Strongly regular points of mappings

R E S E A R C H Open Access

Strongly regular points of mappings

Malek Abbasi¹and Michel Théra^2,3*

Dedicated to the memory of Jonathan Borwein, in recognition of his deep contributions to mathematics and of his lasting friendship. He was one of the pioneers of metric regularity.

*Correspondence:

michel.thera@unilim.fr;

m.thera@federation.edu.au

2XLIM UMR-CNRS 7252, Université de Limoges, 123, Avenue Albert Thomas, Limoges, 87060, France

3Centre for Informatics and Applied Optimisation, School of

Engineering, IT and Physical Sciences, Federation University, Ballarat, Victoria, 3350, Australia Full list of author information is available at the end of the article

Abstract

In this paper, we use arobust lower directional derivativeand provide some sufficient conditions to ensure thestrong regularityof a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system oflinear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

MSC: 49J52; 49J53; 49J99

Keywords: Error bound; Lyusternik theorem; Regular point; Hadamard directional derivative; Hoffman estimate

1 Introduction

Letfbe a mapping acting between the normed spacesXandY, whose norms are denoted by the same symbol · . To estimate the approximate solutions of the equationy=f(x), we seek anerror bound

dist

x,f^–1(y)

≤κy–f(x),

locally, for all (x,y) near (¯x,y¯=f(¯x)), or globally, for allxandy, whereκis some positive constant. The infimum of suchκ is called themodulus of regularityoff. For instance, whenf :R→Ris smooth and verifiesf(x)¯ = 0, it is easily observed that the modulus of regularity off atx¯is exactly|f(x)¯ |^–1. A first approach to the concept of regularity goes back to a celebrated fundamental result proved in 1934 by Lyusternik [1]:

Theorem 1.1(Lyusternik, [1]) Let f be a mapping from a Banach spaceXto a Banach spaceY.Suppose that f is Fréchet differentiable in a neighborhood ofx and that its deriva-¯ tive f(x)is continuous atx and f¯ (x)¯ is surjective.Then,for everyε> 0,there exists r> 0 such that

dist

x,f^–1(0)

≤εx–x,¯

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whenever

x–x ≤¯ r and f(¯x)(x–x) = 0.¯

In other words,the tangent manifold to f^–1(0)is equal tox¯+Kerf(¯x),whereKerf(¯x)is the set of those x such that f(¯x)(x) = 0.

We refer to Dontchev [2] for a nice overview on the Lyusternik theorem and to the fact that the Lyusternik theorem can be easily obtained from the Graves theorem. We also refer to the forthcoming book by Thibault [3].

Theorem 1.2(Graves, [4]) LetXandYbe Banach spaces, x¯∈X, and f :X→Ybe a C¹-mapping whose derivative f(¯x)is onto.Then,there exist a neighborhoodUofx and a¯ constant c> 0such that for every x∈Uandτ> 0withB(¯x,τ)⊂U,

B(¯x,cτ)⊂f B(¯x,τ)

(partial openness property with linear rate).

Ioffe and Tihomirov showed in [5] that the original Lyusternik proof may lead to a stronger result and proved that iff(¯x) is surjective, then there areκ> 0 andδ> 0 such that

dist

x,f^–1(¯y)

≤κf(x) –f(¯x) wheneverx–x¯ <δ. (1.1) Ioffe’s remark leads to a standard definition:

Definition 1.1 Pointx¯∈Xis said to be aregular point of a mappingf :X→Yif the relation (1.1) is satisfied.

In this note, we will callx¯astrongly regular point of f if the inequality

x–x ≤¯ κf(x) –f(¯x), (1.2)

holds locally, for allxbelonging to a neighborhood ofx, where¯ κ> 0 is a positive constant.

Next, we will provide sufficient conditions forx¯to be a strongly regular point. Our results allow us to estimate the constantκ in (1.2). Then, we apply our results to the Hoffman estimate and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the upper limit of the error. In particular, for a finite-dimensional spaceXand a linear (continuous) mappingA:X→X, we prove that the estimate

dist(x,KerA)≤ A(x)

inf{A(u):u∈X,dist(u,KerA) = 1},

holds for allx∈X(Corollary3.2below). We can easily see that this estimate issharpfor injective linear mappings, in the sense that, ifAis an injective linear mapping and

infA(u):u∈X,dist(u,KerA) = 1

<μ,

then there exists somex∈Xsuch thatx=dist(x,KerA) >μ^–1A(x).

Our work is outlined as follows. In Sect.1, we recall the famous Lyusternik theorem and survey briefly its relationship with the concept of metric regularity. In Sect.2, we first introduce the notion ofhomogeneous continuityof mappings. Then, using an appropriate notion of lower directional derivative, we achieve some results ensuring in finite dimen- sion that for a given mapping a point is strongly regular. Finally, in Sect.3, we focus our attention on Hoffman’s estimate of approximate solutions of finite systems of linear in- equalities and prove some similar estimates.

2 Sufficient conditions of regularity via generalized derivative

Throughout the paper, we use standard notations. For a normed spaceX, we denote its norm by · and byX^∗ its (continuous) dual. The symbolSstands for the unit sphere, that is, the set of all points ofXof norm one, whileB(x,r) andB(x,r) denote, respectively, the open and closed balls centered atxwith radiusr. Some other notations are introduced as and when needed.

2.1 Homogeneous continuity We begin with the following definition.

Definition 2.1 LetXandYbe normed spaces andE⊂X. The mappingf :X→Yis said to behomogeneously continuous atx¯∈XonEif for every> 0 there existδ> 0 and 0 <β≤1 such that

x–y<δ ⇒ f(¯x+tx) –f(¯x+ty)<t, for all 0 <t≤βand allx,y∈E.

We are going to provide some sufficient conditions under which a mappingf is homo- geneously continuous. Let us recall that a mappingf :X→Yis said to belocally Lipschitz aroundx¯∈Xif there exist a neighborhoodOofx¯and a real numberλ> 0 such that

f(x) –f(y)≤λx–y, (2.1)

for allx,y∈O.

Lemma 2.1 Suppose thatXandYare normed spaces.If f :X→Yis locally Lipschitz aroundx¯∈X,then f is homogeneously continuous atx on some closed ball¯ B(0,r).

Proof By hypothesis, there exist a constantλ> 0 and a neighborhoodOofx¯ inXsuch that (2.1) holds for allx,y∈O. Chooser> 0 such thatB(¯x,r)⊂O. It follows that

f(x¯+tx) –f(x¯+ty)≤λx¯+tx– (x¯+ty)=tλx–y,

for allx,y∈B(0,r) and all 0≤t≤1. Now for each> 0 take 0 <δ<λ^–1. It follows that x–y<δ ⇒ f(¯x+tx) –f(¯x+ty)<t,

for allx,y∈B(0,r) and all 0 <t≤1. This completes the proof.

Proposition 2.1 LetXandYbe normed spaces,f :X→Ybe a mapping,Ebe a subset ofX equipped with the topology induced by the norm andx¯∈X.If the bifunction f_E:E×(0, 1]→ Y,defined by

f_E(x,t) :=f(¯x+tx) –f(¯x)

t ,

is uniformly continuous(E×(0, 1]equipped with the product topology with the usual linear operations of vector addition and scalar multiplication),then f is homogeneously continu- ous atx on¯ E.

Proof Let> 0. By hypothesis, there existδ,β> 0 such that for allx,y∈Ewithx–y<δ and alls,h∈(0, 1] with|s–h|<βwe havefE(x,s) –fE(y,h)<. It follows that

f(¯x+tx) –f(¯x)

t –f(¯x+ty) –f(¯x) t

<,

for allx,y∈Ewithx–y<δand all 0 <t≤1. Thus f(¯x+tx) –f(¯x+ty)<t,

for allx,y∈Ewithx–y<δand all 0 <t≤1. This completes the proof.

2.2 Generalized derivatives

We recall the definitions of the Hadamard and Gateaux derivatives: TheHadamard direc- tional derivative f_H(¯x)(ν) off atx¯in directionνis defined as

f_H(x)(ν) :=¯ lim

t↓0,μ→ν

f(¯x+tμ) –f(¯x)

t = lim

n→+∞

f(¯x+tnνn) –f(¯x) tn

, where (ν_n) and (t_n) are any sequences such thatν_n→νandt_n→0⁺.

TheGateaux directional derivative f_G(x)(ν) of¯ f atx¯in directionνis defined by f_G(¯x)(ν) :=lim

t↓0

f(¯x+tν) –f(¯x)

t .

The following facts are well known:

• Hadamard differentiability is astrongernotion than Gateaux differentiability, see, e.g., [6]; whenf is Hadamard differentiable atx, it is Gateaux (directional) differentiable at¯

¯

xand, moreover,f_G(x)¯ is continuous;

• For locally Lipschitz mappings in normed spaces, Hadamard and Gateaux directional derivatives coincide.

The following corollary uses Hadamard differentiability and provides another sufficient condition for a mappingf to be homogeneously continuous.

Corollary 2.1 LetXandYbe normed spaces,f :X→Ybe a continuous mapping,E be a compact subset ofX(equipped with the topology induced by the norm)andx¯∈X. If the Hadamard directional derivative of f atx in every direction¯ ν∈Eexists,then f is homogeneously continuous atx on¯ E.

Proof Define the bifunctionf¯_E:E×[0, 1]→Yas f¯_E(ν,t) :=

⎧⎨

⎩

f(¯x+tν)–f(¯x)

t if 0 <t≤1, f_H(¯x)(ν) ift= 0.

Sincef is continuous and the Hadamard directional derivative off atx¯in every direction ν∈Eexists, the bifunction¯f_Eis continuous. SinceE×[0, 1] is compact,f¯_Eis uniformly continuous. It follows that the bifunctionf_E:E×(0, 1]→Y, defined by

f_E(x,t) :=f(¯x+tx) –f(¯x)

t ,

is uniformly continuous. Now apply Proposition2.1.

The following proposition illustrates our main motivation for introducing the homoge- neously continuous mappings.

Proposition 2.2 LetXandYbe normed vector spaces,f :X→Ybe a mapping,Ebe a subset ofXandx¯∈X.If f is homogeneously continuous atx on¯ E,then there existδ> 0 andβ> 0such that

f(¯x+tx) –f(¯x)

t –f(¯x+ty) –f(¯x) t

<,

for all x,y∈Ewithx–y<δand all0 <t≤β.

Proof The proof is obvious; we therefore omit it.

For a mappingf :X→Y, we consider the following notions oflower directional deriva- tiveswhich are crucial to our approach:

f_l(¯x)(ν) :=lim inf

t↓0

f(¯x+tν) –f(¯x)

t ,

f₀(x)(ν) :=¯ lim inf

t↓0,μ→ν

f(¯x+tμ) –f(¯x)

t .

Note that we have

0≤f₀(¯x)(ν)≤f_l(¯x)(ν), (2.2)

for everyν∈X. We shall observe that ifinf_ν∈Sf_l(¯x)(ν) > 0 andf is homogeneously contin- uous atx¯onS, thenf satisfies the property (1.2) above.

2.3 Main results

Throughout the remaining part of the discussion, unless specified otherwise, we assume thatXis afinite-dimensional spaceandYis an arbitrary normed space. We now are com- pletely ready to state the main theorem of the paper. For a positive scalarα∈R, let

Sα:=

x∈X:x=α

=αS.

Theorem 2.1 Let f:X→Ybe homogeneously continuous atx¯∈XonSαfor some positive scalarα.If there exists someκ> 0such thatinf_ν∈Sαf_l(¯x)(ν) >κ,then there existsδ> 0such that

x–x¯ ≤ α

κf(x) –f(x)¯ ,

for all x∈B(¯x,δ).In other words,x is a strongly regular point of f¯ .

Proof Letκ<γ <inf_ν∈Sαf_l(¯x)(ν) and:=γ–κ. Hence, for allν∈Sαthere exists 0 <rν≤1 such that

0<h≤rinfν

f(¯x+hν) –f(¯x)

h >γ. (2.3)

Sincef is homogeneously continuous atx¯onSα, there existθ> 0 andβ> 0 such that ν–μ<θ ⇒

f(¯x+tν) –f(¯x)

t –f(¯x+tμ) –f(¯x) t

<, (2.4)

for allν,μ∈Sαand all 0 <t≤β, by Proposition2.2. Letrˆν:=min{θ,β,rν}for allν∈Sα. Clearly,Sα⊂

ν∈SαB(ν,ˆrν). The compactness ofSαimplies that there existν₁,ν₂, . . . ,νm∈ Sα such thatSα⊂_m

k=1B(νk,rˆν_k). Now letx∈B(¯x,αδ)ˆ \ {¯x}andν:=_x–¯^α_x(x–x), where¯ δˆ:=min{ˆrν_k: 1≤k≤m}. Then,ν∈Sαand thereforeν∈B(ν_s,ˆrνs) for some 1≤s≤m. It follows thatν–ν_s<θ andα^–1x–x¯<β. By (2.4), we deduce that

f(x¯+α^–1x–x¯ν_s) –f(x)¯

α^–1x–x¯ –f(x¯+α^–1x–x¯ν) –f(x)¯ α^–1x–x¯

<.

Hence

f(x) –f(x)¯

α^–1x–x¯ =f(x¯+α^–1x–x¯ν) –f(x)¯

α^–1x–x¯ >f(x¯+α^–1x–x¯ν_s) –f(x)¯ α^–1x–x¯ –

>γ–=κ by (2.3), sinceα^–1x–x¯<rνs. It follows that

x–x ≤¯ α

κf(x) –f(¯x),

for allx∈B(¯x,αδ). Lettingˆ δ:=αδˆcompletes the proof.

Corollary 2.2 Let f :X→Ybe homogeneously continuous atx¯∈XonS.If there exists someκ> 0such thatinfν∈Sf₀(¯x)(ν) >κ,then there existsδ> 0such that

x–x ≤¯ 1

κf(x) –f(¯x), (2.5)

for all x∈B(¯x,δ).

Proof Apply Theorem2.1and (2.2).

Corollary 2.3 Suppose that f :X→Yis locally Lipschitz aroundx¯∈X.If there exists someκ> 0such thatinf_ν∈Sf_l(¯x)(ν) >κ,then there existsδ> 0such that(2.5)holds for all x∈B(¯x,δ).

Proof By Lemma2.1,f is homogeneously continuous atx¯on some closed ballB(0,r). It follows thatf is homogeneously continuous atx¯onSr(sinceSr⊂B(0,r)). The condition inf_ν∈Sf_l(x)(ν) >¯ κimplies thatinf_ν∈Srf_l(x)(ν) >¯ rκ. Now apply Theorem2.1.

Corollary 2.4 Let f : X→Y be a continuous mapping and x¯ ∈X. Assume that the Hadamard directional derivative of f atx in every direction¯ ν∈Sexists. If there exists someκ> 0such thatinf_ν∈Sf_H(¯x)(ν)>κ,then there existsδ> 0such that(2.5)holds for all x∈B(¯x,δ).

Proof By hypothesis,f_H(¯x)(ν) exists for everyν∈S. By continuity off and · , it follows that

f_H(¯x)(ν)=f_G(¯x)(ν)=lim

t↓0

f(¯x+tν) –f(¯x)

t =f_l(¯x)(ν),

for everyν∈S. It follows thatinfν∈Sf_l(¯x)(ν) >κ. SinceXis finite dimensional,Sis com- pact. Hence,f is homogeneously continuous atx¯∈XonS, by Corollary2.1. Now apply

Theorem2.1.

The following example has been considered in [7] (Example 2.1). We shall prove that the origin is a regular point of the involved mappingf once again by Theorem2.1.

Example2.1 Consider the mappingf :R→Rdefined as

f(x) :=

⎧⎨

⎩

|x|(_π² –xsin(¹_x)), x= 0;

0, x= 0.

We haveS={±1}and thereforef is homogeneously continuous at 0 onS. One may easily verify that

f_l(0)(±1) =lim inf

t↓0

| ±t(_π² – (±t)sin(_±t¹))|

t = 2

π.

It follows thatinf_s∈Sf_l(0)(s) =_π² > 0. Hence, if 0 <κ<_π², then there existsδ> 0 such that

|x– 0|=|x| ≤1

κf(x) –f(0)=1 κf(x),

for all|x|<δ, by Theorem2.1. Hence, 0 is a strongly regular point off. Sincef is contin- uous, thus the subsetf^–1(0) is closed and therefore the distance functiondist(·,f^–1(0)) is Lipschitz around 0 (see [8, p. 11]). Hence, 0 is a regular point off.

3 Hoffman’s estimate for the distance to the set of solutions to a system of linear inequalities

Theorem 3.1(Hoffman, 1952, [9,10]) Let x^∗_i,i= 1, 2, . . . ,k be a finite family of linear forms onX^∗.Set

C≤:=

x∈Xsuch that x^∗_i,x

≤0,i= 1, 2, . . . ,k

. (3.1)

Then,there existsκ> 0such that dist(x,C≤)≤κ

(x)

+, (3.2)

where(x) :=max{x^∗_i,x,i= 1, 2, . . . ,k}and[(x)]+:=max((x), 0).

Hoffman’s result is considered as the starting point of the theory of error bounds, theory that has been extended over the years to different contexts. We refer to [3,11–13] and the references therein for the discussion of the fundamental role played by Hoffman bounds and more generally by error bounds in mathematical programming. As described, for ex- ample, in [14], they are used, for instance, in convergence properties of algorithms, in sen- sitivity analysis, in designing solution methods for nonconvex quadratic problems. When C:={x∈X:A(x) = 0,x^∗_i,x ≤0,i= 1, 2, . . . ,k}wherex^∗_i ∈X^∗,i= 1, 2, . . . ,k, are some given functionals andA:X→Yis a linear (continuous) mapping, we have the following result.

Theorem 3.2(Ioffe, 1979, [15]) There exists someκ> 0such that

dist(x,C)≤κ

A(x)+ k

i=1

x^∗_i,x

+

, (3.3)

for all x∈X.

Now letG:=KerA∩(_k

i=1Kerx^∗_i). Then, Theorem3.2yields the following result.

Corollary 3.1 There exists someκ> 0such that

dist(x,G)≤κ

A(x)+ k

i=1

x^∗_i,x

, (3.4)

for all x∈X.

In this section, we apply Theorem2.1and establish similar estimates. We prove that there existsκ¯> 0 such that

dist(x,G)≤ ¯κ

L(x)+ k

i=1

x^∗_i,x

+

,

for allx∈X, whereL:X→Xis a linear mapping withKerL=G. Our results also allow us to evaluate the constantκ¯. The details are as follows.

Proposition 3.1 Let A:X→Ybe a linear mapping and x^∗_i ∈X^∗,i= 1, 2, . . . ,k be given.

Suppose that L:X→Xis a linear mapping such thatKerL=G.Then

dist(x,G)≤ 1 γ

L(x)+ k

i=1

x^∗_i,x

+

, (3.5)

whereγ is a positive real number given by

γ :=inf

L(ν)+ n

i=1

x^∗_i,ν

+:ν∈X,dist(ν,G) = 1

. (3.6)

Proof Let us consider the finite-dimensional quotient spaceM:=_G^X, and denote by [x] the equivalence class containing x inM, that is, [x] :=x+G. We note[x]:=inf{x+y:y∈ G}. Denote byS_Mthe unit sphere ofM(i.e., the elements ofMof norm one). Obviously,

SM=

x∈X:inf

x+y:y∈G

= 1

=

x∈X:dist(x,G}= 1

. (3.7)

Consider the continuous linear mappingL:M→Xdefined asL([x]) :=L(x) for all [x]∈ M. Also for each 1≤i≤kdefine[xi]^∗, [x]:=x^∗_i,xfor all [x]∈M. Obviously, each [xi]^∗ belongs toG^⊥(the orthogonal complement ofG), and hence belongs to the dual ofM (which is isometrically isomorphic toG^⊥[16]). Set

C:=

[x]∈M:L [x]

= 0,

[x_i]^∗, [x]

≤0,i= 1, 2, . . . ,k .

We haveKerL=KerL=Gand thereforeC={[0]}. Now define the mappingf :M→R as

f [x]

:=L [x]+

k i=1

[xi]^∗, [x]

+.

We show that the conditions of Theorem2.1forf at [¯x] = [0] are all satisfied. For all [ν]∈ SM, one has

t↓0,μ→νlim

f([tμ]) –f([0])

t =L

[ν]+ k

i=1

[xi]^∗, [ν]

+.

Hence,f is homogeneously continuous at [0] onSM, by Corollary2.1. We also have

f_l [0]

[ν]

=L [ν]+

k i=1

[xi]^∗, [ν]

+=L(ν)+ n

i=1

x^∗_i,ν

+.

The continuity off implies that the mappingf|S_M (the restriction off toSM) attains its minimum at some [ν0]∈S_M. Then, [ν0] /∈C(note thatC={[0]}) and therefore

L [ν0]+

k i=1

[xi]^∗, [ν0]

+> 0.

It follows thatinf_[ν]∈SMf_l([0])([ν]) > 0. Using (3.7), we obtain

[ν]∈Sinf_Mf_l [0]

[ν]

=inf

L(ν)+ n

i=1

x^∗_i,ν

+:ν∈X,dist(ν,G) = 1

=γ.

Thusγ > 0. Now let 0 <κ<γ. Theorem2.1implies that there exists someδ> 0 such that [x] – [0]=[x]≤1

κ L

[x]+ k

i=1

[x_i]^∗, [x]

+

,

for all [x]∈BM([0],δ). Sincef is sublinear, [x]≤1

κ L

[x]+ k

i=1

[xi]^∗, [x]

+

,

for all [x]∈M. It follows that

dist(x,G)≤1 κ

L(x)+ k

i=1

x^∗_i,x

+

. (3.8)

For allx∈X. Lettingκ→γ in (3.8), we obtain the desired result.

Remark3.1 The existence of the linear mappingL:X→Xdiscussed in Proposition3.1is straightforward. Indeed,Gis a closed subspace ofXandXis separable, thus there exists a (continuous) linear mappingL:X→XwithKerL=G(see [17]). Of course, one can easily defineLdirectly (without using [17]). To see this, suppose thatdimX=nand let {e1, . . . ,ej}be a linearly independent basis for the vector spaceG. By linear algebra, we can extend{e1, . . . ,ej}to get a linearly independent basis forX(sinceGis a subspace ofX).

Let us denote this basis by{e1, . . . ,ej,ej+1, . . . ,en}. Now for everyx:=x1e1+· · ·+xnen∈X, define the mappingL:X→Xas

L(x) := (0, 0, . . . , 0

j

,xj+1, . . . ,xn

n–j

).

One can easily check thatLis well-defined, linear, andKerL=G. Corollary 3.2 Let A:X→Xbe a linear mapping.Then

dist(x,KerA)≤ A(x)

inf{A(ν):ν∈X,dist(ν,KerA) = 1}, for all x∈X.

Proof LetX=Y, andx^∗_i ≡0 for all 1≤i≤k. Then,G=KerA. LettingL:=Ain Proposi-

tion3.1yields the result.

Corollary 3.3 Let A:X→Yand L:X→Xbe linear mappings withKerL=Gand x^∗_i ∈ X^∗,i= 1, 2, . . . ,k,be some given functionals.Then

dist(x,G)≤ L(x)

inf{L(ν)+_n

i=1[x^∗_i,ν]+:ν∈X,dist(ν,G) = 1}, (3.9) for every x∈C≤(see(3.1)above).

Proof Proposition3.1implies that

dist(x,G)≤ 1 γ

L(x)+ k

i=1

x^∗_i,x

+

,

where

γ :=inf

L(ν)+ n

i=1

x^∗_i,ν

+:ν∈X,dist(ν,G) = 1

.

Now letx∈C≤. Thusx^∗_i,x ≤0 for every 1≤i≤k. Hence [x^∗_i,x]+= 0 for every 1≤i≤ k. Then, the above inequality yields

dist(x,G)≤ L(x)

inf{L(ν)+_n

i=1[x^∗_i,ν]+:ν∈X,dist(ν,G) = 1},

for everyx∈C≤. This completes the proof.

Finally, let us make a comparison between the two estimations (3.5) in Proposition3.1 and (3.4) in Corollary3.1described above. First, note that an application of Corollary3.1 (or a direct application of Theorem3.2) withL(described in Proposition3.1) in place of Aand no inequalities immediately produces the following estimate:

dist(x,G)≤κ₀L(x), (3.10)

for allx∈X, whereκ₀is a constant. On the other hand, doing the same replacements in Proposition3.1(i.e., applying Proposition3.1withLin place ofAandLin place of itself without the inequalities) yields

dist(x,G)≤ 1

γ₀L(x), (3.11)

where

γ0:=infL(ν):ν∈X,dist(ν,G) = 1

> 0.

The question is: which of the above estimates (3.10) and (3.11) is better? To answer this question, we need to know the relationship between the coefficientsκ0andγ0stated above.

As long as the value of the constantκ₀in (3.10) is not known, we can’t say which of the estimates (3.10) and (3.11) produces a better result. We can just say that the estimate (3.11)

technicallyis better, since it also allows us to estimate the unknown constantκ₀in (3.10).

Indeed,κ₀≤_γ¹₀.

Another question which may arise is: with the simple estimate (3.11) in hand, what is the necessity of using the estimate (3.5) in Proposition3.1(regarding the inequalities)? To an- swer this question, let’s take a closer look at the estimate (3.5). Indeed, by Proposition3.1, we have

dist(x,G)≤ 1 γ

L(x)+ k

i=1

x^∗_i,x

+

, (3.12)

where γ :=inf

L(ν)+ n

i=1

x^∗_i,ν

+:ν∈X,dist(ν,G) = 1

> 0.

We observe that, on the one hand,L(x) ≤ L(x)+k

i=1[x^∗_i,x]₊and, on the other hand,

1

γ ≤_γ¹₀. As a result, we cannot generally compare the right-hand sides of the estimates (3.11) and (3.12) to determine which is better. Corollary 3.3says that when x∈C≤, it would be better to use (3.12).

Acknowledgements

The authors would like to thank the anonymous referees whose meticulous reading helped us improve the presentation.

Funding

This research benefited from the support of the FMJH Program PGMO and from the support of EDF.

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

Author details

1Department of Mathematics, University of Isfahan, Isfahan, Iran.²XLIM UMR-CNRS 7252, Université de Limoges, 123, Avenue Albert Thomas, Limoges, 87060, France. ³Centre for Informatics and Applied Optimisation, School of Engineering, IT and Physical Sciences, Federation University, Ballarat, Victoria, 3350, Australia.

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Received: 14 April 2021 Accepted: 22 August 2021

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