Math. Program., Ser. A (2015) 153:333–362DOI 10.1007/s10107-014-0806-9

FULL LENGTH PAPER

New fractional error bounds for polynomial systemswith applications to Hölderian stability in optimizationand 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 spaceand the number/degree of the involved polynomials. Our major result extends theexisting 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 weresolve, 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 FellowshipFT130100038. Research of B. S. Mordukhovich was also partially supported by the USA National ScienceFoundation under Grant DMS-1007132 and by the Portuguese Foundation of Science and Technologiesunder Grant MAT/11109. Research of T. S. Pha.m was also partially supported by the Vietnam NationalFoundation for Science and Technology Development (NAFOSTED) under Grant 101.04–2013.07.

G. LiDepartment of Applied Mathematics, University of New South Wales, Sydney 2052, Australiae-mail: g.li@unsw.edu.au

B. S. Mordukhovich (B)Department of Mathematics, Wayne State University, Detroit, MI 48202, USAe-mail: boris@math.wayne.edu

B. S. MordukhovichDepartment of Mathematics and Statistics, King Fahd Universityof Petroleum and Minerals, Dhahran, Saudi Arabia

T. S. Pha.mCenter of Research and Development, Duy Tan University, K7/25, Danang, Quang Trung, Vietnam

T. S. Pha.mDepartment of Mathematics, University of Dalat, 1, Phu Dong Thien Vuong, Dalat, Vietname-mail: sonpt@dlu.edu.vn

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334 G. Li et al.

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. Ourmajor applications concern quantitative Hölderian stability of solution maps for para-meterized polynomial optimization problems and nonlinear complementarity systemswith polynomial data as well as high-order semismooth properties of the eigenvaluesof 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 inmany optimization problems can be described by systems of inequal-ities and equalities

gi (x) ≤ 0, i = 1, . . . , r, and h j (x) = 0, j = 1, . . . , s, (1.1)

where gi , h j : Rn → R for i = 1, . . . , r and j = 1, . . . , s are real-valued functions onR

n . One of the most important issues for (1.1) is the so-called error bounds. Denotingby S the set of solutions to (1.1), recall that this system has a (local) error bound withexponent τ > 0 at x ∈ R

n if there exist a constant c > 0 and an neighborhood U of xsuch that

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|)τ

for all x ∈ U, (1.2)

where d(x, S) signifies the Euclidean distance between x and the set S, and where[α]+ := max{α, 0}. This estimates bounds the distance from an arbitrary point xaround the reference one x to the solution set S via a constant multiple of a com-putable residual function, which measures the violation of the constraint S := {x ∈R

n∣∣ gi (x) ≤ 0, h j (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, inparticular, 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 errorbounds andmetric regularity/subregularity issues 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 themost important and celebrated error bound/metric regularity result is dueto Hoffman [18] who proved, in the case of linear functions gi and h j and solvabilityof system (1.1), the existence of c > 0 such that the error bound (1.2) holds with

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New fractional error bounds for polynomial systems 335

U = Rn and τ = 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 havebeen obtained for convex inequality systems on finite-dimensional Riemannian andHadamard manifolds. For nonconvex inequality and equality systems some local errorbound 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 questionvia its values around the reference point. On the other hand, it is proved in [35,36] byusing the cerebrated Łojasiewicz’s inequality [33] that (1.2) holds with some unknownfractional exponent τ when all gi and h j are polynomials or analytic functions. Fur-thermore, it is stated by the authors of [36] in their concluding remarks that “wehave not been able to obtain explicit formulas for the multiplier or the exponent inthe error bound. We feel that such formulas would be useful for computational andother purposes.” Note to this end that local error bound results with explicit exponentsare indeed important for both theory and applications since they can be used, e.g., toestablish explicit convergence rates of the proximal point algorithm as demonstratedin [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 stabilityof polynomial optimization problems and nonlinear complementarity systems withpolynomial data. There are some important progress along this direction for specialpolynomial systems. For example, as shown in [38], regularity assumptions are notneeded to obtain (1.2) with τ = 1

2 if system (1.1) involves only one quadratic function;see also [45,46] for infinite-dimensional extensions. Moreover, error bound results forsystem (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 establishing error bound results (1.1) withexplicit exponents τ in (1.2). Employing advanced techniques of variational analy-sis and generalized differentiation allows us to derive error bounds for such systemswith exponents explicitly determined by the dimension of the underlying space andthe number/degree of the involved polynomials without any regularity conditions.Besides meeting the aforementioned general goals formulated in [36], in this way weresolve, in particular, a long-standing open question raised in [38] about Hölderianerror bounds with explicit exponents for nonconvex quadratic systems. Furthermore,we apply our error bound results to deriving verifiable conditions for Hölderian stabil-ity of general polynomial optimization problems as well as nonlinear complementarityproblems with polynomial data. As a by-product of our analysis, we give a positiveanswer to another open question raised in [31] about the ρth-order semismoothness ofthe maximum eigenvalue for a symmetric tensor with explicit estimating the exponentρ. Since the concept of symmetric tensors has been well recognized as a high-orderextension of symmetric matrices with various applications to automatic control andimage science [47,51,53], the result obtained is of undoubted importance for furtherapplications to these areas. Note that much of our study on error bounds is in thespirit of [27,28,35,36] being largely motivated by the recent work on nonsmoothextensions of Łojasiewicz’s inequality initiated in [3]. It is worth emphasizing that, asdemonstrated in this paper, generalized differential techniques can be very instrumen-

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tal for revolving applied quantitative issues even for smooth/polynomial systems. Wealso refer the reader to [6,7,22,23,25,30,48,49,58] and the bibliographies therein forother 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 andpolynomial theory, which are widely used in the formulations and proofs of the mainresults below. Section 3 is devoted to establishing major error bounds for polynomialsystems with explicitly calculated exponents. In Sect. 4 we consider some special set-tings for which the exponents in error bounds obtained in Sect. 3 can be significantlysharpen. Section 5 concerns applications of the error bounds established in the previ-ous sections to deriving new results on quantitative Hölderian stability for polynomialoptimization problems and as well as for nonlinear complementarity problems withpolynomial data. Finally, in Sect. 6 we present concluding remarks and discuss somedirections 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〉 := xT y and the Euclidean norm ‖x‖ := (xT x)1/2 for any x, y ∈ R

n , where xT

signifies the vector (as well as matrix) transposition. We use the symbols B(x, ε) andB(x, ε) to denote the open and closed balls, respectively, of the space in question withcenter x and radius ε > 0. Given a set � ⊂ R

n , its interior (resp. closure, boundaryand convex hull) is denoted by int� (resp. cl�, bd�, and co�). Recall also thatN := {1, 2, . . .}.

2 Preliminaries

This section contains the necessary preliminaries needed in the paper. We start withgeneralized 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 usethe symbol z

f→ x to indicate that z → x and f (z) → f (x). Our basic subdiffer-ential of f at x ∈ dom f (known also as the general, or limiting, or Mordukhovichsubdifferential) is defined by

∂ f (x) :={v ∈ R

n∣∣∣ ∃xk

f→ x, vk → v with lim infz→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 subdifferentialof convex analysis

∂ f (x) = {v ∈ R

n∣∣ 〈v, z − x〉 ≤ f (z) − f (x) whenever z ∈ R

n}, x ∈ dom f.

(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-

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New fractional error bounds for polynomial systems 337

tional/extremal principles of variational analysis [41,56]. Note also that ∂ f (x) �= ∅ iff is locally Lipschitzian around x .

Definition 2.1 (Subdifferential slope) Given f : Rn → R and using (2.1), the subd-ifferential slope of f at x ∈ dom f is defined by

m f (x) := inf{‖v‖∣∣ v ∈ ∂ f (x)

}.

We can see directly from the definition that m f (x) = ∞ whenever ∂ f (x) = ∅.Observe also that for f ∈ C1 around x we have ∂ f (x) = {∇ f (x)} and hencem f (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 class C1, and let f (x) := maxi=1,...,l gi (x). Then f is a locally Lipschitzfunction, and we have

m f (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 → R is a polynomial if there is a numberr ∈ N such that

f (x) =∑

0≤|α|≤r

λαxα,

where λα ∈ R, x = (x1, . . . , xn), xα := xα11 · · · xαn

n , αi ∈ N ∪ {0}, and |α| :=∑nj=1 α j . The corresponding constant r is called the degree of f . Recall further that

f : Rn → R is (real) analytic if it can be locally represented on Rn by a convergent

infinite power series, i.e., for all vectors x = (x1, . . . , xn) ∈ Rn there is a neighbor-

hood U of x such that for every x = (x1, . . . , xn) ∈ U we have

f (x) =∞∑

|α|=0

λα(x − x)α.

Amajor property of analytic functions that is most important for this paper is givenby 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 constants c, τ, and ε such that

‖∇ f (x)‖ ≥ c| f (x)|τ for all ‖x‖ ≤ ε.

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As pointed out in [37], it is often difficult to determine the corresponding exponentsτ in Łojasiewicz’s gradient inequality, and they are typically unknown. Some estimatesof the exponent τ in the gradient inequality were derived in [8,15] in the case whenf is a polynomial. To formulate these results, for each n, d ∈ N define the followingtwo 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 numbers n and d andthat this inequality is strict when n ≥ 2 and d ≥ 2.

Lemma 2.3 (Exponent estimates in Łojasiewicz’s gradient inequality for polynomi-als) Let f be a real polynomial on R

n with degree d ∈ N. The following hold:

(i) (cf. [8, Theorem 4.2]) Suppose that f (0) = 0 and ∇ f (0) = 0. Then there existconstants c, ε > 0 such that for all ‖x‖ ≤ ε 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 thatf (0) = 0 and there is δ > 0 with f (x) > 0 for all x ∈ B(0, δ)\{0}. Then thereexist positive constants c, ε such that for all ‖x‖ ≤ ε 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 system withoutany regularity conditions. Let us begin with specifying the definition of local errorbounds.

Definition 3.1 (Local error bounds)We say that system (1.1) has alocal Hölderian(or Hölder type) error bound with exponent τ > 0 at x ∈ R

n if there arepositive constants c and ε such that

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|)τ

for all x with ‖x − x‖ ≤ ε, (3.1)

where S is the solution set for the system (1.1) given by

S := {x ∈ R

n∣∣ gi (x) ≤ 0, i = 1, . . . , r, and h j (x) = 0, j = 0, . . . , s

}. (3.2)

Throughout this paper, to avoid triviality, we always assume that ∅ �= S �= Rn .

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New fractional error bounds for polynomial systems 339

Prior to deriving the main results of this section we present an example illustratingthe 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 ford = 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 h j (x1, . . . , xn) := x j+1 − xd

j for j = 1, . . . , n − 1,

hn(x1, . . . , xn) := xdn , and gi (x) ≡ 0 for all i = 1, . . . , r in (3.1). Then the solution

set S for (3.1) is S = {x ∈ Rn| x = 0}. Take further x = 0 and consider the family

of vectors x(ε) := (ε, εd , . . . , εdn−1) ∈ R

n with ε ∈ (0, 1]. It is easy to see that

d(x(ε), S) =√∑n

i=1 ε2di−1 = O(ε),∑n

j=1 |h j (x)| = εdn, and thus we have

d(x(ε), S

) = O

([ n∑j=1

|h j (x(ε))|] 1

dn)

,

which shows that the exponent τ in (3.1) for this system at x does not exceed d−n .

Our first goal in this section is employing Lemmas 2.2 and 2.3(i) to obtain anonsmooth version of Łojasiewicz’s gradient inequality for maximum functions overfinitely many polynomials with an explicit exponent. It is certainly of its own interestwhile being applied in what follows to deriving error bounds for polynomial systemswith 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 arereal polynomials on R

n with their degrees not exceeded d, and let x ∈ Rn with

f (x) = 0. Then there are numbers c, ε > 0 such that

m f (x) ≥ c | f (x)|1− 1R(n+l−1, d+1) for all x with ‖x − x‖ ≤ ε,

where m f (x) is the subdifferential slope from Definition 2.1, and where the constantR is defined in (2.3).

Proof Without loss of generality, assume that gi (x) = 0 for all i = 1, . . . , l. Then foreach subset I := {i1, . . . , iq} ⊂ {1, . . . , l},we define the function FI : Rn ×R

q−1 →R by

FI (x, λ) :={∑q−1

j=1 λ j gi j (x) +(1 −∑q−1

j=1 λ j

)giq (x) if q ≥ 2,

gi1(x) if q = 1,

which is clearly a polynomial on Rn+q−1 with degree at most d + 1 and F(x, λ) = 0for all λ ∈ R

q−1. Define further the set P ⊂ Rq−1 by

P :=⎧⎨⎩λ ∈ R

q−1∣∣∣ λ j ≥ 0,

q−1∑j=1

λ j ≤ 1

⎫⎬⎭ .

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340 G. Li et al.

Then, there exist numbers cI > 0 and εI > 0 for which we have

‖∇FI (x, λ)‖ ≥ cI |FI (x, λ)|1− 1R(n+q−1, d+1) whenever ‖x − x‖ ≤ εI and λ ∈ P.

(3.3)To verify (3.3), by standard compactness arguments, we only need to check that foreach λ ∈ P there are numbers c(λ) > 0 and ε(λ) > 0 such that

‖∇F(x, λ)‖ ≥ c(λ) |F(x, λ)|1− 1R(n+q−1, d+1) for all ‖x −x‖ ≤ c(λ), ‖λ−λ‖ ≤ ε(λ).

Indeed, since F(x, λ) = 0 for all λ ∈ P, it is obvious if ‖∇F(x, λ)‖ �= 0, while in theremaining case of ‖∇F(x, λ)‖ = 0 this inequality follows from Lemma 2.3(i).

Let c := min{cI | I ⊂ {1, . . . , l}, I �= ∅} > 0 and ε := min

{εI | I ⊂

{1, . . . , l}, I �= ∅} > 0. Pick an arbitrary point x in Rn with ‖x − x‖ ≤ ε and

denote I = I (x) := {i | gi (x) = f (x)}. Lemma 2.2 tells us that there are numbersλi ≥ 0 for i ∈ I such that

∑i∈ I λi = 1 and

m f (x) =∥∥∥∥∥∥∑i∈ I

λi∇gi (x)

∥∥∥∥∥∥ .

Let us renumerate the index set I as I = {i1, . . . , iq0}, where q0 signifies its cardinality.Then

FI (x, λi1 , . . . , λiq0−1) =q0∑

j=1

λi j gi j (x) =∑i∈ I

λi gi (x) =∑

i∈I (x)

λi gi (x) = f (x).

Furthermore, we have the representations

‖∇FI (x, λi1 , . . . , λiq0−1 )‖ =∥∥∥∥∥∥⎛⎝ q0∑

j=1

λi j ∇gi j (x), gi1 (x) − giq0(x), . . . , giq0−1 (x) − giq0

(x)

⎞⎠∥∥∥∥∥∥

=∥∥∥

q0∑j=1

λi j ∇gi j (x)

∥∥∥ =∥∥∥ ∑

i∈I (x)

λi∇gi (x)

∥∥∥ = m f (x),

which, being combined with inequality (3.3), allow us to conclude that

m f (x) = ‖∇FI (x, λi1 , . . . , λiq0−1)‖≥ cI |FI (x, λi1 , . . . , λiq0−1)|1−

1R(n+q0−1, d+1)

= cI | f (x)|1− 1R(n+q0−1, d+1) ,

≥ c | f (x)|1− 1R(n+l−1, d+1)

and thus to complete the proof of the theorem. ��

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New fractional error bounds for polynomial systems 341

Employing further the nonsmooth Łojasiewicz’s inequality of Theorem 3.3 leads usto effective error bounds of polynomial systems with explicit exponents. To proceed,we need the following lemma on error bounds for locally Lipschitz functions takenfrom [46, Corollary 2].

Lemma 3.4 (Sufficient condition for error bounds of Lipschitz functions) Let f : Rn

→ R be locally Lipschitzian around x ∈ bd S f , where S f = {x | f (x) ≤ 0}. Assumethat there are numbers c, ε > 0 such that τm f (x) ≥ c | f (x)|1−τ for all x with‖x − x‖ ≤ ε and x /∈ S f . Then we have

d(x, S f ) ≤ 1

c

[f (x)

]τ+ whenever ‖x − x‖ ≤ ε

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 polynomialsystems, type I) Let gi as i = 1, . . . , r and h j as j = 1, . . . , s be real polynomials onR

n with degree at most d, and let S be the solution set (3.2). Then there are numbersc, ε > 0 such that

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|) 1

R(n+r+s,d+1)

whenever ‖x − x‖ ≤ ε,

where the quantity R is defined in (2.3).

Proof The conclusion is rather straightforward if either x ∈ int S or x /∈ S. To proceedwith the remaining case of x ∈ bd S, 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 vector e := (e1, . . . , es) ∈ {−1, 1}s and define thefunction

fe(x) := max{0, g1(x), . . . , gr (x), e1h1(x), . . . , eshs(x)

}, x ∈ R

n,

which is the maximum of r + s + 1 polynomials with degree at most d and withfe(x) = 0. Employing Theorem 3.3 gives us numbers c(e) > 0 and ε(e) > 0 suchthat

m fe (x) ≥ c(e)| fe(x)|1− 1R(n+r+s,d+1) whenever ‖x − x‖ ≤ ε(e).

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342 G. Li et al.

Let c := mine∈{−1,1}s c(e) > 0 and ε := mine∈{−1,1}s ε(e) > 0. Take any x with‖x − x‖ ≤ ε and f (x) > 0. Then for each j = 1, . . . , s we have that either h j (x) �=−h j (x) or h j (x) < f (x). It allows us to find e ∈ {−1, 1}s so that f (x) = fe(x) andm f (x) = m fe (x). This gives us the estimate

m f (x) = m fe (x) ≥ c(e) | fe(x)|1− 1R(n+r+s,d+1) ≥ c | f (x)|1− 1

R(n+r+s,d+1) ,

which completes the proof of the theorem by applying Lemma 3.4. ��Employing another technique (somewhat similar to [35,36]) and Lemma 2.3(i), the

next theorem provides a local error bound with an explicit exponent for polynomialsystems, which is different from that in Theorem 3.5. The idea of the proof is to usecertain slack variables to convert the polynomial system (1.1) into a single polynomialand then apply Lemma 2.3(i).

Theorem 3.6 (Local error bounds with explicit fractional exponents for polynomialsystems, type II) Let gi as i = 1, . . . , r and h j as j = 1, . . . , s be real polynomials onR

n with degree at most d, and let S be given in (3.2). Then there are numbers c, ε > 0such that

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|) 2

R(n+r,2d)

whenever ‖x − x‖ ≤ ε,

where the quantity R is defined in (2.3).

Proof Similarly to the proof of Theorem 3.5, we only need to examine the case ofx ∈ bd S. Define the polynomial θ : Rn × R

r → R by

θ(x, z) :=r∑

i=1

(gi (x) + z2i

)2 +s∑

j=1

h j (x)2

and note that its degree does not exceed 2d. Consider the set S := {(x, z) ∈R

n × Rr | θ(x, z) = 0} and the continuous mapping φ(x) := (

√[−g1(x)]+, . . . ,√[−gr (x)]+) on Rn .

Since x ∈ S, we have θ(x, φ(x)) = 0 and ∇θ(x, φ(x)) = 0. ApplyingLemma 2.3(i) to θ gives us positive numbers ε0 and c0 such that ‖∇θ(x, z)‖ ≥c0 θ(x, z)1−

1R(n+r,2d) for all ‖(x, z)−(x, φ(x))‖ ≤ ε0. Let c :=c−1

0 . Then Lemma 3.4ensures the estimate

d((x, z), S

) ≤ c θ(x, z)1

R(n+r,2d) for all (x, z) with ‖(x, z) − (x, φ(x))‖ ≤ ε0

2.

(3.4)By continuity of φ we find 0 < ε < ε0

4 such that ‖φ(x)−φ(x)‖ ≤ ε04 whenever ‖x −

x‖ ≤ ε, which clearly implies the inequality

‖(x, φ(x)) − (x, φ(x))‖ ≤ ε0

2whenever ‖x − x‖ ≤ ε. (3.5)

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New fractional error bounds for polynomial systems 343

Now let x be an arbitrary vector satisfying ‖x − x‖ ≤ ε. There exists a point (x, z) ∈ Ssuch that ‖(x, φ(x))−(x, z)‖ = d

((x, φ(x)), S

).Bydefinition of S wehave θ (x, z) =

0, and hence gi (x) = −z2i ≤ 0, i = 1, . . . , r, and h j (x) = 0, j = 1, . . . , s. Thisimplies that x ∈ S. Therefore

d(x, S) ≤ ‖x − x‖ ≤ ‖(x, φ(x)) − (x, z)‖ = d((x, φ(x)), S

)≤ c θ(x, φ(x))

1R(n+r,2d)

= c

( r∑i=1

[gi (x)]2+ +s∑

j=1

h j (x)2) 1

R(n+r,2d)

≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|) 2

R(n+r,2d)

,

where the third inequality follows from (3.4) and (3.5) while the last equality followsfrom the fact that gi (x) + [−gi (x)]+ = [gi (x)]+. This justifies the claimed errorbound. ��Remark 3.7 (Comparing the two types of local error bounds) It is worth noting thatthe two types of local error bounds obtained in Theorems 3.5 and 3.6 are generallyindependent from each other. Recall that R(p, q) = q(3q − 3)p−1 in the setting ofTheorem 3.6. Consider, e.g., the case of n = 3, r = 4, s = 1, and d = 2. Then

2R(n+r,2d)

= 2R(7,4) = 1

4·96 and 1R(n+r+s,d+1) = 1

R(8,3) = 13·67 ; thus we have in this

case that 2R(n+r,2d)

< 1R(n+r+s,d+1) . On the other hand, letting n = r = 1, s = 2, and

d = 2, we get that 2R(n+r,2d)

= 2R(2,4) = 1

18 and 1R(n+r+s,d+1) = 1

R(4,3) = 14·92 ; so it

gives 2R(n+r,2d)

> 1R(n+r+s,d+1) .

As a consequence of the theorem, we now establish some globalized error boundresults with explicit exponents for polynomial systems of type (1.1) over compact sets.

Corollary 3.8 (Hölderian error bounds with explicit exponents for polynomial sys-tems over compact sets) Let gi , h j , and S be as in Theorem 3.6. Then for any compactset K ⊂ R

n there is a number c > 0 such that

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|)τ

for all x ∈ K , (3.6)

where the exponent τ 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 with τ given by (3.7).

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344 G. Li et al.

Proof Combining the results of Theorem 3.5 and Theorem 3.6, for every x ∈ Rn we

can find numbers ε(x) > 0 and c(x) > 0 such that

d(x, S) ≤ c(x)

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|)τ

whenever ‖x − x‖ ≤ ε(x).

Then the conclusion follows by using standard compactness arguments. ��Let us mention that the authors of [38] established a Hölder error bound with

exponent τ = 12 over compact sets for a single quadratic function and then raised the

question about the possibility to extend this result to finitely many quadratic functions.They actually conjectured that a Hölder error bound would hold over compact setsfor nonconvex quadratic systems with exponent τ = 1

2p with p denoting the numberof quadratic functions involved in the system. Now we provide a partial answer fortheir conjecture by showing that such an error bound holds with a larger while exactlycalculated exponent.

Corollary 3.9 (Hölderian error bounds over compact sets for nonconvex quadraticsystems) Let r, s ∈ N, let gi as i = 1, . . . , r and h j as j = 1, . . . , s be quadraticfunctions on R

n, and let S be defined in (3.2). Then for any compact set K ⊂ Rn there

is a number c > 0 such that the error bound inequality (3.6) holds with the explicitexponent τ calculated 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 Corollary 3.8 with d =2 and formula (3.7) for calculating τ .��

Next we show that the globalized version of the Hölderian error bound result fromCorollary 3.8 over compact set cannot be generally extended to the global one overthe whole space Rn . The following example was used in [11] in the case of d = 2,

Example 3.10 (Failure of global error bounds for polynomial systems) Let d be anyeven number. Define the polynomial function h : R2 → R by h(x) := (x1x2 − 1)d +(x1 − 1)d . The solution set here is S = {x ∈ R

2| h(x) = 0} = {(1, 1)}. The globalversion of the error bound in Corollary 3.8 is as follows: there are numbers c, τ > 0such that

d(x, S) ≤ c |h(x)|τ for all x ∈ Rn . (3.8)

To show that (3.8) fails, consider a sequence xk = ( 1k , k) for h(xk) = (1 − 1k )d → 1

and d(xk, S) → ∞ as k → ∞. Then the global error bound (3.8) is obviously violatedalong this sequence.

We conclude this section by establishing (as yet another consequence of the mainresults above) a Hölder-type regularity property for two nonconvex semi-algebraicsets, i.e., subsets of Rn that can be described by finitely many equality and inequality

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New fractional error bounds for polynomial systems 345

constraints given by polynomials. We say that the pair of sets {Q, T } has the boundedHölderian regularity property with exponent τ > 0 if for each compact set K there isa constant c > 0 such that

d(x, Q ∩ T ) ≤ c(d(x, Q) + d(x, T )

)τ whenever x ∈ K . (3.9)

For convex setswith τ = 1 in (3.9) this property reduces to the so-calledbounded linearregularity of [2], which is an important concept of convex analysis and optimizationwith various applications; in particular, to convergence rates of alternative projectionalgorithms [2,5]. Observe that in real algebraic geometry properties of this type arereferred to as separation of semi-algebraic sets and go back to Łojasiewicz [34]. Thefollowing corollary ensures the bounded Hölderian regularity of nonconvex semi-algebraic sets with explicit calculating the Hölder exponent τ in (3.9).

Corollary 3.11 (Bounded Hölderian regularity of semi-algebraic sets) Let g(m)i for

i = 1, . . . , rm and h(m)j for j = 1, . . . , sm, m = 1, 2, be real polynomials on R

n withdegree at most d ≥ 2. Consider the two semi-algebraic sets in R

n defined by

Q := {x ∈ R

n∣∣ g(1)

i (x) ≤ 0, i = 1, . . . , r1, h(1)j (x) = 0, j = 1, . . . , s1

},

T := {x ∈ R

n∣∣ g(2)

i (x) ≤ 0, i = 1, . . . , r2, h(2)j (x) = 0, j = 1, . . . , s2

}.

Then for any compact set K ⊂ Rn there is a constant c > 0 such that the bounded

Hölder regularity property (3.9) holds with the exponent τ calculated in (3.7), wherer := r1 + r2 and s := s1 + s2.

Proof Define the real-valued functions

fQ(x) :=r1∑

i=1

[g(1)

i (x)]+ +

s1∑j=1

|h(1)j (x)|,

fT (x) :=r2∑

i=1

[g(2)

i (x)]+ +

s2∑j=1

|h(2)j (x)|,

and observe that f −1Q (0) = Q, f −1

T (0) = T , and ( fQ + fT )−1(0) = Q ∩ T . SinceK is compact, we have that M := max{maxx∈K d(x, Q),maxx∈K d(x, Q)} < ∞ andthat the set K0 = K + M B(0, 1) is also compact. It follows from Corollary 3.8 thatthere is a constant c > 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 set K0, i.e., there is a constantL > 0 for which x, y ∈ K0,

| fQ(x) − fQ(y)| ≤ L‖x−y‖, | fT (x) − fT (y)| ≤ L‖x−y‖ whenever x, y ∈ K0.

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346 G. Li et al.

Now pick any x ∈ K and find y ∈ Q, z ∈ T such that d(x, Q) = ‖x − y‖ andd(x, T ) = ‖x − z‖. Since y, z ∈ K0, we get the estimates

| fQ(x)| = | fQ(x) − fQ(y)| ≤ L‖x − y‖ = Ld(x, Q),

| fT (x)| = | fT (x) − fT (z)| ≤ L‖x − 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 derivefor them Hölderian error bounds with sharper explicit exponents in comparison withgeneral 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) consistsof only finitely many points. We now show that the fractional exponent τ in Corol-lary 3.8 on the Hölderian error bound over compact sets can be significantly sharpenfor such systems.

Theorem 4.1 (sharper error bounds over compact sets for systems with finitely manysolutions) Let gi as i = 1, . . . , r and h j as j = 1, . . . , s be real polynomials on R

n

with degree at most d, and let the solution set (3.2) consist of finitely many points.Then for any compact set K ⊂ R

n there is a constant c > 0 such that we have theerror bound

d(x, S) ≤ c

( r∑i=1

[gi (x)]+ +s∑

j=1

|h j (x)|) 2

κ(n+r,2d)

for all x ∈ K ,

where the quantity κ > 0 is defined in (2.3).

Proof The proof follows on the same lines as that of Theorem 3.6, by usingLemma 2.3(ii) instead of Lemma 2.3(i) and by employing a standard compactnessargument. We omit the details. ��

4.2 Polynomial systems with simple equalities

In this subsectionwe sharpen exponents in error bounds for another type of polynomialsystems. Recall that a polynomial f with degree d is simple if it can be written as

f (x) = γ∏i∈I

(xi − ai )αi , (4.1)

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New fractional error bounds for polynomial systems 347

where I ⊂ {1, . . . , n}, γ �= 0, ai ∈ R, and αi ∈ N for i ∈ I with∑

i∈I αi = d.Note that a simple polynomial system may have infinitely many solutions. Consider,e.g., the function f (x1, x2) = x31 , which is a simple polynomial with the solution set{(x1, x2) ∈ R

2| 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 → R be areal simple polynomial of degree d, and let S := {x ∈ R

n | h(x) = 0}. Then there isa constant c > 0 such that

d(x, S) ≤ c |h(x)| 1d for all x ∈ Rn .

Proof Representing h in form (4.1), we have S = ⋃i∈I

{x ∈ R

n∣∣ xi = ai } and arrive

at

|h(x)| = |γ |∏i∈I

|xi − ai |αi ≥ |γ |(mini∈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 Lemma 4.2.

Indeed, consider the function h(x) := (x1x2 − 1)d + (x1 − 1)d , which is not asimple polynomial. Then it follows from Example 3.10 that the global error bound ofLemma 4.2 fails. The next example shows that this global error bound can also failfor 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 polynomials h1(x1, x2) := x21 and h2(x1, x2) := (x1 − 2)x2 withdegree d = 2. Then we have

S = {x = (x1, x2) ∈ R

2∣∣ h1(x) = 0, h2(x) = 0

} = {(0, 0)}

for the solution set. If the global error bound of Lemma 4.2 holds, then there is c > 0such that

d(x, S) ≤ c(|h1(x)| + |h2(x)|) 12 for all x ∈ R

2. (4.2)

Consider the sequence of xk := (1, k) as k ∈ N and 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|) 1

2 = 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) =√

x21 + x22 ≤ (x21 + |x1 − 2| · |x2|

) 12 for all (x1, x2) ∈ B(0, 1).

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348 G. Li et al.

The next theorem establishes a sharpen error bound over compact sets for simplepolynomial systems.

Theorem 4.4 (Sharper error bounds over compact sets for systems of simple polyno-mials) Let h j : Rn → R as j = 1, . . . , s be simple real polynomials of degree at mostd, let S := {x ∈ R

n| h j (x) = 0, j = 1, . . . , s} �= ∅, and let K be a compact set inR

n. Then there is a constant c > 0 such that

d(x, S) ≤ c

( s∑j=1

|h j (x)|) 1

d

for all x ∈ K .

Proof By (4.1) we represent each simple polynomial h j by h j (x) = γ j∏

i∈I j(xi −

ai j )αi j , j = 1, . . . , s, where I j ⊂ {1, . . . , n}, γ j �= 0, ai j ∈ R, and αi j ∈ N for

i ∈ I j with∑

i∈I jαi j = d. Since K is compact, it suffices to show that for each point

x ∈ K , there are constants c, ε > 0 such that

d(x, S) ≤ c

( s∑j=1

|h j (x)|) 1

d

for all x ∈ B(x, ε).

Without loss of generality we suppose that x ∈ S. Then for each j = 1, . . . , s considerthe index set I j (x) := {i ∈ I j | xi − ai j = 0} and define the polynomial

h j (x) := γ j

∏i∈I j (x)

(xi − ai j )αi j .

Let ε > 0 be such that for all x ∈ B(x, ε) we have

|xi − ai j | > 3ε whenever i /∈ I j (x), j = 1, . . . , s. (4.3)

It follows from the above relationships that

M := minj=1,...,s

minx∈B(x,ε)

|γ j |∏

i∈I j \I j (x)

|xi − ai j |αi j > 0.

By further shrinking ε if necessary, we can assume that |h j (x)| ≤ 1 for all x ∈B(x, ε), j = 1, . . . , s. Taking any j = 1, . . . , s, consider the sets S j := {x ∈R

n| h j (x) = 0} and find by Lemma 4.2 positive constants c j > 0 ensuring the errorbounds

d(x, S j ) ≤ c j |h j (x)|1

d j whenever x ∈ Rn with d j := deg h j ≤ d. (4.4)

Given now an arbitrary vector x0 ∈ B(x, ε), we get by the constructions above thatfor each j = 1, . . . , s there exists i( j) ∈ I j (x) such that the linear function x �→xi( j) − ai( j) j divides the polynomial h j and d(x0, S j ) = |x0i( j) − ai( j) j |. Denote

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New fractional error bounds for polynomial systems 349

Z := {x ∈ B(x, ε)| xi( j) = ai( j) j for all j = 1, . . . , s}. By the definition of Z andby (4.3) it is not hard to see that x ∈ Z ⊂ S and

d(x0, Z) ≤s∑

j=1

|x0i( j) − ai( j) j |.

These imply together with (4.4) the following estimates:

d(x0, S) ≤ d(x0, Z) ≤s∑

j=1

|x0i( j) − ai( j) j | =s∑

j=1

d(x0, S j )

≤ c j |h j (x)|1

d j ≤s∑

j=1

c j |h j (x0)| 1d ≤s∑

j=1

c j

∣∣∣∣h j (x0)

M

∣∣∣∣1d

≤(

maxj=1,...,s

c j

)(1

M

) 1d

s∑j=1

|h j (x0)| 1d ,

where the fourth inequality follows due to |h j (x)| ≤ 1 for all x ∈ B(x, ε) as j =1, . . . , s, and the fifth one follows by the definition of M . Since the function t �→t1d is concave on R+, we get for each t j ≥ 0 that 1

s

∑sj=1 t

1dj ≤

(∑sj=1

1s t j

) 1d

.

Consequently, it gives us for all x0 ∈ B(x, ε) the desired estimate

d(x0, S) ≤(

maxj=1,...,s

c j

)(1

M

) 1d

s1−1d

( s∑j=1

|h j (x0)|) 1

d

,

which thus completes the proof of the theorem. ��

5 Applications: higher-order stability analysis

The main aim of this section is to apply the error bound results derived above to quan-titative stability of two important classes of parametric variational systems playinga crucial role in optimization theory and applications, namely problems of polyno-mial optimization and nonlinear complementarity with polynomial data. In contrastto first-order stability results related to Lipschitzian stability, we concentrate hereon higher-order issues unified around Hölderian stability with fractional exponents.Based on our error bound analysis and advanced tools of generalized differentiation,we establish general results in this direction and their specifications with explicit cal-culations of Hölder exponents.

Let us begin with Hölderian stability of optimal solution maps in polynomial opti-mization.

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350 G. Li et al.

5.1 Hölderian stability in polynomial optimization

Consider the following parameterized polynomial optimization problem:

(POP)u maxx∈Rn

f (x, u)

subject to gi (x) ≤ 0, i = 1, . . . , r,h j (x) = 0, j = 1, . . . , s,

where u ∈ Rl is the perturbation parameter, where x �→ f (x, u) is a real polynomial

on Rn with degree at most d for each fixed u ∈ R

l , and where gi as i = 1, . . . , r andh j as j = 1, . . . , s are all real polynomials on R

n with degree at most d. For eachu ∈ R

l denote the solution set of (P O P)u by S(u).Let u ∈ R

l . We are interested in behavior of the solution map S : Rl ⇒ Rn when

its argument u changes around the reference point u. The following assumptions areimposed:

• Assumption 1: The set K := {x ∈ Rn| gi (x) ≤ 0 as i = 1, . . . r and h j (x) =

0 as j = 1, . . . , s} is compact and the function f is continuous on Rn × R

l .• Assumption 2: There are constants L , δ > 0 such that

‖ f (x, u) − f (x, u)‖ ≤ L‖u − u‖ (5.1)

for all x ∈ K and for all u with ‖u − u‖ ≤ δ.

The class of polynomial optimization problems (P O Pu) satisfying Assumptions 1and 2 covers a number of remarkable models. To illustrate, we mention the two impor-tant subclasses as follows.

Subclass 1: Polynomial Optimization with Tilt/Canonical Perturbations

Consider the parametric polynomial optimization problems with tilt/canonical pertur-bations defined by

maxx∈Rn

p(x) + uT x

subject to ‖x‖2 = 1,

where p is a polynomial of degree d ≥ 2 on Rn , and where u ∈ R

n . Denotingf (x, u) := p(x) + uT x , it is easy to see that both Assumptions 1 and 2 are satisfied.

Subclass 2: Maximum Eigenvalues of Symmetric Tensors

Recall that an mth-order n-dimensional tensor A consists of nm real entries given by

A = (Ai1i2...im ), Ai1i2...im ∈ R, 1 ≤ i1, i2, . . . , im ≤ n.

We say that the tensorA is symmetric if the values ofAi1i2...im are invariant under anypermutation of the indices {i1, i2, . . . , im}.Whenm = 2, a symmetric tensor is nothingbut a symmetric matrix. The concept of symmetric tensor is a multilinear extension

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New fractional error bounds for polynomial systems 351

of symmetric matrices and has recently found numerous applications in polynomialoptimization, automatic control, image science, etc.; see, e.g., [47,51,53].

Let m be an even number and let T := {A| A is an mth-order n-dimensionalsymmetric tensor}, which is a vector space under the addition and multiplicationdefined as follows: for any t ∈ R and any tensors A = (Ai1,...,im )1≤i1,...,im≤n andB = (Bi1,...,im )1≤i1,...,im≤n we have

A + B := (Ai1,...,im + Bi1,...,im )1≤i1,...,im≤n and tA := (tAi1,...,im )1≤i1,...,im≤n .

Further, for each A,B ∈ T define the inner product and norm by 〈A,B〉T :=∑ni1,...,im=1Ai1,...,imBi1,...,im and ‖A‖T := (〈A,A〉T )

12 , respectively. We say that

λ ∈ R is an eigenvalue of A and that x ∈ Rn\{0} is the eigenvector corresponding to

λ if the pair (x, λ) satisfies

n∑i2,...,im=1

Ai i2...im xi2 . . . xim = λxm−1i for all i = 1, . . . , n,

where x⊗m is the mth-order n-dimensional symmetric rank one tensor induced by x ,i.e.,

(x⊗m)i1...im = xi1 . . . xim for all i1, . . . , im ∈ {1, . . . , n}.

Observe that a symmetric tensor always has finitely many eigenvalues [31], and wemay consider the maximum eigenvalue ofA defined by λ1(A) := max{λ ∈ R | λ is aneigenvalue of A}. Note also that a symmetric tensor uniquely determines a real mthdegree homogeneous polynomial function by

〈A, x⊗m〉T :=n∑

i1,...,im=1

Ai1i2...im xi1 . . . xim

for all x = (x1, . . . , xn) ∈ Rn . It can be verified (see, e.g., [31,51]) that the maxi-

mum eigenvalue λ1(A) is the optimal value of the following polynomial optimizationproblem:

(P)A maxx∈Rn

〈A, x⊗m〉T

subject ton∑

i=1

xmi = 1.

Letting now f (x,A) := 〈A, x⊗m〉T with (x,A) ∈ Rn × T , it is not hard to check

that both Assumptions 1 and 2 are satisfied.To derive next our major sensitivity result for polynomial optimization problems,

we denote byφ(u) := max

x∈Kf (x, u), u ∈ R

l , (5.2)

the optimal value function in (POPu).

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352 G. Li et al.

Lemma 5.1 (Optimal value functions in polynomial optimization) Under the validityof Assumptions 1 and 2, the optimal value function (5.2) in (P O Pu) is continuous onR

l .

Proof Let uk be a sequence in Rl such that uk → u. To show that limk→∞ φ(uk) =

φ(u), choose xk ∈ K so that φ(uk) = maxx∈K f (x, uk) = f (xk, uk). Since K iscompact, we may assume that there exists x ∈ K such that xk → x . It follows fromthe continuity of f that limk→∞ φ(uk) = limk→∞ f (xk, uk) = f (x, u). Picking anarbitrary vector x ∈ K , we get that φ(uk) = f (xk, uk) ≥ f (x, uk). This implies inturn that

f (x, u) = limk→∞ f (xk, uk) ≥ lim

k→∞ f (x, uk) = f (x, u) for all x ∈ K .

Thus f (x, u) = φ(u), which completes the proof of the lemma. ��Now we are ready to establish the quantitative Hölderian stability of polynomial

optimization.

Theorem 5.2 (Hölder continuity of solutionmaps in polynomial optimization)Underthe validity of Assumptions 1 and 2, for any fixed u ∈ R

l the solution map S : Rl ⇒ Rn

in (P O Pu) satisfies the following Hölderian stability property at u: there are constantsc, δ > 0 such that we have

S(u) ⊂ S(u) + c ‖u − u‖τB(0, 1) whenever ‖u − u‖ ≤ δ (5.3)

with the explicit exponent

τ = max{ 1

R(n + r + s + 1, d + 1),

2

R(n + r, 2d)

}

Proof Note that for any fixed u ∈ Rl the solution set to (P O Pu) is represented as

S(u) := {x ∈ R

n∣∣ gi (x) ≤ 0 as i = 1, . . . , r, h j (x) = 0 as

j = 1, . . . , s, and φ(u) − f (x, u) = 0}.

It is easy to see that gi , h j , and φ(u) − f (·, u) are all polynomials on Rn with degreeat most d. Define

�u(x) :=(

r∑i=1

[gi (x)

]+

)+⎛⎝ s∑

j=1

|h j (x)|⎞⎠+ |φ(u) − f (x, u)|

and observe that S(u) = {x ∈ Rn| �u(x) = 0}.Let u be an arbitrary point inRl . Since

K is compact (Assumption 1), it follows from Corollary 3.8 that there is a constantc0 > 0 such that

d(x, S(u)

) ≤ c0 �u(x)τ for all x ∈ K .

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New fractional error bounds for polynomial systems 353

Next we employ Assumption 2 and find numbers L > 0 and δ > 0 such thatthe estimate (5.1) holds for all x ∈ K and u ∈ R

l with ‖u − u‖ ≤ δ. Denote

c := (2β−1L)τ with β := c− 1

τ

0 > 0. For any y ∈ S(u) we select now z ∈ S(u)

satisfying ‖y − z‖ = d(y, S(u)). To finish the proof, it suffices to show that

‖y − z‖ ≤ c ‖u − u‖τ . (5.4)

To see this, note that |φ(u)− f (y, u)| = �u(y) ≥ βd(y, S(u)

) 1τ = β‖y − z‖ 1

τ . Sincez ∈ S(u), we have that f (z, u) = φ(u) ≥ f (y, u), and hence

‖y − z‖ 1τ ≤ β−1|φ(u) − f (y, u)| = β−1( f (z, u) − f (y, u)

). (5.5)

Furthermore, it follows from y ∈ S(u) that f (z, u) ≤ f (y, u), and therefore (5.1)gives us the relationships

f (z, u) − f (y, u) = (f (z, u) − f (y, u)

)+ ( f (z, u) − f (z, u))+ ( f (y, u) − f (y, u)

)≤ (

f (z, u) − f (z, u))+ ( f (y, u) − f (y, u)

)≤ 2L‖u − u‖ as y, z ∈ K

implying together with (5.5) that ‖y − z‖ 1τ ≤ β−1

(f (z, u)− f (y, u)

) ≤ 2β−1L‖u −u‖. Thus

d(y, S(u)

) = ‖y − z‖ ≤ c ‖u − u‖τ ,

which justifies (5.4) and completes the proof of the theorem. ��Remark 5.3 (Comparison with Lipschitzian stability) If τ = 1 in (5.3), then we getthe upper Lipschitz property of S at u in the sense of Robinson [55], which morerestrictive than the Hölderian/fractional one established in Theorem 5.2 for generalproblems of polynomial optimization.We refer the reader to the books [41,56] and thebibliographies therein for such a Lipschitzian stability, its robust (around the referencepoint) version, and their further Lipschitzian type extensions.

As a consequence of the Hölderian stability in Theorem 5.2 we now show that themaximum eigenvalue function over the mth-order n-dimensional symmetric tensorspace T is at least ρth-order semismooth with the fractional quantity ρ calculated by

ρ := max{ 1

R(n + 2, m + 1),

2

R(n, 2m)

}, (5.6)

where R is taken from (2.3). This answers the following open question raisedin [31], where the authors showed that the maximum eigenvalue of an mth-ordern-dimensional symmetric tensor is always ρth-order semismooth for some ρ > 0 andposed the question about the possibility to give an estimate for the constant ρ. Notethat the order of semismoothness plays an important role in establishing convergence

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354 G. Li et al.

rates of nonsmooth Newton methods in solving nonsmooth equations; see [4,31,52]for more details.

To proceed, recall the definition of semismoothness, which goes back to [39] forscalar functions; see [52] for its extension to the vector case and important applica-tions to the generalized Newton method. We also present the ρth-order version ofsemismoothness on tensor spaces, which is used in what follows.

Definition 5.4 (Semismoothness) Let f : T → R be locally Lipschitzian around anddirectionally differentiable at the point in question. Then it is semismooth atA ∈ T if

f (A + �A) − f (A) − 〈V (�A),�A〉T = o(‖�A‖T )

for all V (�A) ∈ co ∂ f (A + �A).

Furthermore, f : T → R is ρth-order semismooth atA ∈ T with some ρ ∈ (0, 1]if

f (A + �A) − f (A) − 〈V (�A),�A〉T = O(‖�A‖1+ρ

T )

for all, V (�A) ∈ co ∂ f (A + �A).

When ρ = 1, f is called strongly semismooth at A. We also say that f semi-smooth (resp. ρth-order semismooth) on T if it is semismooth (resp. ρth-ordersemismooth) at every A ∈ T .

It easily follows from Definition 5.4 that the classes of semismooth and ρth-ordersemismooth functions is closed with respect to summation. The next result taken from[57, Theorem 3.7] provides a convenient tool for dealing with ρth-order semismooth-ness.

Lemma 5.5 (Equivalent description of ρth-order semismoothness) Let f : T → R

be locally Lipschitzian and directionally differentiable on a neighborhood of A. Thenf is ρth-order semismooth at A with ρ ∈ (0, 1] if and only if for any point A + �Aof differentiability of f we have

f (A + �A) − f (A) − ∇ f (A + �A)�A = O(‖�A‖1+ρ).

Nowwe are ready to derive the aforementioned result on the ρth-order semismooth-ness of the maximum eigenvalue function λ1 with the explicit calculation of ρ.

Theorem 5.6 (ρth-order semismoothness of maximum eigenvalue functions) Let Abe an mth-order n-dimensional symmetric tensor with an even number m. Then themaximum eigenvalue function λ1 is at least ρth-order semismooth at A, where theexponent ρ is explicitly calculated in (5.6).

Proof Recall that λ1(A) is the optimal value of the problem (PT )A defined above,i.e.,

λ1(A) = max{⟨A, x⊗m ⟩

T

∣∣∣n∑

i=1

xmi = 1

},

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New fractional error bounds for polynomial systems 355

where x⊗m is the rankone tensor inducedby x . Observe that the functionλ1 is Lipschitzcontinuous, convex, and thus directionally differentiable at the points in question.Denote by E1(A) the solution set of (PT )A, i.e., E1(A) = {A : ⟨A, x⊗m

⟩T =

λ1(A),∑n

i=1 xmi = 1}. By Danskin’s Theorem [9] we get

∂λ1(A) = co{

x⊗m∣∣ x ∈ E1(A)

}. (5.7)

It follows from Theorem 5.2 the existence of constants c, δ > 0 such that

E1(B) ⊂ E1(A) + c ‖B − A‖ρ

T B(0, 1) (5.8)

for all B ∈ T with ‖B − A‖T ≤ δ. Consider further an mth-order n-dimensionalsymmetric tensor �A such that 0 < ‖�A‖T ≤ ε and that λ1 is differentiable atA+�A; the existence of such a tensor follows from the classical Rademacher theoremdue to the Lipschitz continuity of λ1; see, e.g., [56]. This implies that ∂λ1(A + �A)

is a singleton. Then we get from (5.7) that

∂λ1(A+�A) = {∇λ1(A+�A)} = {

(w�A)⊗m} for some w�A ∈ E1(A+�A).

To complete the proof of the theorem by employing Lemma 5.5, it remains to showthat

λ1(A + �A) − λ1(A) − 〈(w�A)⊗m,�A〉T = O(‖�A‖1+ρ

T ). (5.9)

Since the mapping x �→ x⊗m from Rn to T is local Lipschitz, there is L > 0 with

‖x⊗m − y⊗m‖T ≤ L‖x − y‖ for all x, y ∈{

x ∈ Rn∣∣∣

n∑i=1

xmi = 1

}. (5.10)

Select v ∈ E1(A) so that ‖w�A−v‖ = d(w�A, E1(A)). Then inclusion (5.8) impliesthat ‖w�A − v‖ ≤ c‖�A‖ρ

T . It follows from (5.7) that v⊗m ∈ ∂λ1(A), which givesus by (2.2) the estimate

λ1(A + �A) − λ1(A) ≥ 〈v⊗m,�A〉T .

Then by using (5.10) we get the relationships

λ1(A + �A) − λ1(A) − 〈(w�A)m,�A〉T ≥ 〈v⊗m,�A〉T − 〈(w�A)m,�A〉T≥ −‖v⊗m − (w�A)⊗m‖T ‖�A‖T≥ −L‖v − w�A‖ ‖�A‖T≥ −Lc‖�A‖1+ρ

T . (5.11)

On the other hand, it follows from ∇λ1(A + �A) = {(w�A)m} and the convexity ofλ1 that

〈(w�A)m,−�A〉T = 〈(w�A)m,A − (A + �A)〉T ≤ λ1(A) − λ1(A + �A),

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356 G. Li et al.

which implies that λ1(A + �A) − λ1(A) − 〈(w�A)m,�A〉T ≤ 0. Combining thiswith (5.11), we arrive at (5.9) and complete the proof of the theorem. ��Remark 5.7 (matrix case) In the special case ofm = 2, a symmetric tensorA is nothingbut an (n × n) symmetric matrix. It follows from [57] that in this case the maximumeigenvalue function is strongly semismooth at A, i.e., it is ρth-order semismoothwith ρ = 1. However, our general result in Theorem 5.6 shows that A is merely

ρth-order semismooth with ρ = max{

1R(n+2,3) ,

2R(n,4)

}. Thus, although our order

estimate works for general tensors, it may not be tight in particular settings. This callsfor further improvements of the order semismoothness result obtained in the generaltensor case.

5.2 Hölderian stability of complementarity systems with polynomial data

This subsection is devoted to the study of Hölderian stability with explicit exponentfor the class of parameterized nonlinear complementarity problems described by

(NC P) xT F(x, u) = 0, x ≥ 0, and F(x, u) ≥ 0,

where x ∈ Rn and u ∈ R

p. In what follows we assume that each component of themapping F(x, u) = (F1(x, u), . . . , Fn(x, u)) is a polynomial on R

n+p with degreed. It has been well recognized that nonlinear complementarity systems under consid-eration constitute an important class of optimization-related problems with numerouspractical applications to, e.g., economics and engineering; see [14,37].

For each u ∈ Rp we define the solution set S(u) to (NCP) by

S(u) := {x ∈ R

n∣∣ xT F(x, u) = 0, x ≥ 0, and F(x, u) ≥ 0

}(5.12)

and say that the set-valuedmapping S : Rp ⇒ Rn isHölder calmwith exponent τ > 0

at (u, x) ∈ gph S if there are positive numbers c, ε, and δ such that

S(u) ∩ B(x, ε) ⊂ S(u) + c‖u − u‖τB(0, 1) whenever ‖u − u‖ ≤ δ. (5.13)

Note that for τ = 1 this property reduces to the (Lipschitz) calmness of multifunctions(a graphical localization of Robinson’s upper Lipschitz property in (5.3) with τ = 1)and has been widely studied in the literature; see, e.g., [56] and the references therein.

Theorem 5.8 (Hölder calmness of solution maps for NCP) Let S : Rp ⇒ Rn be the

solution map (5.12) for (NC P), and let (u, x) ∈ gph S. Then S is Hölder calm at

(u, x) with the explicit exponent τ = max{

1R(3n+1,d+1) ,

2R(3n,2d)

}.

Proof Since S(u) is the solution set for a polynomial system, we apply to it the localbound estimate fromCorollary 3.8with r = 2n and s = 1 finding in this way constantsc0, ε > 0 such that

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New fractional error bounds for polynomial systems 357

d(x, S(u)

) ≤ c0

( n∑i=1

[− xi]+ +

n∑i=1

[− Fi (x, u)]+ +

∣∣∣∣n∑

i=1

xi Fi (x, u)

∣∣∣∣)τ

(5.14)

whenever ‖x − x‖ ≤ ε. Considering the function

h(x, u) :=n∑

i=1

[− xi]+ +

n∑i=1

[− Fi (x, u)]+ +

∣∣∣n∑

i=1

xi Fi (x, u)

∣∣∣,

we see that h is locally Lipschit around (x, u) with nonnegative values and that therepresentation S(u) = {x ∈ R

n| h(x, u) = 0} holds. Fix δ > 0 and denote by L > 0some Lipschitz constant of the function h on the set B(x, ε) × B(u, δ), i.e.,

|h(x, u) − h(x ′, u′)| ≤ L(‖x − x ′‖ + ‖u − u′‖)for all (x, u), (x ′, u′) ∈ B(x, ε) × B(u, δ). (5.15)

Taking further any y = (y1, . . . , yn) ∈ S(u) ∩ B(x, ε) and using (5.14), we get therelationships

d(y, S(u)

) ≤ c0

( n∑i=1

[−yi ]+ +n∑

i=1

[− Fi (y, u)]+ +

∣∣∣n∑

i=1

xi Fi (a, u)

∣∣∣)τ

= c0h(y, u)τ

≤ c0

(h(y, u) + L‖u − u‖

)τ

= c0Lτ‖u − u‖τ ,

where the second inequality follows by (5.15) while the last equality is due to y ∈ S(u)

and so h(y, u) = 0. Thus justifies the Hölder calmness (5.13) of map (5.12) andcompletes the proof of the theorem. ��

In the same way as the classical local Lipschitzian behavior of set-valued mappingsaround the reference point is a robust version of Robinson’s upper Lipschitz property(see Remark 5.3), the robust counterpart of calmness in (5.13) with τ = 1 is knownas the Lipschitz-like (also as pseudo-Lipschitz or Aubin) property of S around (u, x),which corresponds to the case of τ = 1 in the relationship

S(u1) ∩ B(x, ε) ⊂ S(u2) + c‖u1 − u2‖τB(0, 1)

whenever ‖ui − u‖ ≤ δ as i = 1, 2 (5.16)

with some positive constants ε, δ, and c. The Lipschitz-like property of general mul-tifunctions has been extensively studied and applied in variational analysis and opti-mization; see, e.g., the books [41,56] and their commentaries. We particularly referthe reader to the recent paper [17] and the bibliography therein, where construc-tive characterizations of the Lipschitz-like property are obtained in terms of theinitial data for solution maps to parameterized variational systems, including the

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358 G. Li et al.

complementarity ones from (5.12), on the basis of the coderivative/Mordukhovichcriterion from [40] and [56, Theorem 9.40]. Now we derive, for the first timein the literature, a verifiable condition ensuring the robust Hölderian stability(5.16) with any exponent τ > 0 for polynomial complementarity systems as in(NCP).

To proceed, define via the initial data of (NCP) the function

f (x, u) := max{max1≤i≤n

{− xi}, max

1≤i≤n

{− Fi (x, u)},

∣∣∣n∑

i=1

xi Fi (x, u)

∣∣∣} (5.17)

and consider the two index subsets given by

I0(x, u) := {i ∈ {1, . . . , n}∣∣ − xi = f (x, u)

}and

I<(x, u) := {i ∈ {1, . . . , n}∣∣ − Fi (x, u) = f (x, u)

}.

By ei wedenote an element ofRn whose i th coordinate is 1 and all the other coordinatesare 0.

Theorem 5.9 (Robust Hölderian stability of solution maps for NCP) Let (u, x) ∈gph S for the solution map (5.12), and let τ > 0. Suppose that there exist positivenumbers c, δ, and ε such that for all x ∈ R

n and u ∈ Rp with ‖x − x‖ ≤ ε,

‖u − u‖ ≤ δ, and f (x, u) > 0 we have

inf{∥∥∥ ∑

i∈I<(x,u)

αi +∑

i∈I<(x,u)

βi ∇x Fi (x, u) + γ( n∑

i=1

xi ∇x Fi (x, u) +n∑

i=1

Fi (x, u)ei

)∥∥∥∣∣∣ ∑

i∈I0(x,u)

αi +∑

i∈I<(x,u)

βi + |γ | = 1, αi ≥ 0, βi ≥ 0, γ ∈ R

}≥ c| f (x, u)|1−τ .

Then the solution map (5.12) has the robust Hölder stability property (5.16) withexponent τ .

Proof Let fi (x, u) := −xi and fi+n(x, u) := −Fi (x, u) for i = 1, . . . , n,f2n+1(x, u) := ∑n

i=1 xi Fi (x, u), and f2n+2(x, u) := −∑ni=1 xi Fi (x, u). Then each

fi is a real polynomial on Rn+p with degree at most d + 1. It follows from the defin-itions that the function f from (5.17) is represented as

f (x, u) = max1≤i≤2n+2

fi (x, u).

For any fixed u ∈ Rp with ‖u − u‖ ≤ δ, write for convenience fi

u(x) := fi (x, u)

as i = 1, . . . , 2n + 2 and f u(x) := f (x, u) whenever x ∈ Rn . Then f u(x) =

max1≤i≤2n+2 f ui (x). Considering further the index set I (x) := {i | f u

i (x) = f u(x)},we deduce from the assumption made and Lemma 2.2 that

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New fractional error bounds for polynomial systems 359

m f u (x) = inf{∥∥∥ ∑

i∈I (x)

λi ∇ f ui (x)

∥∥∥ ∣∣∣ ∑i∈I (x)

λi = 1, λi ≥ 0}

≥ inf{∥∥∥−

∑i∈I<(x,u)

αi −∑

i∈I<(x,u)

βi ∇x Fi (x, u)

+(γ1 − γ2)( n∑

i=1

xi ∇x Fi (x, u) +n∑

i=1

Fi (x, u)ei

)∥∥∥∣∣∣ ∑

i∈I0(x,u)

αi +∑

i∈I<(x,u)

βi + γ1 + γ2 = 1, αi ≥ 0, βi ≥ 0, γi ≥ 0}

= inf{∥∥∥ ∑

i∈I<(x,u)

αi

∑i∈I<(x,u)

βi ∇x Fi (x, u) + γ( n∑

i=1

xi ∇x Fi (x, u) +n∑

i=1

Fi (x, u)ei

)∥∥∥∣∣∣ ∑

i∈I0(x,u)

αi +∑

i∈I<(x,u)

βi + |γ | = 1, αi ≥ 0, βi ≥ 0, γ ∈ R

}

≥ c | f (x, u)|1−τ = c | f u(x)|1−τ whenever ‖x − x‖ ≤ ε and f (x, u) > 0,

where the third equality follows from the fact that f (x, u) > 0 and hence γ1γ2 = 0.Employing now Lemma 3.4 ensures that for each u with ‖u − u‖ ≤ δ and for all xwith ‖x − x‖ ≤ ε/2 we have

d(x, S(u)

) = d(x, {x | f u(x) ≤ 0})

≤ 1

c

[f (x, u)

]τ+

≤ 1

c

( n∑i=1

[− xi]+ +

n∑i=1

[− Fi (x, u)]+ +

∣∣∣n∑

i=1

xi Fi (x, u)

∣∣∣)τ

. (5.18)

Consider the function h(x, u) := ∑ni=1[−xi ]+ +∑n

i=1[−Fi (x, u)]+ + |∑ni=1 xi Fi

(x, u)| and note that it is nonnegative and Lipschitz continuous on B(x, ε2 ) × B(u, δ)

with some constant L , i.e.,

|h(x, u) − h(x ′, u′)| ≤ L(‖x − x ′‖ + ‖u − u′‖)for all (x, u), (x ′, u′) ∈ B(x,

ε

2) × B(u, δ). (5.19)

Observing that S(u) = {x ∈ Rn| h(x, u) = 0} and picking any u1, u2 ∈ R

p with‖ui − u‖ ≤ δ as well as any y ∈ S(u1) ∩ B(x, ε

2 ), we deduce from (5.18) that

d(y, S(u2)

) ≤ 1

c

( n∑i=1

[− yi]+ +

n∑i=1

[− Fi (y, u2)]+ +

∣∣∣n∑

i=1

xi Fi (y, u2)

∣∣∣)τ

= 1

ch(y, u2)

τ

≤ 1

c

(h(y, u1) + L‖u2 − u1‖

)τ

= 1

cLτ‖u2 − u1‖τ ,

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360 G. Li et al.

where the second inequality holds by (5.19) while the last equality is a consequenceof y ∈ S(u1) and hence h(y, u1) = 0. This justifies the claimed Hölder continuity ofthe solution map (5.12). ��

6 Concluding remarks

In this paper we employ advanced techniques of variational analysis and generalizeddifferentiation to extended, in particular, the local and global error bounds in [8] from asingle polynomial to general polynomial systems with explicitly calculated exponents.Besides being of their own interest, these results are important for convergence ratesof numerical algorithms. The obtained error bounds are applied to Hölderian stabilityof solution maps for polynomial optimization problems and their tensor eigenvaluespecifications as well as for parameterized nonlinear complementarity systems withpolynomial data. In this way we resolve, in particular, some open questions posted inthe literature.

Nevertheless, many significant issues in these directions still needs further investi-gation. Some of them are indicated in the text; see, e.g., Remark 5.7. It would be alsoimportant to identify remarkable classes of polynomial systems for which the generallocal and global error bounds can be sharpened. On the other hand, it is appealingto extend the proposed techniques and the results obtained on Hölderian stability topolynomial optimization problems with perturbations not only in the cost function butalso in the constraint functions as well.

Furthermore, in contrast to Lipschitzian stability, its higher-order Hölderian coun-terpart seems to be largely uninvestigated in variational analysis and optimization; inparticular, for polynomial systems considered in the paper. Among the most impor-tant and challenging issues of further research related to the context of our Sect. 5we mention the desired developments of Hölderian tilt and full stability of optimalsolutions to extend the original Lipschitzian frameworks proposed in [50] and [24],respectively; see [42,43] and the references therein for recent Lipschitzian type resultsin these directions.

Acknowledgments The authors are gratefully indebted to the referees and the handling Associate Editorfor their helpful remarks, which allowed us to significantly improved the original presentation.

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