ≤and x∈/ Sf. Then we have
d(x,Sf)≤ 1 c
f(x)τ
+ whenever x−x ≤ 2. Now we are ready to derive the first error bound result of this paper.
Theorem 3.5 (Local error bounds with explicit fractional exponents for polynomial systems, type I)Let gias i=1, . . . ,r and hjas j =1, . . . ,s be real polynomials on Rnwith degree at most d, and let S be the solution set(3.2). Then there are numbers c, >0such that
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
1
R(n+r+s,d+1)
whenever x−x ≤,
where the quantity R is defined in(2.3).
Proof The conclusion is rather straightforward if eitherx∈intSorx∈/ S. To proceed with the remaining case ofx∈ bdS, define the function f:Rn→R+by
f(x):=max
[g1(x)]+, . . . ,[gr(x)]+,|h1(x)|, . . . ,|hs(x)|
and easily verify the representations f(x)=max
[g1(x)]+, . . . ,[gr(x)]+,|h1(x)|, . . . ,|hs(x)|
=max
0,g1(x), . . . ,gr(x),h1(x), . . . ,hs(x),−h1(x), . . . ,−hs(x) with f(x)=0. Form further the vectore:=(e1, . . . ,es)∈ {−1,1}s and define the function
fe(x):=max
0,g1(x), . . . ,gr(x),e1h1(x), . . . ,eshs(x) , x ∈Rn, which is the maximum ofr +s+1 polynomials with degree at most d and with
fe(x)=0. Employing Theorem3.3gives us numbersc(e) >0 and(e) >0 such that
mfe(x)≥c(e)|fe(x)|1−R(n+r+s,d+1)1 whenever x−x ≤(e).
Letc := mine∈{−1,1}sc(e) > 0 and := mine∈{−1,1}s(e) > 0. Take any x with x−x ≤and f(x) >0. Then for each j =1, . . . ,swe have that eitherhj(x)=
−hj(x)orhj(x) < f(x). It allows us to finde¯∈ {−1,1}s so that f(x)= fe¯(x)and mf(x)=mfe¯(x). This gives us the estimate
mf(x)=mfe¯(x)≥c(¯e)|fe¯(x)|1−R(n+r+s,d+1)1 ≥c|f(x)|1−R(n+r+s,d+1)1 , which completes the proof of the theorem by applying Lemma3.4.
Employing another technique (somewhat similar to [35,36]) and Lemma2.3(i), the next theorem provides a local error bound with an explicit exponent for polynomial systems, which is different from that in Theorem3.5. The idea of the proof is to use certainslack variablesto convert the polynomial system (1.1) into a single polynomial and then apply Lemma2.3(i).
Theorem 3.6 (Local error bounds with explicit fractional exponents for polynomial systems, type II)Let gias i =1, . . . ,r and hjas j=1, . . . ,s be real polynomials on Rnwith degree at most d, and let S be given in(3.2). Then there are numbers c, >0 such that
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
2
R(n+r,2d)
whenever x−x ≤,
where the quantity R is defined in(2.3).
Proof Similarly to the proof of Theorem3.5, we only need to examine the case of x∈ bdS. Define the polynomialθ:Rn×Rr →Rby
θ(x,z):=
r i=1
gi(x)+z2i2
+ s
j=1
hj(x)2
and note that its degree does not exceed 2d. Consider the set S := {(x,z) ∈ Rn×Rr | θ(x,z) = 0}and the continuous mapping φ(x) := (!
[−g1(x)]+, . . . ,
![−gr(x)]+)onRn.
Since x ∈ S, we have θ(x, φ(x)) = 0 and ∇θ(x, φ(x)) = 0. Applying Lemma 2.3(i) to θ gives us positive numbers 0 and c0 such that ∇θ(x,z) ≥ c0θ(x,z)1−R(n+1r,2d) for all (x,z)−(x, φ(x)) ≤0.Letc:=c−01. Then Lemma3.4 ensures the estimate
d
(x,z),S
≤cθ(x,z)R(n+1r,2d) for all (x,z) with (x,z)−(x, φ(x)) ≤ 0
2. (3.4) By continuity ofφwe find 0< < 40 such thatφ(x)−φ(x) ≤ 40 whenever x− x ≤, which clearly implies the inequality
(x, φ(x))−(x, φ(x)) ≤0
whenever x−x ≤. (3.5)
Now letxbe an arbitrary vector satisfyingx− ¯x ≤. There exists a point(x, z)∈S such that(x, φ(x))−(x, z) =d
(x, φ(x)),S
.By definition ofSwe haveθ( x, z)= 0,and hencegi(x)= −z2i ≤0, i =1, . . . ,r, and hj(x)=0, j =1, . . . ,s.This implies thatx ∈S. Therefore
d(x,S)≤ x−x ≤ (x, φ(x))−( x, z) =d
(x, φ(x)),S
≤cθ(x, φ(x))R(n+1r,2d)
=c r
i=1
[gi(x)]2++ s
j=1
hj(x)2
R(n+r,2d)1
≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
2
R(n+r,2d)
,
where the third inequality follows from (3.4) and (3.5) while the last equality follows from the fact that gi(x)+ [−gi(x)]+ = [gi(x)]+. This justifies the claimed error
bound.
Remark 3.7 (Comparing the two types of local error bounds)It is worth noting that the two types of local error bounds obtained in Theorems3.5and3.6are generally independentfrom each other. Recall thatR(p,q)=q(3q −3)p−1in the setting of Theorem3.6. Consider, e.g., the case ofn = 3,r = 4,s = 1, andd = 2. Then
2
R(n+r,2d) = R(72,4) = 4·196 and R(n+r+1s,d+1) = R(81,3) = 3·167; thus we have in this case that R(n+2r,2d) < R(n+r+1s,d+1). On the other hand, lettingn =r =1,s=2, and d =2, we get that R(n+2r,2d) = R(22,4) = 181 and R(n+r+1s,d+1) = R(41,3) = 4·192; so it gives R(n+2r,2d) > R(n+r+1s,d+1).
As a consequence of the theorem, we now establish someglobalizederror bound results with explicit exponents for polynomial systems of type (1.1) overcompactsets.
Corollary 3.8 (Hölderian error bounds with explicit exponents for polynomial sys- tems over compact sets)Let gi, hj, and S be as in Theorem3.6. Then for any compact set K ⊂Rnthere is a number c>0such that
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
τ
for all x ∈K, (3.6)
where the exponentτ is calculated as τ :=max
" 1
R(n+r+s,d+1), 2 R(n+r,2d)
#
= max
" 1
(d+1)(3d)n+r+s−1, 1 d(6d−3)n+r−1
#. (3.7)
In particular, the local Hölderian error bound(3.1)holds withτ given by(3.7).
Proof Combining the results of Theorem3.5and Theorem3.6, for everyx∈Rnwe can find numbers(x) >0 andc(x) >0 such that
d(x,S)≤c(x) r
i=1
[gi(x)]++ s
j=1
|hj(x)|
τ
whenever x−x ≤(x).
Then the conclusion follows by using standard compactness arguments.
Let us mention that the authors of [38] established a Hölder error bound with exponentτ =12over compact sets for a single quadratic function and thenraised the questionabout the possibility to extend this result tofinitely many quadratic functions.
They actually conjectured that a Hölder error bound would hold over compact sets for nonconvex quadratic systems with exponentτ = 21p withpdenoting the number of quadratic functions involved in the system. Now we provide apartial answerfor their conjecture by showing that such an error bound holds with alargerwhileexactly calculated exponent.
Corollary 3.9 (Hölderian error bounds over compact sets for nonconvex quadratic systems)Let r,s ∈ N, let gi as i =1, . . . ,r and hj as j = 1, . . . ,s be quadratic functions onRn, and let S be defined in(3.2). Then for any compact set K ⊂Rnthere is a number c>0such that the error bound inequality(3.6)holds with the explicit exponentτcalculated by
τ =max
" 1
R(n+r+s,3), 2 R(n+r,4)
#=max
" 1
3·6n+r+s−1, 1 2·9n+r−1
#.
Proof It follows from Corollary3.8withd=2 and formula (3.7) for calculatingτ. Next we show that the globalized version of the Hölderian error bound result from Corollary3.8over compact setcannotbe generally extended to theglobalone over the whole spaceRn. The following example was used in [11] in the case ofd =2, Example 3.10 (Failure of global error bounds for polynomial systems)Letd be any even number. Define the polynomial functionh:R2→Rbyh(x):=(x1x2−1)d+ (x1−1)d.The solution set here isS = {x ∈R2| h(x)=0} = {(1,1)}. The global version of the error bound in Corollary3.8is as follows: there are numbersc, τ >0 such that
d(x,S)≤c|h(x)|τ for all x∈Rn. (3.8) To show that (3.8) fails, consider a sequencexk =(k1,k)forh(xk)=(1−1k)d →1 andd(xk,S)→ ∞ask→ ∞. Then the global error bound (3.8) is obviously violated along this sequence.
We conclude this section by establishing (as yet another consequence of the main results above) a Hölder-type regularity property for two nonconvex semi-algebraic sets, i.e., subsets ofRnthat can be described by finitely many equality and inequality
constraints given by polynomials. We say that the pair of sets{Q,T}has thebounded Hölderian regularity propertywith exponentτ >0 if for each compact setK there is a constantc>0 such that
d(x,Q∩T)≤c
d(x,Q)+d(x,T)τ
whenever x∈ K. (3.9) For convex sets withτ =1 in (3.9) this property reduces to the so-calledbounded linear regularityof [2], which is an important concept of convex analysis and optimization with various applications; in particular, to convergence rates of alternative projection algorithms [2,5]. Observe that in real algebraic geometry properties of this type are referred to asseparationof semi-algebraic sets and go back to Łojasiewicz [34]. The following corollary ensures the bounded Hölderian regularity of nonconvex semi- algebraic sets with explicit calculating the Hölder exponentτ in (3.9).
Corollary 3.11 (Bounded Hölderian regularity of semi-algebraic sets)Let gi(m) for i =1, . . . ,rmand h(jm)for j=1, . . . ,sm, m=1,2, be real polynomials onRnwith degree at most d≥2. Consider the two semi-algebraic sets inRndefined by
Q:=
x∈Rng(i1)(x)≤0, i =1, . . . ,r1, h(j1)(x)=0, j=1, . . . ,s1 , T :=
x∈Rng(i2)(x)≤0, i =1, . . . ,r2, h(j2)(x)=0, j =1, . . . ,s2 . Then for any compact set K ⊂Rn there is a constant c>0such that the bounded Hölder regularity property(3.9)holds with the exponentτ calculated in(3.7), where r:=r1+r2and s:=s1+s2.
Proof Define the real-valued functions fQ(x):=
r1
i=1
gi(1)(x)
++
s1
j=1
|h(j1)(x)|,
fT(x):=
r2
i=1
gi(2)(x)
++
s2
j=1
|h(j2)(x)|,
and observe that fQ−1(0)= Q, fT−1(0)=T, and(fQ+ fT)−1(0)=Q∩T. Since K is compact, we have thatM :=max{maxx∈Kd(x,Q),maxx∈Kd(x,Q)}<∞and that the setK0=K +MB(0,1)is also compact. It follows from Corollary3.8that there is a constantc>0 such that
d(x,Q∩T)≤c
fQ(x)+ fT(x)τ
for all x∈K0. (3.10) On the other hand, it is easy to see that the functions fQ, fT are locally Lipschitzian, and so they are Lipschitz continuous on the compact setK0,i.e., there is a constant L >0 for whichx,y∈ K0,
|fQ(x)− fQ(y)| ≤Lx−y, |fT(x)− fT(y)| ≤Lx−y whenever x,y∈ K0.
Now pick any x ∈ K and find y ∈ Q,z ∈ T such thatd(x,Q) = x−y and d(x,T)= x−z. Sincey,z∈ K0, we get the estimates
|fQ(x)| = |fQ(x)− fQ(y)| ≤Lx−y = Ld(x,Q),
|fT(x)| = |fT(x)− fT(z)| ≤Lx−z =Ld(x,T).
Combining them with (3.10) completes the proof of the corollary.
4 Hölderian error bounds with sharper exponents
In this section we study two particular classes of polynomial systems and derive for them Hölderian error bounds with sharper explicit exponents in comparison with general results of Sect.3.
4.1 Polynomial systems with finitely many solutions
This subsection deals with polynomial systems (1.1) whose solution sets (3.2) consists of onlyfinitely many points. We now show that the fractional exponentτ in Corol- lary3.8on the Hölderian error bound over compact sets can be significantly sharpen for such systems.
Theorem 4.1 (sharper error bounds over compact sets for systems with finitely many solutions)Let gi as i =1, . . . ,r and hj as j =1, . . . ,s be real polynomials onRn with degree at most d, and let the solution set (3.2)consist of finitely many points.
Then for any compact set K ⊂Rn there is a constant c>0 such that we have the error bound
d(x,S)≤c r
i=1
[gi(x)]++ s
j=1
|hj(x)|
2
κ(n+r,2d)
for all x∈ K,
where the quantityκ >0is defined in(2.3).
Proof The proof follows on the same lines as that of Theorem 3.6, by using Lemma2.3(ii) instead of Lemma2.3(i) and by employing a standard compactness
argument. We omit the details.
4.2 Polynomial systems with simple equalities
In this subsection we sharpen exponents in error bounds for another type of polynomial systems. Recall that a polynomial f with degreedissimpleif it can be written as
f(x)=γ$
(xi−ai)αi, (4.1)
where I ⊂ {1, . . . ,n},γ = 0,ai ∈ R, and αi ∈ Nfori ∈ I with
i∈Iαi = d.
Note that a simple polynomial system may have infinitely many solutions. Consider, e.g., the function f(x1,x2)=x13, which is a simple polynomial with the solution set {(x1,x2)∈R2| f(x1,x2)=0} = {0} ×R.
We begin with a particular case when the polynomial system involves one simple polynomial equality.
Lemma 4.2 (Global error bound for one simple polynomial)Let h: Rn → Rbe a real simple polynomial of degree d, and let S:= {x ∈Rn|h(x)=0}.Then there is a constant c>0such that
d(x,S)≤c|h(x)|1d for all x ∈Rn. Proof Representinghin form (4.1), we haveS=%
i∈I
x∈Rnxi =ai}and arrive at
|h(x)| = |γ|$
i∈I
|xi−ai|αi ≥ |γ|(min
i∈I |xi−ai|)d= |γ|
d(x,S)d
, x∈Rn.
This readily ensures the claimed error bound.
It is worth noting that simple polynomial assumption is essential in Lemma4.2.
Indeed, consider the function h(x) := (x1x2−1)d +(x1 −1)d, which is not a simple polynomial. Then it follows from Example3.10that the global error bound of Lemma4.2fails. The next example shows that this global error bound can also fail for simple polynomial systems involving more than one simple polynomial.
Example 4.3 (Failure of global error bound for general simple polynomial systems) Consider the two polynomialsh1(x1,x2) :=x21andh2(x1,x2) :=(x1−2)x2 with degreed =2. Then we have
S=
x=(x1,x2)∈R2h1(x)=0, h2(x)=0 = {(0,0)}
for the solution set. If the global error bound of Lemma4.2holds, then there isc>0 such that
d(x,S)≤c
|h1(x)| + |h2(x)|1
2 for all x ∈R2. (4.2) Consider the sequence ofxk:=(1,k)ask∈Nand observe that
d(xk,S)=!
1+k2, h1(xk)=1, and h2(xk)= −k.
Then it follows from the error bound (4.2) that√
1+k2≤c(1+ | −k|)12 =c(1+ k)12 for all k ∈ N,which is a contradiction. It is worth noting in this example we have the following local error bound:
d(x,S)=
x12+x22≤
x12+ |x1−2| · |x2|1
2 for all (x1,x2)∈B(0,1).
The next theorem establishes a sharpen error bound over compact sets for simple polynomial systems.
Theorem 4.4 (Sharper error bounds over compact sets for systems of simple polyno- mials)Let hj:Rn→Ras j=1, . . . ,s be simple real polynomials of degree at most d, let S:= {x ∈Rn|hj(x)=0, j =1, . . . ,s} = ∅, and let K be a compact set in Rn. Then there is a constant c>0such that
d(x,S)≤c s
j=1
|hj(x)|
1
d
for all x∈K.
Proof By (4.1) we represent each simple polynomialhj byhj(x)=γj&
i∈Ij(xi− ai j)αi j, j = 1, . . . ,s,where Ij ⊂ {1, . . . ,n},γj = 0,ai j ∈ R, andαi j ∈ Nfor i ∈Ij with
i∈Ijαi j =d. SinceKis compact, it suffices to show that for each point x∈K,there are constantsc, >0 such that
d(x,S)≤c s
j=1
|hj(x)|
1
d
for all x∈B(x, ).
Without loss of generality we suppose thatx∈ S. Then for eachj =1, . . . ,sconsider the index setIj(x):= {i ∈ Ij|xi−ai j =0}and define the polynomial
hj(x):=γj
$
i∈Ij(x)
(xi −ai j)αi j.
Let >0 be such that for allx∈B(x, )we have
|xi−ai j|>3 whenever i∈/ Ij(x), j =1, . . . ,s. (4.3) It follows from the above relationships that
M := min
j=1,...,s min
x∈B(x,)|γj| $
i∈Ij\Ij(x)
|xi−ai j|αi j >0.
By further shrinking if necessary, we can assume that|hj(x)| ≤ 1 for all x ∈ B(x, ), j = 1, . . . ,s. Taking any j = 1, . . . ,s, consider the sets Sj := {x ∈ Rn|hj(x)=0}and find by Lemma4.2positive constantscj >0 ensuring the error bounds
d(x,Sj)≤cj|hj(x)|d j1 whenever x∈Rn with dj :=deghj ≤d. (4.4) Given now an arbitrary vectorx0 ∈ B(x, ), we get by the constructions above that for each j = 1, . . . ,s there existsi(j) ∈ Ij(x)such that the linear functionx → x( ) −a( ) divides the polynomial h and d(x0,S ) = |x0 −a( ) |. Denote
