R E S E A R C H
Open Access
Second-order optimality conditions for
nonlinear programs and mathematical
programs
Ikram Daidai
**Correspondence:
[email protected] Department of Mathematics, Faculté des Sciences et Techniques, University Cadi Ayyad, B.P. 549, Marrakech, Morocco
Abstract
It is well known that second-order information is a basic tool notably in optimality conditions and numerical algorithms. In this work, we present a generalization of optimality conditions to strongly convex functions of order
γ
with the help of first-and second-order approximations derived from (Optimization 40(3):229-246, 2011) and we study their characterization. Further, we give an example of such a function that arises quite naturally in nonlinear analysis and optimization. An extension of Newton’s method is also given and proved to solve Euler equation with second-order approximation data.Keywords: strong convexity of order
γ
; second-order approximation;C1,1functions; Newton’s method1 Introduction
The concept of approximations of mappings was introduced by Thibault []. Sweetser [] considered approximations by subsets of the space of continuous linear mapsL(X,Y), whereXandY are Banach spaces, and Ioffe [] by the so-called fans. This approach was revised by Jourani and Thibault []. Another approach belongs to Allali and Amahroq []. Following the same ideas, Amahroq and Gadhi [, ] have established optimality condi-tions to some optimization problems under set-valued mapping constraints.
In this work, we explore the notion of strongly convex functions of orderγ; see, for in-stance, [–] and references therein. Letf be a mapping from a Banach spaceXintoR, and letC⊂Xbe a closed convex set. It is well known that the notion of strong convex-ity plays a central role. On the one hand, it ensures the existence and uniqueness of the optimal solution for the problem
(P) min
x∈Cf(x).
On the other hand, iff is twice differentiable, then the strong convexity off implies that its Hessian matrix is nonsingular, which is an important tool in numerical algorithms. Here we adopt the definition of a second-order approximation [] to detect some equivalent properties of strongly convex functions of orderγ and to characterize the latter.
more, for aC,functionf on a finite-dimensional setting, we show some simple facts. We
also provide an extension of Newton’s method to solve an Euler equation with second-order approximation data.
The rest of the paper is written as follows. Section contains basic definitions and pre-liminary results. Section is devoted to mains results. In Section , we point out an ex-tension of Newton’s method and prove its local convergence.
2 Preliminaries
LetXandY be two Banach spaces. We denote byL(X,Y) the set of all continuous linear mappings fromXintoY, byB(X×X,Y) the set of all continuous bilinear mappings from X×XintoY, and byBY the closed unit ball ofYcentered at the origin.
Throughout this paper,X∗andY∗denote the continuous duals ofXandY, respectively, and we write·,·for the canonical bilinear forms with respect to the dualitiesX∗,Xand
Y∗,Y.
Definition ([]) Letf be a mapping fromXintoY,x¯∈X. A set of mappingsAf(x¯)⊂ L(X,Y) is said to be a first-order approximation off atx¯if there existδ> and a function r:X→Rsatisfyinglimx→¯xr(x) = such that
f(x) –f(x¯)∈Af(x¯)(x–x¯) +x–x¯r(x)BY ()
for allx∈ ¯x+δBX.
It is easy to check that Definition is equivalent to the following: for allε> , there exists
δ> such that
f(x) –f(x¯)∈Af(x¯)(x–x¯) +εx–x¯BY ()
for allx∈ ¯x+δBX.
Remark IfAf(x¯) is a first-order approximation off atx¯, then () means that for any
x∈ ¯x+δBX, there existA(x)∈Af(x¯) andb∈BY such that
f(x) –f(x¯) =A(x)(x–x¯) +εx–x¯b.
Hence, for anyx∈B(x¯,δ) andA(x)∈Af(x¯),
f(x) –f(x¯) –A(x)(x–x¯)≤εx–x¯. ()
IfAf(x¯) is norm-bounded (resp. compact), then it is called a bounded (resp. compact)
first-order approximation. Recall thatAf(x¯) is a singleton if and only iff is Fréchet differentiable
atx¯.
Proposition ([]) Let f :Rp→Rbe a locally Lipschitz function atx¯.Then the Clarke
subdifferential of f atx¯,
∂cf(x¯) :=co
lim∇f(xn) :xn∈dom∇f and xn→ ¯x
, ()
is a first-order approximation of f atx¯.
In [], it is also shown that whenf is a continuous function, it admits as an approximation the symmetric subdifferential defined and studied in [].
The next proposition shows that Proposition holds also whenf is a vector-valued tion. Let us first recall the definition of the generalized Jacobian for a vector-valued func-tion (see [, ] for more details) and the definifunc-tion of upper semicontinuity.
Definition The generalized Jacobian of a functiong:Rp→Rqatx¯, denoted∂ cg(x¯), is
the convex hull of all matricesMof the form
M= lim
n→+∞Jg(xn),
wherexn→ ¯x,gis differentiable atxnfor alln, andJgdenotes theq×pusual Jacobian
matrix of partial derivatives.
Definition A set-valued mappingF:Rp⇒Rqis said to be upper semicontinuous at a
pointx¯∈Rpif, for everyε> , there existsδ> such that F(x)⊂F(x¯) +εB
for everyx∈Rpsuch thatx–x¯<δ.
Proposition Let g:Rp→Rqbe a locally Lipschitz function atx¯.Then the generalized
Jacobian∂cg(x¯)of g atx is a first-order approximation of g at¯ x¯.
Proof Since the set-valued mapping∂cg(·) is upper semicontinuous, for allε> , there
existsr> such that
∂cg(x)⊂∂cg(x¯) +εBL(Rp,Rq) for allx∈ ¯x+rBRp.
We may assume that g is Lipschitzian in x¯+rBRp. Letx∈ ¯x+rBRp. We apply [],
Prop. .., to derive that there exitsc∈]x,x¯[ such that
g(x) –g(x¯)∈∂cg(c)(x–x¯)⊂∂cg(x¯)(x–x¯) +εBL(Rp,Rq)(x–x¯).
Since
BL(Rp,Rq)(x–x¯)⊂ x–x¯BRq,
we have
g(x) –g(x¯)∈∂cg(x¯)(x–x¯) +εx–x¯BRq,
Recall that a mappingf :X→Y is said to be C, at x¯ if it is Fréchet differentiable in
neighborhood ofx¯and if its Fréchet derivative∇f(·) is Lipschitz atx¯.
Letx¯∈Rp, and letf :Rp→Rbe aC,function atx¯. The generalized Hessian matrix of
f atx¯was introduced and studied by Hiriart-Urruty et al. [] is the compact nonempty convex set
∂Hf(x¯) :=colim∇f(xn) : (xn)∈dom∇f and xn→ ¯x
, ()
wheredom∇f is the effective domain of∇f(·).
Corollary Letx¯∈Rp,and f:Rp→Rbe a C,function atx¯.Then,∇f admits∂ Hf(x¯)as
a first-order approximation atx¯.
Definition ([]) We say thatf :X→Y admits a second-order approximation atx¯ if there exit two setsAf(x¯)⊂L(X,Y) andBf(x¯)⊂B(X×X,Y) such that
(i) Af(x¯)is a first-order approximation off atx¯;
(ii) For allε> , there existsδ> such that
f(x) –f(x¯)∈Af(x¯)(x–x¯) +Bf(x¯)(x–x¯)(x–x¯) +εx–x¯BY
for allx∈ ¯x+δBX.
In this case the pair (Af(x¯),Bf(x¯)) is called a second-order approximation off atx¯. It is
called a compact second-order approximation ifAf(x¯) andBf(x¯) are compacts.
EveryCmappingf :X→Yatx¯admits (∇f(x¯),∇f(x¯)) as a second-order approxima-tion, where∇f(x¯) and∇f(x¯) are, respectively, the first- and second-order Fréchet
deriva-tives off atx¯.
Proposition ([]) Let f :Rp→Rbe a C,function atx¯.Then f admits(∇f(x¯), ∂
Hf(x¯))
as a second-order approximation atx¯.
Proposition Let f :X→Y be a Fréchet-differentiable mapping.If(∇f(x¯),Bf(x¯))is a
bounded second-order approximation of f atx¯.Then∇f(·)is stable atx¯,that is,there exist c,r> such that
∇f(x) –∇f(x¯)≤cx–x¯ ()
for all x∈ ¯x+rBX.
To derive some results forγ-strong convex functions, the following notions are needed. Definition ([]) Letγ > . We say that a mapf:X→R∪ {+∞}isγ-strongly convex if there existc≥ andg: [, ]→R+satisfying
g() =g() = and lim θ→
g(θ)
and such that
fθx+ ( –θ)y≤θf(x) + ( –θ)f(y) –cg(θ)x–yγ ()
for allθ∈[, ] andx,y∈X.
Of course, whenc= ,fis called a convex function. Otherwise,fis saidγ-strongly convex. This class has been introduced by Polyak [] whenγ = andg(θ) =θ( –θ) and studied by many authors. Recently, a characterization ofγ-strongly convex functions has been shown in []. For example, iff isCandγ≥, then () is equivalent to
∇f(x),y–x≤f(y) –f(x) – c
γy–x
γ
, ∀x,y∈X. ()
Letf:X→R∪ {+∞}andx¯∈domf :={x∈X,f(x) < +∞}(the effective domain off). The Fenchel-subdifferential off atx¯is the set
∂Fenf(x¯) =
x∗∈X∗:x∗,y–x¯≤f(y) –f(x¯),∀y∈X. ()
Letγ > andc> . The (γ,c)-subdifferential off atx¯is the set
∂(γ,c)f(x¯) =
x∗∈X∗:x∗,y–x¯≤f(y) –f(x¯) –c¯x–yγ,∀y∈X. ()
For more details on (γ,c)-subdifferential, see []. Note that ifx∈/ domf, then∂(γ,c)f(x¯) =
∂Fenf(x¯) =∅. Clearly, we have∂(γ,c)f(x¯)⊂∂Fenf(x¯). Note that the Fenchel-subdifferential
defined by () coincides with the Clarke subdifferential offatx¯if the functionfis convex. We also need to recall the following definitions.
Definition ([]) We say that a mapf :X→R∪ {+∞}is -paraconvex if there exists
c> such that
fθx+ ( –θ)y≤θf(x) + ( –θ)f(y) +cmin(θ, –θ)x–y ()
for allθ∈[, ] andx,y∈X.
It has been proved in [] that iff is aCmapping, then () is equivalent to
∇f(x),y–x≤f(y) –f(x) +cy–x, ∀x,y∈X. ()
3 Main results
In this section, we obtain the main results of the paper related to strongly convex functions of orderγ defined by ()-(). We begin by showing some interesting facts of functions that admit a first-order approximation.
For any subsetAofX∗, we define the support function ofAas
It is well known that, for any convex functionf:X→R∪{+∞}, the ‘right-hand’ directional derivative atxindomf (the domain off ) exists and, for eachh∈X, is
d+f(x)(h) = lim
t→+
f(x+th) –f(x)
t .
Theorem Letx¯∈X.If f :X→R∪{+∞}is convex and continuous atx and if¯ Af(x¯)⊂X∗
is a convex w(X∗,X)-closed approximation of f atx¯,then
∂(γ,c)f(x¯)⊂Af(x¯).
Proof By the definition ofAf(x¯), there existδ> andr:X→Rwithlimx→¯xr(x) = such
that, for allx∈ ¯x+δBX,t∈],δ[, andh∈X, there existA∈Af(x¯) andb∈[–, ] satisfying
f(x¯+th) –f(x¯)
t –hr(x¯+th)b=A,h ≤s
Af(x¯);h
.
By lettingt→+the directional derivative off atx¯satisfies
d+f(x¯)(h)≤sAf(x¯);h
, ∀h∈X. ()
Using [], Prop. ., we get
s∂Fenf(x¯);h
≤sAf(x¯);h
.
Since∂(γ,c)f(x¯)⊂∂Fenf(x¯), we deduce that
s∂(γ,c)f(x¯);h
≤sAf(x¯);h
.
Hence we conclude that∂(γ,c)f(x¯)⊂Af(x¯).
Proposition Let f :X→R∪ {+∞}be aγ-strongly convex function.Assume thatAf(x¯)
is a compact approximation atx¯.ThenAf(x¯)∩∂(γ,c)f(x¯)=∅.
Proof Letd∈Xbe fixed and definexn:=x¯+nd. Using Definition , we get, fornlarge
enough,An∈Af(x¯) andbn∈[–, ] such that
nAn,d=f x¯+ nd
–f(x¯) –
ndr(xn)bn.
Byγ-strong convexity we obtain
nAn,d ≤ n
f(x¯+d) –f(x¯)–cg n
dγ–
ndr(xn)bn.
By the compactness ofAf(x¯), extracting a subsequence if necessary, we may assume that
there existsA∈Af(x¯) such thatAn,d → A,d; and hence we obtain
Assume thatA∈Af(x¯)∩∂(γ,c)f(x¯). By the separation theorem there existsh∈Xwithh=
such that
min
A∈Af(x)¯
A,h> sup
x∗∈∂(γ,c)f(x)¯
x∗,h.
Lett> sufficiently small, so that
min
A∈Af(x)¯
A,h>f(x¯+th) –f(x¯)
t ,
in contradiction with relation () by takingd=th.
Following a result by Rademacher, which states that a locally Lipschitzian function be-tween finite-dimensional spaces is differentiable (Lebesgue) almost everywhere, we can prove the following result.
Proposition Letγ ≥,x¯∈Rp,and let f :Rp→Rbe continuous atx¯.Assume that f is
aγ-strongly convex function.Then∂cf(x¯) =∂(γ,c)f(x¯).
Proof Obviously, we have∂(γ,c)f(x¯)⊂∂cf(x¯). Now let A∈∂cf(x¯). For alln, there exists
xn∈dom∇f such thatxn→ ¯xand∇f(xn)→A. Sincef isγ-strongly convex and Fréchet
differentiable atxnfor alln∈N, it follows by () that
∇f(xn),y–xn
≤f(y) –f(xn) –cy–xnγ, ∀y∈Rp,∀n∈N.
Lettingn→+∞, we get
A,y–x¯ ≤f(y) –f(x¯) –cy–x¯γ, ∀y∈Rp,
which means that∂cf(x¯)⊂∂(γ,c)f(x¯).
Corollary Letγ ≥,x¯∈Rp,and let f :Rp→Rbe continuous atx¯.Assume that f is a
γ-strongly convex function.Then,for allε> ,there exists r> such that
f(x) –f(x¯)∈∂(γ,c)f(x¯)(x–x¯) +εx–x¯BR ()
for all x∈ ¯x+rBRp,which means that∂(γ,c)f(x¯)is a first-order approximation of f atx¯.
Proof It is clear that∂cf(x¯) is a first-order approximation of atx¯. We end the proof by
Propositions and .
The converse of Proposition holds if () is valid for anyA∈Af(x) andx∈X.
Proof Definexθ:=θu+ ( –θ)vforθ∈[, ] andu,v∈X. Let us takeA∈Af(xθ). Then
A,u–xθ ≤f(u) –f(xθ) –cu–xθγ.
Multiplying this inequality byθ, we obtain
a θ( –θ)A,u–v ≤θf(u) –θf(xθ) –c( –θ)γθu–vγ. In a similar way, since
A,v–xθ ≤f(v) –f(xθ) –cv–xθγ,
we get
a –θ( –θ)A,u–v ≤( –θ)f(v) – ( –θ)f(xθ) –c( –θ)θγu–vγ. We deduce by addition of (a) and (a) that
f(xθ)≤θf(u) + ( –θ)f(v) –cg(θ)u–vγ for allu,v∈X,
whereg(θ) = ( –θ)θγ+ ( –θ)γθ, so thatf isγ-strongly convex. The next results are devoted to presenting some useful properties of the generalized Hessian matrix for aC,function in the finite-dimensional setting and a characterization
ofγ-strongly convex functions with the help of a second-order approximation.
Proposition Letx¯∈X,and let f :X→R∪ {+∞}be convex and Fréchet differentiable atx¯.Suppose that f admits(∇f(x¯),Bf(x¯))as a second-order approximation atx and that¯ Bf(x¯)is compact.Then there exists B∈Bf(x¯)such that
sup
B∈Bf(x)¯
Bd,d ≥, ∀d∈X. ()
If f is-strongly convex,then we obtain
sup
B∈Bf(x)¯
Bd,d ≥cd, ∀d∈X, ()
for some c> .
Proof We prove only the case wheref is convex. In a similar way, we can prove the other case. Letd∈Xandε> be fixed. We get fornlarge enoughBn∈Bf(x¯) andbn∈[–, ]
such that
f x¯+ nd
–f(x¯) = n
∇f(x¯),d+
nBnd,d+ε
nd
b n.
Sincef is convex, we obtain
By the compactness ofBf(x¯), extracting a subsequence if necessary, we may assume that
there exitsB∈Bf(x¯) such thatBnconverges toB; therefore
Bd,d ≥,
and hence
sup
B∈Bf(x)¯
Bd,d ≥, ∀d∈X.
WhenXis a finite-dimensional space, we get the following essential result.
Proposition Let f:Rp→Rbe a C,function atx¯.Assume that f isγ-strongly convex.
Then,for any B∈∂Hf(x¯),we have the following inequality:
Bd,d ≥cdγ, ∀d∈Rp, ()
for some c> .
Proof It is clear that (∇f(x¯),∂
Hf(x¯)) is a second-order approximation off atx¯. Now let
B∈∂Hf(x¯), so that there exists a sequence (xn)∈dom∇fsuch thatxn→ ¯xand∇f(xn)→
B. Sincef isγ-strongly convex, there existsc> such that
∇f(x n)d,d
≥cdγ, ∀d∈Rp,∀n∈N.
Lettingn→+∞, we have
Bd,d ≥cdγ, ∀d∈Rp.
The preceding result shows thatγ-strongly convex functions enjoy a very desirable prop-erty for generalized Hessian matrices. In fact, in this case, any matrixB∈∂Hf(x¯) is invert-ible. The next result proves the converse of Proposition . Let us first recall the following characterization of l.s.c.γ-strongly convex functions.
Theorem (Amahroq et al. []) Let f:X→R∪ {+∞}be a proper and l.s.c.function. Then f isγ-strongly convex iff∂cf isγ-strongly monotone,that is,there exists a positive
real number c such that,for all x,y∈X,x∗∈∂cf(x),and y∗∈∂cf(y),we have
x∗–y∗,x–y≥cx–yγ.
We are now in position to state our main second result.
Theorem Let f :Rp→Rbe a C,function.Assume that∂
Hf(·)satisfies relation()at
any x∈Rp.Then f isγ-strongly convex.
Proof Lett∈[, ] andu,v∈Rp. Defineϕ:R→Ras
so thatϕ(t) :=∇f(u+t(v–u)),v–u. By the Lebourg mean value theorem [] there existst∈], [ such that
ϕ() –ϕ()∈∂cϕ(t).
By using calculus rules it follows that
ϕ() –ϕ()∈∂cϕ(t)⊂∂Hf
u+t(v–u)
(v–u)(v–u).
Hence, there existsBt∈∂Hf(u+t(v–u)) such that∇f(v) –∇f(u),v–u=Bt(v–u),v–
u. The result follows from Theorem .
Hiriart-Urruty et al. [] have presented many examples of C, functions. The next
proposition shows another example of aC,function.
Theorem Let f :H→Rbe continuous on a Hilbert space H.Suppose that f is convex (or-strongly convex)and that–f is-paraconvex.Then f is Fréchet differentiable on H, and for some c> ,we have that
∇f(x) –∇f(y)≤cx–y for all x,y∈H. ()
Proof Letx∈X. Clearly,f is locally Lipschitzian atx. Now letx∗ andx∗ be arbitrary
elements of∂cf(x) and∂c(–f)(x), respectively. By [], Thm. ., there existsc> such
that∂c(–f)(x) =∂(,c)(–f)(x), and for anyy∈Hand positive realθ, we have
(a) θx∗,y≤–f(x+θy) +f(x) +cθy
and
a θx∗,y≤f(x+θy) –f(x).
Adding (a) and (a), we get
θx∗+x∗,y≤cθy, and hence
x∗+x∗,y≤cθy.
Lettingθ →, we havex∗+x∗,y ≤, so thatx∗= –x∗. Sincex∗ andx∗are arbitrary in
∂cf(x) and∂c(–f)(x), it follows that∂cf(x) is single-valued. Put∂cf(x) ={p(x)}. Since
(a) and (a) hold for anyθ> andy∈H, we deduce that, forθ= ,
p(x),y
≤f(x+y) –f(x)
and
f(x+y) –f(x) –
p(x),y
Hence, for ally= , we obtain
|f(x+y) –f(x) –p(x),y|
y ≤cy. ()
Lettingy → in (), we conclude thatf is Fréchet differentiable atx. Now since –f
is -paraconvex andf is Fréchet differentiable, we may prove that there existsc> such that
–∇f(x),y–x≤–f(y) +f(x) +cx–y for allx,y∈H. ()
For everyz∈H, we have that
–f(z)≥–f(x) +∇f(x),x–∇f(x),z–cx–z.
Thus
–f(z)≥f∗∇f(x)–∇f(x),z–cx–z,
so that
f∗∇f(y)≥∇f(y),z–f(z),
f∗∇f(y)≥∇f(y),z+f∗∇f(x)–∇f(x),z–cx–z,
and hence
f∗∇f(y)–f∗∇f(x)–∇f(y) –∇f(x),x
≥∇f(y) –∇f(x),z–x–cx–z
≥sup
z∈H
∇f(y) –∇f(x),z–x–cx–z.
This means that, for allx,y∈H,
f∗∇f(y)–f∗∇f(x)–∇f(y) –∇f(x),x≥
c∇f(y) –∇f(x)
.
Changing the roles ofxandy, we obtain
f∗∇f(x)–f∗∇f(y)–∇f(x) –∇f(y),y≥
c∇f(x) –∇f(y)
.
So by addition we get
∇f(x) –∇f(y),x–y≥
c∇f(x) –∇f(y)
. ()
Consequently, by the Cauchy-Schwarz inequality we obtain
4 Newton’s method
The aim of this section is to solve the Euler equation
∇f(x) = ()
by Newton’s method. The classic assumption is thatf :Rp→RaC mapping and the
Hessian matrix∇f(x) off atxis nonsingular. Here we prove the convergence of a natural
extension of Newton’s method to solve () assuming that∇f(·) admitsβf(·) as a
first-order approximation. Clearly, iff :Rp→Ris aC,mapping, then using Corollary , we
obtain that∇f(·) admits∂Hf(·) as a first-order approximation.
This algorithm has been proposed by Cominetti et al. [] with C,data. Only some ideas were given, but it remains as an open question to state results on rate of conver-gence and local converconver-gence of that algorithm. In the sequel,f :Rp →Ris a Fréchet-differentiable mapping such that its Fréchet derivative admits a first-order approximation, andx¯is a solution of ().
Algorithm(M) Starting from an arbitrary pointx∈B(x¯,r), generate the sequences (xk)
and (hk) as follows:
(i) hk∈Rpis a solution of∈ ∇f(xk) +βf(x¯)(hk), and
(ii) xk+=xk+hk.
Theorem Let f :Rp→Rbe a Fréchet-differentiable function,andx be a solution of¯
().Letε,r,K> be such that∇f(·)admitsβf(x¯)as a first-order approximation atx¯
such that,for each x∈BRp(x¯,r),there exists an invertible element B(x)∈Bf(x)satisfying
B(x)– ≤K andξ:=εK< .Then the sequence(xk)generated by Algorithm(M)is well
defined for every x∈BRp(x¯,r)and converges linearly tox with rate¯ ξ.
Proof Since∇f(x¯) = , we have
xk+–x¯=B(xk)–
∇f(x¯) –∇f(xk) +B(xk)(xk–x¯)
.
We inductively obtain that
xk+–x¯ ≤K∇f(x¯) –∇f(xk) +B(xk)(xk–x¯).
Thus
xk+–x¯ ≤ξxk–x¯,
which means thatxk+∈BRp(x¯,r), and we havexk+–x¯ ≤ξkx–x¯. Therefore the
whole sequence (xk) is well defined and converges tox¯.
Now let us consider the following algorithm under less assumptions.
Algorithm(M) Starting from an arbitrary pointx∈Rp, generate the sequences (xk)
and (hk) as follows:
(i) hk∈Rpis a solution of∈ ∇f(xk) +βf(x)(hk), and
Theorem Let U be an open set ofRp,x
∈U,and f :Rp→Rbe a Fréchet-differentiable
function on U.Letε,r,K> be such that∇f(·)admitsβf(x)as a strict first-order
approx-imation at xsuch that,for each x∈BRp(x,r),there exists a right inverse of B(x)∈βf(x),
denoted byB˜(x),satisfying ˜B(x)(·) ≤K · andξ:=εK< .
If∇f(x) ≤K–( –ξ)r and∇f is continuous,then the sequence (xk)generated by
Algorithm (M)is well defined and converges to a solutionx of¯ ().Moreover,we have
xk–x¯ ≤rξkfor all k∈Nand¯x–x ≤ ∇f(x)K( –ξ)–<r.
Proof We prove by induction that xk ∈ x +rBRp, xk+ –xk ≤Kξk∇f(x), and
∇f(xk) ≤ξk∇f(x) for allk∈N. Fork= , these relations are obvious. Assuming
that they are valid fork<n, we get
xn–x ≤ n–
k=
xk+–xk ≤K∇f(x)
∞
k=
ξk
≤K∇f(x)( –ξ)–<r.
Thusxn∈x+rBRpand since∇f(xn–) +B(xn–)(xn–xn–) = , from Algorithm (M) we
have
∇f(xn)≤∇f(xn) –∇f(xn–) –B(xn–)(xn–xn–)≤εxn–xn– ≤ξn∇f(x)
and
xn+–xn ≤Kξn∇f(x).
Sinceξ< , the sequence (xn) is a Cauchy sequence and hence converges to somex¯∈Rp
withx–x¯<r. Since∇f is a continuous function, we get∇f(x¯) = .
5 Conclusions
In this paper, we investigate the concept of first- and second-order approximations to gen-eralize some results such as optimality conditions for a subclass of convex functions called strongly convex functions of orderγ. We also present an extension of Newton’s method to solve the Euler equation under weak assumptions.
Acknowledgements
The author wishes to express his heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The author read and approved the final manuscript.
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References
1. Allali, K, Amahroq, T: Second order approximations and primal and dual necessary optimality conditions. Optimization40(3), 229-246 (1997)
2. Thibault, L: Subdifferentials of compactly Lipschitzian vector-valued functions. Ann. Mat. Pura Appl. (4)125, 157-192 (1980)
3. Sweetser, TH: A minimal set-valued strong derivative for set-valued Lipschitz functions. J. Optim. Theory Appl.23, 539-562 (1977)
4. Ioffe, AD: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc.266, 1-56 (1981)
5. Jourani, A, Thibault, L: Approximations and metric regularity in mathematical programming in Banach space. Math. Oper. Res.18(2), 390-401 (1993)
6. Amahroq, T, Gadhi, N: On the regularity condition for vector programming problems. J. Glob. Optim.21(4), 435-443 (2001)
7. Amahroq, T, Gadhi, N: Second order optimality conditions for the extremal problem under inclusion constraints. J. Math. Anal. Appl.285(1), 74-85 (2003)
8. Amahroq, T, Daidai, I, Syam, A:γ-strongly convex functions,γ-strong monotonicity of their presubdifferential and γ-subdifferentiability, application to nonlinear PDE. J. Nonlinear Convex Anal. (2017, to appear)
9. Crouzeix, JP, Ferland, JA, Zalinescu, C:α-convex sets and strong quasiconvexity. SIAM J. Control Optim.22, 998-1022 (1997)
10. Lin, GH, Fukushima, M: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl.118, 67-80 (2003)
11. Polyak, BT: Introduction to Optimization. Optimization Software, New York (1987). Translated from the Russian, with a foreword by Dimitri P. Bertsekas
12. Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14, 877-898 (1976) 13. Vial, J-P: Strong convexity of sets and functions. J. Math. Econ.9(1-2), 187-205 (1982).
doi:10.1016/0304-4068(82)90026-X
14. Vial, J-P: Strong and weak convexity of sets and functions. Math. Oper. Res.8(2), 231-259 (1983) 15. Z˘alinescu, C: On uniformly convex functions. J. Math. Anal. Appl.95(2), 344-374 (1983)
16. Mordukhovich, BS, Shao, YH: On nonconvex subdifferential calculus in Banach spaces. J. Convex Anal.2(1-2), 211-227 (1995)
17. Clarke, FH: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 18. Clarke, FH: On the inverse function theorem. Pac. J. Math.64(1), 97-102 (1976)
19. Hiriart-Urruty, J-B, Strodiot, J-J, Nguyen, VH: Generalized Hessian matrix and second-order optimality conditions for problems withC1,1data. Appl. Math. Optim.11(1), 43-56 (1984)
20. Jourani, A: Subdifferentiability and subdifferential monotonicity ofγ-paraconvex functions. Control Cybern.25(4), 721-737 (1996)
21. Phelps, RR: Convex Functions, Monotone Operators and Differentiability Lecture Notes in Mathematics, vol. 1364. Springer, Berlin (1989)
22. Lebourg, G: Valeur moyenne pour gradient généralisé. C. R. Acad. Sci. Paris281, 795-797 (1975)
23. Cominetti, R, Correa, R: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim.
