Higher order error bound for the difference of two functions

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R E S E A R C H

Open Access

Higher-order error bound for the

difference of two functions

Hui Huang

1*

_{and Mengxue Xia}

1

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Yunnan University, Kunming, P.R. China

Abstract

Error bounds play an important role in the research of mathematical programming. Using some techniques of nonsmooth analysis, we establish some results on the existence of higher-order error bounds for diﬀerence functions with set constraints.

MSC: 49J52; 90C31

Keywords: Error bound; Diﬀerence function; Ekeland variational principle

1 Introduction

LetXbe a Banach space, and letΩbe a nonempty closed convex subset ofX. Letf :X→

R∪ {+∞}be a proper lower semicontinuous function. We assume that

S=x∈Ω|f(x)≤0=∅.

Leta∈S,τ> 0, andλ> 0. We say thatf has aλ-order local error boundτ ataif there existsδ> 0 such that

d(x,S)λ≤f(x)₊, ∀x∈a+δBX, (1.1)

where BX denotes the closed unit ball of X, d(x,S) =inf{x–y|y∈S}, and [f(x)]+=

max{f(x), 0}. We say thatf has aλ-order global error boundτ ifa+δBX in (1.1) can be replaced by the whole spaceX.

Error bounds play an important role in convergence and perturbation analysis of some algorithms and mathematical programming [1–3]. In the last twenty years, many re-searchers studied error bounds and obtained a lot of interesting results; see the survey papers [2,3] and the references therein. However, these results are mainly concerned with error bounds in the caseλ= 1. Recently, Huang [4] considered higher-order error bounds for strongly convex multifunctions. Huang [5] and Huang and Li [6] also con-sidered mixed-order error bounds for gamma paraconvex multifunctions. Zheng and Ng [7] studied Hölder weak sharp minimizers, which closely relate to error bounds for lower semicontinuous functions.

Recall that a functionf:X→R∪ {+∞}is said to be a difference (DC) function iff is the difference of two (convex) functions. Many nonconvex optimization problems are differ-ence structure optimization problems. In 2012, Le Thi, Pham Dinh, and Ngai [8] studied

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error bounds for DC functions inRn_{. In 2014, Huang and Li [}₉_{] studied error bounds for} DC multifunctions. In 2016, Van Hang and Yao [10] established sufficient conditions for the existence of error bounds for difference functions and applications. However, all re-sults mentioned in this paragraph are concerned withλ= 1. It is natural for us to consider higher-order error bounds for difference functions.

The rest of this paper is organized as follows. In Sect.2, we give some notions. In Sect.3, we establish suﬃcient and necessary conditions for the existence of higher-order error bounds for diﬀerence functions with set constraints in terms of nonsmooth analysis tools.

2 Preliminaries

LetXbe a real Banach space, and letτ,λ> 0. We say thatf :X→R∪ {+∞}is aλ-order strongly convex function with modulusτ if for allx,y∈Xandt∈(0, 1),

τt(1 –t)x–yλ₊_f_tx_{+ (1 –}_t₎_y_≤_tf₍_x_{) + (1 –}_t₎_f₍_y_).

In the particular caseτ= 0, strongly convex functions reduce to convex functions. We say thatfis proper if its domaindom(f) :={x∈X|f(x) < +∞} =∅. Letx0∈dom(f). We say that

f is lower semicontinuous atx0if

lim inf

x→x0

f(x)≥f(x0).

Letu∈X. We deﬁne

f(x0;u) =lim inf

t→0+

f(x0+tu) –f(x0)

t .

It is known that

f(x0;u) =f(x0;u) = lim

t→0+

f(x0+tu) –f(x0)

t

iff is a convex function. For a convex functionf andx¯∈dom(f), the subdiﬀerential off

atx¯is deﬁned as

∂f(x¯) =x∗∈X∗|x∗,x–x¯ ≤f(x) –f(x¯),∀x∈X.

LetΩ be a closed convex subset ofX, and leta∈Ω. The tangent cone of Ω atais deﬁned as

T(Ω,a) =v∈X|∃tn> 0,tn→0+,vn∈X,vn→vsuch thata+tnvn∈Ω

.

LetA⊂Rn_and_x_∈_Rn_{. The projection of}_x_on_A_{is deﬁned as}

PA(x) =y∈A|x–y=d(x,A).

The limiting normal cone ofAata∈A[11] is deﬁned as

N(A;a) =v∈Rn|∃xk∈Rn,xk→a,tk> 0,uk∈PA(xk) such thattk(xk–uk)→v

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Letx¯∈Xandδ> 0. ByB(x¯,δ) we denote the open ball with center atx¯and radiusδand bySXthe unit sphere ofX. Bybdry(S) we denote the boundary of a setS.

3 Main results

In this section, we assume thatXis a real Banach space unless stated otherwise,Ω⊂X

is a nonempty closed convex set,g,h:X→R∪ {+∞}are two proper functions, andS=

{x∈Ω|g(x) –h(x)≤0} =∅. We first establish sufficient conditions for the existence of higher-order error bounds for difference functions.

Theorem 3.1 Let g:X→R∪{+∞}be a lower semicontinuous function,and let h:X→R be a continuous function.Letτ > 0,δ> 0,λ> 0,andx¯∈S.Suppose that,for each x∈Ω\S such thatx–x¯<δ,there exists u∈T(Ω;x)∩SXsuch that

g(x;u) +τd(x,S)1λ≤_h₍_x_;_u_). _(3.1)

Then

τ

4·21λ

d(x,S)1+λλ≤g(x) –h(x)

+, ∀x∈Ω∩B

¯

x,2 3δ

. (3.2)

Proof Suppose on the contrary that (3.2) is not true. Then there existsx˜∈Ω such that

˜x–x¯<2₃δand

g(x˜) –h(x˜)₊< τ 4·2λ1

d(x˜,S)1+λλ. (3.3)

Sinceinfx∈Ω[g(x) –h(x)]+= 0, it follows from (3.3) that

g(x˜) –h(x˜)₊< inf

x∈Ω

g(x) –h(x)₊+ τ 4·2λ1

d(x˜,S)1+λλ_.

By the completeness ofXand the closedness ofΩwe know thatΩ is a complete metric space with respect to the metric induced by the norm ofX. By the Ekeland variational principle [12] there existsx0∈Ωsuch that

x0–x˜ ≤

1

2d(x˜,S) (3.4)

and

g(x0) –h(x0)

+<

g(x) –h(x)₊+ τ 2·2λ1

d(x˜,S)λ1x–x₀, ∀x∈_Ω\ {x₀}. (3.5)

From (3.4) we have thatx0∈/S, and sog(x0) –h(x0) > 0. Letu∈T(Ω;x0)∩SX. SinceΩis a convex set, there existst0> 0 such thatx0+tu∈Ωfor allt∈(0,t0). Sinceg–his lower

semicontinuous atx0, there existst1∈(0,t0) such that

g(x0+tu) –h(x0+tu)

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for allt∈(0,t1). It follows from (3.5) that

g(x0) –h(x0) <g(x0+tu) –h(x0+tu) +

τ

2·21λ

d(x˜,S)1λ_x₀₊_tu_–_x₀

for allt∈(0,t1), that is,

h(x0+tu) –h(x0) <g(x0+tu) –g(x0) +

τ

2·21λ

d(x˜,S)1λ_t

sinceu= 1. Therefore

h(x0+tu) –h(x0)

t <

g(x0+tu) –g(x0)

t +

τ

2·2λ1

d(x˜,S)1λ.

Takinglim infast→0+_{, we get}

h(x0;u)≤g(x0;u) +

τ

2·21λ

d(x˜,S)1λ_. _(3.6)

By (3.4) we have

d(x˜,S)≤d(x0,S) +˜x–x0 ≤d(x0,S) +

1 2d(x˜,S),

and sod(x˜,S)≤2d(x0,S). This inequality and (3.6) imply that

h(x0;u)≤g(x0;u) +

τ

2·21λ

21λ_d₍_x₀_,_S₎λ1 ₌_g₍_x₀_;_u_{) +}τ

2d(x0,S)

1

λ_. _(3.7)

By (3.4) we get

x0–x¯ ≤ x0–x˜+˜x–x¯ ≤

1

2d(x˜,S) +˜x–x¯

≤1

2˜x–x¯+˜x–x¯<δ.

(3.8)

Inequalities (3.7) and (3.8) are a contradiction to (3.1). The proof is completed.

We now give necessary conditions for the existence of higher-order error bounds for diﬀerence functions.

Theorem 3.2 Let g,h:X→Rbe two continuous convex functions.Letτ> 0,δ> 0,λ> 0,

andx¯∈S.Suppose that

τd(x,S)1+λλ≤_g₍_x_{) –}_h₍_x₎

+, ∀x∈Ω∩B(x¯,δ). (3.9)

Then,for each x∈Ω\S such thatx–x¯<δ

2,there exist a∈S and u∈T(Ω;x)∩SXsuch

that

g(x;u) +τ 2d(x,S)

1

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Proof Letx∈Ω\Sbe such thatx–x¯<δ₂. Takea∈bdry(S) such thatx–a< 2d(x,S). Then

x–a< 2d(x,S)≤2x–x¯<δ.

Asa∈bdry(S), we haveg(a) –h(a) = 0. By (3.9), τ

2d(x,S)

1

λx–a ≤_τd(x,S) 1+λ

λ ≤g(x) –h(x) –g(a) –h(a),

that is,

h(x) –h(a) +τ 2d(x,S)

1

λ_x_–_a ≤_g₍_x_{) –}_g₍_a_). _(3.11)

Letx∗∈∂g(x) anda∗∈∂h(a). Then

x∗,a–x ≤g(a) –g(x), a∗,x–a ≤h(x) –h(a). It follows from (3.11) that

a∗, x–a

x–a

+τ 2d(x,S)

1

λ ≤

x∗, x–a

x–a

.

Denoteu:=a_a–_–x_x. Thenu∈T(Ω;x)∩SXsinceΩis a convex set. The last inequality im-plies that

– max

a∗∈∂h(a)

a∗,u +τ 2d(x,S)

1

λ ≤_– _max

x∗∈∂g(x)

x∗,u.

Sinceh(a;u) =maxa∗∈∂h(a)a∗,u andg(x;u) =maxx∗∈∂g(x)x∗,u, from this inequality it

follows that

–h(a;u) +τ 2d(x,S)

1

λ≤_–_g₍_x_;_u_).

Therefore (3.10) is veriﬁed.

Corollary 3.1 Let g :X→Rbe a continuous convex function, and letλ> 0.Then the following two statements are equivalent:

(i) There existτ > 0andδ> 0such that

τd(x,S)1+λλ≤_g₍_x₎

+, ∀x∈Ω∩B(x¯,δ);

(ii) There existτ> 0andδ> 0such that,for eachx∈Ω\Swithx–x¯<δ,there existsu∈T(Ω;x)∩SXsuch that

g(x;u)≤–τd(x,S)1λ.

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Theorem 3.3 Let g:X→Rbe aλ-order strongly convex function with modulusτ,and let h:X→Rbe a convex function.If for each x∈Ω\S,there exists y∈bdry(S)such that

g(y;x–y) +h(x;y–x)≥0, (3.12)

then

τd(x,S)λ≤g(x) –h(x)₊, ∀x∈Ω.

Proof Letx∈Ω. Without loss of generality, we may assume thatx∈Ω\S. Takey∈bdry(S) such that (3.12) holds. Lett∈(0, 1). Sinceg is aλ-order strongly convex function with modulusη, we have

τt(1 –t)x–yλ≤tg(x) + (1 –t)g(y) –gtx+ (1 –t)y, that is,

τ(1 –t)x–yλ≤g(x) –g(y) –g(y+t(x–y)) –g(y)

t .

Lettingt→0+_{, we get}

τx–yλ≤g(x) –g(y) –g(y;x–y). (3.13) Sincehis a convex function, we have

0≤h(y) –h(x) –h(x;y–x). (3.14)

Adding (3.13) and (3.14), we have

τx–yλ_≤_g₍_x_{) –}_h₍_x_{) –}_g₍_y_{) –}_h₍_y₎_–_g₍_y_;_x_–_y_{) +}_h₍_x_;_y_–_x₎

≤g(x) –h(x), (3.15)

due to assumption (3.12) and the equalityg(y) –h(y) = 0. Sincey∈bdry(S), it follows from (3.15) that

τd(x,S)λ≤g(x) –h(x) =g(x) –h(x)₊. We now give an example to illustrate Theorem3.3.

Example3.1 Letg,h:R→Rbe deﬁned as

g(x) =x2, h(x) =x, ∀x∈R.

Clearly,g is a second-order strongly convex function with modulusτ = 1. It is easy to calculate thatS={x∈R|g(x) –h(x)≤0}= [0, 1]. TakeΩ= (–∞, 1]. Letx∈Ω\S= (–∞, 0). There existsy= 0∈bdry(S) such that

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All conditions of Theorem3.3are satisﬁed. By Theorem3.3we have

d(x,S)2≤g(x) –h(x)₊, ∀x∈Ω.

Theorem 3.4 Take X=Rn_,_{and let f} _:_Rn_→_R_{be an mth-order smooth function}₍_{m is a}

positive integer number),δ> 0,S:={x∈Rn_|_f₍_x₎_≤₀_}₌_∅_,_and_x_¯_∈_bdry₍_S_)._{Suppose that}_,

for each x∈B(x¯,δ)\S,there exists u∈PS(x)such that

f(p)(u)(x–u)p≥0, p= 1, 2, . . . ,m– 1,

and

f(m)(x¯)vm> 0, ∀v∈N(S,x¯)\ {0}. (3.16)

Then there existτ>andη> 0such that

τd(x,S)m_≤_f₍_x₎

+, ∀x∈B(x¯,η).

Proof Suppose on the contrary that there exists a sequence{xk} ⊂B(x¯,δ) withxk→ ¯xsuch that

d(xk,S)m

k >

f(xk)₊. (3.17)

Clearly,xk∈/Sfor allk. By the assumption we can takeuk∈PS(xk) such that

f(p)(uk)(xk–uk)p≥0, p= 1, 2, . . . ,m– 1. (3.18)

Clearly,xk–uk=d(xk,S). Lettingk→ ∞, we haveuk→ ¯x. By (3.17) we get

xk–ukm

k >f(xk) –f(uk) (3.19)

sincef(uk) = 0. By the Taylor theorem,

f(xk) –f(uk) =f(uk)(xk–uk) + 1 2!f

₍_u_k)(_x

k–uk)2+· · · +f

(m)₍_θ

kxk+ (1 –θk)uk)

m! (xk–uk)

m _(3.20)

for someθk∈(0, 1), and (3.18)–(3.20) imply that

xk–ukm

k ≥f

₍_u_k)(_x

k–uk) + 1 2!f

₍_u_k)(_x

k–uk)2+· · · +f

(m)₍_θ

kxk+ (1 –θk)uk)

m! (xk–uk) m

≥f(m)(θkxk+ (1 –θk)uk)

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that is,

m!

k ≥f

(m)_θ

kxk+ (1 –θk)uk

xk–uk

m

. (3.21)

Sinceuk∈PS(xk) and{x_xk–uk

k–uk}is bounded, without loss of generality, we may assume that

xk–uk

xk–uk→v∈N(S,x¯). Lettingk→ ∞in (3.21), we have f(m)(x¯)vm≤0.

This contradicts (3.16). The proof is completed.

We now give an example to illustrate Theorem3.4.

Example3.2 Letf :R2→Rbe deﬁned by

f(ξ1,ξ2) =ξ12+ξ22, ∀x= (ξ1,ξ2)∈R2.

Clearly,S={(0, 0)}. Takex¯= (0, 0)∈bdry(S). ThenN(S,x¯) =R2_{. Let}_x_∈_R2_\_S_{. There exists}

u= (0, 0)∈PS(x) such thatf(u)(x–u) = 0 and

f(x¯)v2= 2v2₁+v2₂> 0, ∀v= (v1,v2)∈N(S,x¯)\ {0}.

All conditions of Theorem3.4are satisﬁed. By Theorem3.4there existsτ> 0 (τ= 1) such that

τd(x,S)2≤f(x)₊, ∀x∈R2. 4 Conclusion

In this paper, we establish two existence theorems of higher-order error bounds for dif-ference functions and an existence theorem of higher-order error bounds formth-order smooth functions. Moreover, the coeﬃcients in Theorems3.1–3.4can be calculated. It is interesting for us to consider higher-order error bounds for DC multifunctions.

Acknowledgements

The authors are greatly indebted to the reviewers and the Editor for their valuable comments.

Funding

The ﬁrst author was supported by the National Natural Science Foundation of China (Grant No. 11461080).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have made the same contribution. Both authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

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