Journal of Inequalities in Pure and
Applied Mathematics
EPI-DIFFERENTIABILITY AND OPTIMALITY CONDITIONS FOR
AN EXTREMAL PROBLEM UNDER INCLUSION CONSTRAINTS
volume 4, issue 2, article 41,
2003.
T. AMAHROQ, N. GADHI AND PR H. RIAHI
Université Cadi Ayyad,
Faculté des Sciences et Techniques,
B.P. 549, Marrakech, Maroc.
E-Mail: amahroq@fstg-marrakech.ac.ma
Université Cadi Ayyad,
Faculté des Sciences Samlalia,
B.P 2390 Marrakech,
Morocco.
E-Mail: n.gadhi@ucam.ac.ma
E-Mail: riahi@ucam.ac.ma
Received 4 August, 2002;
accepted 12 March, 2003.
Communicated by: A.M. Rubinov
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
086-02
Quit
Abstract
In this paper, we establish first-order optimality conditions for the problem of
minimizing a function f on the solution set of an inclusion 0 ∈ F (x) where f
and the support function of a set-valued mapping F are epi-differentiable at x.
2000 Mathematics Subject Classification: Primary 58C05; Secondary 58C20.
Key words: Optimality Conditions, Epi-convergence, Epi-differentiability, Support
function.
Contents
1
2
3
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Epi-differentiability of the Support Function of a Set-valued
Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
1.
Introduction
It is well known that epi-convergence of functions is coming to the fore as
the correct concept for many situations in optimization. The strong feature of
epi-convergence is that it corresponds to a geometric concept of approximation
much like the one used in classical differential analysis (see [2]). Derivatives
defined in terms of it therefore have a certain “robustness” that can be advantageous. Our principal objective is to give necessary and sufficient optimality
conditions of type Ferma for the optimization problem
(P )
Maximize f (x) subject to 0 ∈ F (x)
where f is a function from a reflexive Banach space X into R ∪ {+∞} and F
is a set valued map defined from X into another reflexive Banach space Y.
The organization of the paper is as follows. Section 2 contains basic definitions and preliminaries that are widely used in the sequel. In Section 3 we study
the epi-differentiability of the support function of F defined by CF (y ∗ , x) :=
inf y∈F (x) hy ∗ , yi for every y ∗ ∈ Y ∗ . Section 4 is devoted to the optimality conditions and also for an application in mathematical programming problems.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
2.
Preliminaries
Let F be a mapping defined in X with nonempty, closed and convex values in
Y, and let X ∗ be the topological dual of X.
In all the sequel, we denote the domain of F and its graph by, respectively,
Dom(F ) := {x ∈ X : F (x) 6= ∅} ,
Gr(F ) := {(x, y) ∈ X × Y : y ∈ F (x)} .
Let us recall some definitions.
Definition 2.1. A set-valued mapping F is said to be locally Lipschitz at x if
there exists α > 0 and r > 0 such that
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
F (x) ⊂ F (x0 ) + α kx − x0 k BY
for all x and x0 in x + rBX , where BX indicates the unit ball centered at the
origin in space X.
Let A be an arbitrary nonempty subset of X. The notions of contingent cone
and tangent cone to A at a point x ∈ A will be used frequently in this paper.
The contingent cone to A at x is
Title Page
Contents
JJ
J
II
I
Go Back
K (S, x) := {v ∈ X : ∃ (tn ) & 0, ∃vn → v : x + tn vn ∈ S, ∀n} .
The tangent cone to A at x is
k(A, x) := v ∈ X : ∀tn → 0+ , ∃vn → v
Close
Quit
x + tn vn ∈ A ∀n .
Page 4 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Definition 2.2. A set-valued map F is said to be proto-differentiable at (x, y) ∈
Gr(F ) if the contingent cone K(Gr(F ), (x, y)) coincides with the tangent cone
k(Gr(F ), (x, y)).
The proto-derivative is thus the set-valued map denoted by DFx,y, the graph
of which is the common set
Gr(DFx,y ) = K(GrF, (x, y)) = k(GrF, (x, y)).
For more details, see [1] and [9].
Lemma 2.1. Let F be a Lipschitz set valued map at x and y ∈ F (x), one has
i) lim sup Dt Fx,y (v) = lim sup Dt Fx,y (v) ,
(t,v)→(0+ ,v)
ii)
iii)
t→0+
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
lim inf Dt Fx,y (v) = lim sup Dt Fx,y (v) ,
(t,v)→(0+ ,v)
lim+
(t,v)→(0 ,v)
t→0+
Dt Fx,y (v) = lim sup Dt Fx,y (v) ,
t→0+
with Dt Fx,y (v) = t−1 (F (x + tv) − y).
Proof. i) The inclusion ” ⊃ ” is trivial. Let us prove the opposite inclusion. Consider any w ∈ lim sup Dt Fx,y (v) , there exists (tn , vn , wn ) →
Title Page
Contents
JJ
J
II
I
Go Back
(t,v)→(0+ ,v)
(0+ , v, w) such that y +tn wn ∈ F (x + tn vn ) . As F is Lipschitz at x, there
exists n0 ∈ N such that for any n ≥ n0 one has
Close
F (x + tn vn ) ⊂ F (x + tn v) + αtn kvn − vk BY .
Page 5 of 26
Quit
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Since y + tn wn ∈ F (x + tn vn ) , there exist (w
en ) ⊂ Y and (bn ) ⊂ BY
such that for any n ≥ n0 we get y + tn wn = y + tn w
en + αtn kvn − vk bn
and y + tn w
en ∈ F (x + tn v) . This implies that (tn , w
en ) → (0+ , w) and
w
en ∈ Dtn Fx,y (v) . Hence w ∈ lim sup Dt Fx,y (v) .
t→0+
Dt Fx,y (v) .
ii) It suffices to prove that lim sup Dt Fx,y (v) ⊂ lim inf
+
(t,v)→(0+ ,v)
t→0
Let w ∈ lim sup Dt Fx,y (v) and let tn & 0+ . Then there exists wn → w
(t,v)→(0+ ,v)
such that y + tn wn ∈ F (x + tn v) . Considering a sequence vn → v; and
using the Lipschitz property of F at x, there exist n0 ∈ N, (w
en ) ⊂ Y
and (bn ) ⊂ BY such that y + tn wn = y + tn w
en + αtn kvn − vk bn and
y + tn w
en ∈ F (x + tn vn ) for all n ≥ n0 . Then w
en ∈ lim sup Dtn Fx,y (vn )
t→0+
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
with w
en → w.
Title Page
iii) It is a direct consequence of i) and ii).
Contents
Proposition 2.2. Let F be a set valued map from X into Y and (x, y) ∈ Gr(F ).
If F is Lipschitz at x then F is proto-differentiable at (x, y) with a protoderivative DFx,y (v) if and only if for every v ∈ X
JJ
J
II
I
Go Back
DFx,y (v) = lim sup Dt Fx,y (v) exists.
t→0+
Proof. i) Fix (v, w) ∈ K(Gr F, (x, y)). There exists (tn , vn , wn ) → (0+ , v, w)
such that wn ∈ Dtn Fx,y (vn ) ; that is w ∈ lim inf
Dt Fx,y (e
v ). Using
+
Close
Quit
Page 6 of 26
(t,e
v )→(0 ,v)
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Lemma 2.1 one has w ∈
lim inf Dt Fx,y (e
v ). Consequently, for any
(t,e
v )→(0+ ,v)
(tn , ven ) → (0+ , v) there exists w
en → w such that (x, y) + tn (e
vn , w
en ) ∈
Gr(F ). This implies that (v, w) ∈ k(Gr F, (x, y)), and hence the protodifferentiability of F at (x, y).
ii) Fix w ∈ lim inf
Dt Fx,y (v) and let tn & 0. From Lemma 2.1, one has
+
t→0
w ∈
lim sup Dt Fx,y (v 0 ) and consequently (v, w) ∈ K(Gr F, (x, y)).
(t,v 0 )→(0+ ,v)
Thus by proto-differentiability (v, w) ∈ k(Gr F, (x, y)). Then there exists
(vn , wn ) → (v, w) such that y + tn wn ∈ F (x + tn vn ). Using the Lipschitz
property of F at x, there exists w
en → w such that w
en ∈ Dtn Fx,y (v) .
Hence w ∈ lim sup Dt Fx,y (v); and the proof is finished.
t→0+
In order to define the epi-derivative, as introduced by Rockafellar [8], let us
recall the notion of epi-convergence and some of its main properties; for more
details see [8].
A sequence of functions ϕn from X into R ∪ {+∞} is said to be τ -epiconverging for a topology τ from X, and we denote by τ -elm ϕn its τ -epi-limit,
if the two following conditions hold

τ
ϕ(x) ≤ lim sup ϕn (xn ),
 ∀ xn −→ x
τ
 ∃ xn −→ x
n→+∞
ϕ(x) ≥ lim sup ϕn (xn ).
n→+∞
A family of functions (ϕt )t>0 is said to be epi-converging to ϕ when t & 0, if
for every sequence tn & 0, the sequence of functions (ϕtn ) epi-converges to ϕ.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
When s designs the strong topology of X and w its weak sequential topology,
a sequence of functions (ϕn ) is said to be Mosco-epi-converging to ϕ if (ϕn )
s-epi-converges and w-epi-converges to ϕ, that is
w
∀vn → v ϕ (v) ≤ lim inf ϕn (vn ) ,
∃vn → v ϕ (v) ≥ lim sup ϕn (vn )
Finally, recall that a sequence of subsets (Cn ) τ -converges in the sense of
Kuratawski-Painlevé to another subset C if the indicator functions δCn of Cn
τ -epi-converge to δC . Note, see [2], that if X is reflexive and (Cn ), C is a sequence of closed convex subsets, then (Cn ) Mosco-converges to C if and only
if d(x, Cn ) → d(x, C) for all x ∈ X.
Let f be a function defined from X into R ∪ {+∞} , finite at a point x. The
function f is (Mosco-) epi-differentiable at x if the difference quotient functions
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
(∆t f )x (·) := t−1 (f (x + t·) − f (x)) ; t > 0,
Contents
0
0
have the property that the (Mosco-) epi-limit function fx exists and fx (0) >
0
−∞. Note that the epigraph of fx is the proto-derivative of the epigraph of f at
(x, f (x)).
By ∂e f (x), the epi-gradient of f at x, we denote the set of all vectors x∗ ∈
0
∗
X satisfying fx (v) ≥ hx∗ , vi for all v ∈ X.
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
3.
Epi-differentiability of the Support Function of
a Set-valued Mapping
In the remainder of this paper we assume that X is reflexive and we denote by
δA∗ (·) the support function of a subset A of X
δA∗ (x∗ ) := sup hx∗ , xi
for every x∗ ∈ X ∗ .
x∈A
It is easy to see that for two subsets A and B of X
∗
δA+B
(·) = δA∗ (·) + δB∗ (·),
A ⊂ B =⇒ δA∗ (·) ≤ δB∗ (·),
and if A and B are closed convex then the last implication is an equivalence.
For the following,
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
• F will be a set-valued mapping from X into Y with a closed convex setvalues.
• NF (x) (y) designs the normal cone to F (x) at y ∈ F (x), i.e.
NF (x) (y) := {y ∗ ∈ Y ∗ : hy ∗ , y − yi ≤ 0 for all y ∈ F (x)}
= y ∗ ∈ Y ∗ : hy ∗ , yi = δF∗ (x) (y ∗ ) .
n
o
• We denote by YF∗ := y ∗ ∈ Y ∗ : δF∗ (x) (y ∗ ) < +∞ the barrier cone of F.
It is easy to prove that when F is locally Lipschitz, see [5], the set YF∗ does
not depend on x.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Definition 3.1. A function f : X → R ∪ {+∞} is said to be (Mosco-) epiregular at x if the (Mosco-) epi-derivative of f exists at x and coincides with
the directional derivative of f at x.
Lemma 3.1. Let (x, y) ∈ Gr F. Suppose that F is Lipschitz at x, then the
∗
function ψt (v) := δD
(y ∗ ) is equi-locally Lipschitz.
t Fx,y (v)
Proof. Fix v ∈ X. As F is Lipschitz at x, there exist α > 0 and r > 0 such that
for any t ∈ ]0, r] and any v, v 0 ∈ v + rBX
F (x + tv) ⊂ F (x + tv 0 ) + αt kv − v 0 k BY .
Consequently (Dt F )x,y (v) ⊂ (Dt F )x,y (v 0 ) + α kv − v 0 k BY . Hence
|ψt (v) − ψt (v 0 )| ≤ α ky ∗ k kv − v 0 k
for any t ∈ ]0, r] and any v, v 0 ∈ v + rBX .
The proof is thus complete.
Proposition 3.2. Let (x, y) ∈ Gr F. Suppose that Y is reflexive and that for
every sequence tn & 0
(3.1)
d (·, Dtn Fx,y (v)) −→ d (·, DFx,y (v)) .
Then:
i) the set-valued map NDtn Fx,y (v) graph-converges to NDFx,y (v) ,
ii) there exists w ∈ DFx,y (v) , y ∗ ∈ NDFx,y (v) (w) , wn ∈ Dtn Fx,y (v) and
∗
yn∗ ∈ NDtn Fx,y (v) (wn ) such that (wn , yn∗ ) −→ (w, y ∗ ) and δD
(yn∗ ) −→
tn Fx,y (v)
∗
δDF
(y ∗ ).
x,y (v)
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Motivated by the article of Demyanov, Lemaréchal and Zowe [4], where the
authors approximate F under the assumption that
∗
−1
∗
∗
∗
∗
∗
(y
)
:=
lim
t
δ
(y
)
−
δ
(y
)
δDF
F (x+th)
F (x)
x,y (·)
t&0,h→d
exists (as an element of R) for every y ∗ ∈ Rp , we give in Theorem 3.3 sufficient
conditions insuring the existence of this derivative.
Theorem 3.3. Let (x, y) ∈ Gr F. Suppose that Y is reflexive, F is directionally
Lipschitz at x and that NF (x) (y) 6= ∅. Suppose also that the condition (3.1)
∗
is satisfied for each v and that the function δD
(·) is equi-Lipschitz. i.e.
tn Fx,y (v)
∃β ≥ 0 such that
T. Amahroq, N. Gadhi and
Pr H. Riahi
δC∗ n (v) (yn∗ ) ≥ δC∗ n (v) (y ∗ ) − β kyn∗ − y ∗ k .
Then for all y ∗ ∈ NF (x) (y), the function f (x) := δF∗ (x) (y ∗ ) = sup hy ∗ , yi is
y∈F (x)
∗
epi-regular at x, with δDF
(y ∗ ) as its epi-derivative.
x,y (·)
Proof. Let tn & 0. By definition of f one has
∗
∗
t−1
n [f (x + tn v) − f (x)] = δDtn Fx,y (v) (y ).
Setting Cn (v) := Dtn Fx,y (v) , C (v) := DFx,y (v) and Ψn (v) := δC∗ n (v) (y ∗ ),
then condition (3.1) permits us to conclude that δCn (v) Mosco-converges to
δC(v) . Using Attouch’s theorem [2], we conclude that δC∗ n (v) Mosco-converges
to δC∗ . Hence
w∗
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 26
∗
(y ∗ ) ≤ lim inf δC∗ n (v) (yn∗ ),
a) for any yn∗ → y ∗ one has δC(v)
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
s
∗
b) there exists yn∗ → y ∗ such that δC(v)
(y ∗ ) ≥ lim sup δC∗ n (v) (yn∗ ).
Let us prove that
∗
i) for any vn → v one has Ψ (v) := δC(v)
(y ∗ ) ≤ lim inf Ψn (vn ),
ii) there exists vn → v such that Ψ (v) ≥ lim sup Ψn (vn ).
i) Let vn → v. From Lemma 3.1, there exists n0 ∈ N such that for any
n ≥ n0
Ψn (vn ) ≥ Ψn (v) − α ky ∗ k kvn − vk .
Letting n → +∞ we get lim inf Ψn (vn ) ≥ lim inf Ψn (v). Finally, using
a), we have lim inf Ψn (vn ) ≥ Ψ (v) . The result is thus proved.
s
ii) Considering b), there exists yn∗ → y ∗ such that Ψ (v) ≥ lim sup δC∗ n (v) (yn∗ ).
Since δC∗ n (v) (·) is equi-Lipschitz, there exists β ≥ 0 such that
δC∗ n (v) (yn∗ ) ≥ δC∗ n (v) (y ∗ ) − β kyn∗ − y ∗ k .
Hence lim sup Ψn (vn ) ≤ Ψ(v). Moreover, since F is directionally Lipschitz at x on has
∗
lim δ(D
(y ∗ ) =
t F )x,y (v)
t→0+
lim
(t,v 0 )→(0+ ,v)
∗
∗
δ(D
0 (y )
t F )x,y (v )
Thus
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
∗
∗
∗
∗
δ(DF
)x,y (v) (y ) = lim+ δ(Dt F )x,y (v) (y )
=
t→0
fx0 (v)
= lim+ t−1 [f (x + tv) − f (x)].
Quit
Page 12 of 26
t→0
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
The proof of the theorem is complete.
Remark 3.1. When, instead of (3.1), we assume the Mosco-Proto-differentiability
of F at (x, y) ∈ Gr F ; we can justify the Mosco-epi-regularity of f at x.
w
Indeed, suppose that there exists vn → v such that Dtn Fx,y (vn ) does not
w
Mosco-converge to DFx,y (v) . Consequently, there exists zn → z such that
w
zn ∈ t−1
/ DFx,y (v) . Thus (vn , zn ) → (v, z) ,
n (F (x + tn vn ) − y) and z ∈
(vn , zn ) ∈ t−1
/ DFx,y (v) ; which contradicts F Moscon (Gr F − (x, y)) and z ∈
Proto-differentiable at x.
Theorem 3.4. Let f : X → R ∪ {+∞} be a function epi-differentiable at x
and let g : X → R ∪ {+∞} be a function epi-regular at x. Then f + g is
epi-differentiable at x.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Proof. Setting
an (v) := t−1 [f (x + tn v) − f (x)], bn (v) := t−1 [g(x + tn v) − g(x)]
Title Page
Contents
and
−1
cn (v) := t [(f + g)(x + tn v) − (f + g)(x)].
JJ
J
w
i) Let vn → v. Since f and g are epi-differentiable at x, we have
lim inf cn (vn ) ≥ lim inf an (vn ) + lim inf bn (vn ) ≥ a(v) + b(v).
n→∞
n→∞
Go Back
n→∞
s
s
ii) Since f and g are epi-differentiable at x, there exist vn1 → v and vn2 → v
such that
b(v) ≥ lim inf bn (vn2 ) and a(v) ≥ lim inf an (vn1 ).
n→∞
II
I
Close
Quit
Page 13 of 26
n→∞
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Consequently
lim inf cn (vn1 ) ≤ lim inf an (vn1 ) + lim inf bn (vn1 ) ≤ a(v) + lim inf bn vn1 .
n→∞
n→∞
n→∞
n→∞
Using the epi-regularity of g at x,
lim inf bn (vn1 ) = lim inf bn (vn2 ) = lim inf bn (v).
n→∞
n→∞
n→∞
Then
lim inf cn (vn1 ) ≤ a(v) + b(v).
n→∞
The conclusion is thus immediate, that is, f + g is epi-differentiable at x.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
4.
Optimality Conditions
Fix y ∗ ∈ Y ∗ and let CF (y ∗ , x) := inf hy ∗ , yi ,
y∈F (x)
NF (x) (y) = {y ∗ ∈ Y ∗ : hy ∗ , yi = CF (y ∗ , x)}
and
YF∗ = {y ∗ ∈ Y ∗ : CF (y ∗ , x) > −∞} .
Let I(x) := {y ∗ ∈ YF∗ such that CF (y ∗ , x) = dF (x)} . In particular, if x ∈ C
we have I(x) := {y ∗ ∈ YF∗ such that CF (y ∗ , x) = 0} . Consider, when CF (y ∗ , ·)
is epi-differentiable at x
DF (x) := {d ∈ X : ∀y ∗ ∈ I(x), ∀x∗ ∈ ∂e CF (y ∗ , ·)(x)
and
n
HF (x) := d ∈ X : ∀y ∗ ∈ I(x)
hx∗ , di ≤ 0}
o
0
CF (y ∗ , ·)x̄ (d) ≤ 0 .
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
Definition 4.1. x is said to be regular if there exist two reals λ, r > 0 such that
−
d(0, F (x)) ≥ λ d(x, F (0))
for all x ∈ x + rBX .
JJ
J
II
I
Go Back
For every x ∈ X, let dF (x) := d(0, F (x)).
Proposition 4.1. The following inclusions are always true
K(F − (0), x) ⊂ HF (x) ⊂ DF (x).
Close
Quit
Page 15 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Proof. Let d ∈ K(F − (0), x). There exists (tn , dn ) → (0+ , d) such that x +
0
tn dn ∈ F − (0). Consider y ∗ ∈ I(x) and x∗ ∈ X ∗ such that CF (y ∗ , ·)x̄ (v) ≥
0
hx∗ , vi for all v ∈ X. All we have to show is that CF (y ∗ , ·)x̄ (d) ≤ 0. Indeed,
since 0 ∈ F (x + tn dn ) we have CF (y ∗ , x + tn dn ) ≤ 0. But CF (y ∗ , x) = 0,
0
∗
∗
consequently CF (y ∗ , ·)x̄ (d) ≤ lim inf t−1
n [CF (y , x + tn dn ) − CF (y , x)] ≤ 0.
n→∞
0
Then CF (y ∗ , ·)x̄ (d) ≤ 0 and hx∗ , di ≤ 0. The proof is thus complete.
The following lemma will play a very crucial role in the remainder of the
paper.
Lemma 4.2. Let ∂dF be the Clarke subdifferential. We assume F to have the
following properties.
i. F is Lipschitz at x,
ii. ∂e CF (y ∗ , ·) (x) is upper semicontinuous (at (y ∗ , x)) when X ∗ , Y ∗ are endowed with the weak-star topology and X with the strong topology, that
w∗
w∗
is, if x∗n ∈ ∂e CF (yn∗ , ·) (xn ) where x∗n → x∗ in X ∗ , yn∗ → y ∗ in Y ∗ and
xn → x in X, then x∗ ∈ ∂e CF (y ∗ , ·) (x) .
Then
∂dF (x) ⊂ co {∂e CF (y ∗ , x) : y ∗ ∈ I(x) ∩ B∗Y } .
Proof. To prove the lemma, we need the following result of Thibault [11].
Let h : X → R be a locally Lipschitzian function, H a subset of X such that
X/H is Haar-nul set and at every x ∈ H, the function h is Gateaux differentiable and has Gateaux differential ∇h (x) . Then we have
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 26
1. h (x, v) = max {hx∗ , vi : x∗ ∈ LH (h, x)} for every v ∈ X,
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
2. ∂h (x) = {coLH (h, x)} ,
where LH (h, x) = lim sup ∇h (xn ) : xn ∈ H, xn → x and the “limit ”
n→∞
*
of {∇h (xn )} is in the weak topology.
Now, from Christensen’s Theorem [3] applied to the locally Lipschitzian
function dF it follows that there exists a subset M ⊂ X such that dF is Gateaux
differentiable on M and X/M is a Haar-nul set.
For every xn ∈ M, yn∗ ∈ I (xn ) ∩ B∗Y , v ∈ X we have
dF (xn + εv) − dF (xn )
h∇dF (xn ) , vi = lim
ε→0
ε
∗
CF (yn , xn + εv) − CF (yn∗ , xn )
≤ lim
ε→0
ε
CF (yn∗ , xn + εk v) − CF (yn∗ , xn )
≤ lim sup
.
εk
k→∞
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Since CF is epi-derivable, there exists vk → v such that
CF0 (yn∗ , ·)xn (v) ≥ lim sup
k→∞
CF (yn∗ , xn
+ εk v k ) −
εk
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
Contents
CF (yn∗ , xn )
.
On the other hand, CF is α kyn∗ k-Lipschitz. Consequently
CF (yn∗ , xn + εk vk ) − CF (yn∗ , xn ) + αεk kyn∗ k kvk − vk
h∇dF (xn ) , vi ≤ lim sup
εk
k→∞
∗
αεk kyn k kvk − vk
≤ CF0 (yn∗ , ·)xn (v) + lim sup
εk
k→∞
0
∗
≤ CF (yn , ·)xn (v) .
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Hence
∇dF (xn ) ∈ ∂e CF (yn∗ , ·)xn .
(4.1)
It is easily seen that the set-valued map x 7→ I (x)∩B∗Y is upper semi-continuous.
Moreover, since ∂e CF (y ∗ , ·) (x) is upper semicontinuous, the set valued-map
x 7→ G (x) defined by
G (x) := {x∗ : x∗ ∈ ∂e CF (y ∗ , x) and y ∗ ∈ I (x) ∩ B∗Y }
is upper semi-continuous as well.
From (4.1), we have
LM (dF , x) ⊂ G (x) .
The lemma now follows from the second part of the mentioned result of Thibault
and the compactness of the set G (x) (in the weak topology).
In the following, we give necessary optimality conditions of type Fermat.
Throughout the reminder of the paper, we assume that the function f is epidifferentiable, the support function CF (z ∗ , ·) of the set valued-map F is epidifferentiable and that ∂e CF (y ∗ , ·) (x) is upper semicontinuous.
Theorem 4.3. Consider problem (P ). Suppose also that x is regular.
If x̄ is a solution of (P ) then
(4.2)
for all v ∈ DF (x).
0
fx (v) ≤ 0
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
We begin by giving an important lemma which we shall use later on.
Lemma 4.4. If x is regular then
HF (x) = DF (x) = K(F − (0), x̄).
Proof. All we have to show is that DF (x) ⊂ K(F − (0), x̄). Let d ∈ DF (x) with
d 6= 0. Without loss of generality we can assume that kdk = 1. As x is regular,
we can fix r > 0 and λ > 0 such that
dF (x) ≥ λd(x, F − (0))
(4.3)
for all x ∈ x + rBX .
Let tn & 0. Since ∂dF (·) is upper semicontinuous, there exists rn > 0 such
that
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
∂dF (x) ⊂ ∂dF (x) + tn λB
(4.4)
for all x ∈ x + rn BX .
Setting µn := min (r, rn , tn ) , and by Lebourg’s Mean value Theorem [7], we
can assert that for any n ∈ N there exist xn ∈ [x, x + µn d] and x∗n ∈ ∂dF (xn )
such that
dF (x + µn d) − dF (x) = hx∗n , µn di
≤ sup hyn∗ , µn di = µn
∗ ∈∂d (x )
yn
n
F
sup
∗ ∈∂d (x )
yn
n
F
Observe that xn ∈ [x, x + µn d] ⊂ x + rn B. From (4.4) we get
sup
∗ ∈∂d (x )
yn
n
F
hyn∗ , di ≤
sup
∗ ∈∂d (x)
yn
F
hyn∗ , di + tn λ.
hyn∗ , di .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
By virtue of Lemma 4.2,
∂dF (x) ⊂ co {∂e CF (y ∗ , x) : y ∗ ∈ I(x) ∩ B∗Y }
and consequently dF (x + µn d) ≤ µn tn λ. Taking account of (4.3), we deduce
d(x + µn d, F − (0)) ≤ µn tn < 2µn tn .
This implies the existence of a sequence (vn ) such that for n large enough,
x + µn vn ∈ F − (0) and kx + µn d − (x + µn vn )k = µn kvn − dk < 2µn tn .
Hence d ∈ K(F − (0), x). This ends the proof of the lemma.
−
Proof of Theorem 4.3. Let v ∈ DF (x). By virtue of Lemma 4.4, v ∈ K(F (0), x̄),
hence there exist (tn ) & 0 and vn → v such that x + tn vn ∈ F − (0). Since x is
a solution of (P ), there exists n0 ∈ N such that for any n ≥ n0 f (x + tn vn ) ≤
f (x). From the epi-differentiability of f at x, one gets
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
0
fx (v) ≤ lim inf (∆tn f )x (vn ) ≤ 0.
n→∞
The proof is thus complete.
Contents
Remark 4.1. Without the regularity of x, the proof of Theorem 4.3 permit to get
0
0
fx (v) ≤ 0 ≤ CF (y ∗ , ·)x̄ (v)
Title Page
for every v ∈ K(F − (0), x̄) and every y ∗ ∈ I(x).
JJ
J
II
I
By virtue of the complexity of F − (0), we were forced to adopt DF (x) instead of
K(F − (0), x̄) in Theorem 4.3.
Go Back
Theorem 4.5. Consider problem (P ) and let us assume that dim (X) < +∞.
Suppose that x is regular and that f is Lipschitz at x.
0
Then x̄ is a solution of (P ) whenever fx (v) < 0 for any v ∈ DF (x)\ {0} .
Quit
Close
Page 20 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Proof. Assume the contrary, that is, that the statement of Theorem 4.5 is not
true. Then there exists a sequence (xn ) ⊂ F − (0) satisfying xn → x and
xn − x
. Since dim (X)
f (xn ) > f (x) ∀n. Let tn := kxn − xk and vn :=
tn
is finite, there exists v ∈ X with kvk = 1 and a subsequence noted another
time (vn ) such that vn → v. From the epi-differentiability of f at x, there exists
ven → v such that
0
fx (v) ≥ lim sup
n→+∞
f (x + tn ven ) − f (x)
.
tn
Setting an := t−1
en ) − f (x)] , bn := t−1
n [f (x + tn vn ) − f (x)] and
n [f (x + tn v
cn := bn − an . We have that f is Lipschitz at x, lim sup cn = 0 and that bn ≥ 0;
n→+∞
0
consequently fx (v) ≥ lim sup an ≥ lim sup bn − lim sup cn ≥ 0. This conflicts
n→+∞
n→+∞
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
n→+∞
with v ∈ DF (x)\ {0} and the theorem follows.
Title Page
0
fh,x
f is said to be hypo-differentiable at x, with
as its hypo-derivative, if
0
−f is epi-differentiable at x. In this case fh,x = −(−f )0x .
Theorem 4.6. Consider problem (P ) and let us assume that dim (X) < +∞.
Suppose that x is regular and that f is hypo-differentiable at x.
0
Then x̄ is a solution of (P ) whenever fh,x
(v) < 0 for any v ∈ DF (x)\ {0} .
Proof. The argument is slightly similar to that used above, but we give it for the
convenience of the reader. Assume the contrary, that is, that the statement of
Theorem 4.5 is not true. Then there exists a sequence (xn ) ⊂ F − (0) satisfying
xn − x
xn → x and f (xn ) > f (x) ∀n. Let tn := kxn − xk and vn :=
. Since
tn
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
dim (X) is finite, there exists v ∈ X with kvk = 1 and a subsequence noted
another time (vn ) such that vn → v. From the hypo–differentiability of f at x,
0
(−f ) (x + tn vn ) − (−f ) (x)
≤ 0.
n→+∞
tn
(−f )x (v) ≤ lim inf
0
Hence fh,x
(v) ≥ 0. This is a contradiction since v ∈ DF (x)\ {0} .
As an application for the above results, we are concerned with the mathematical programming problem
(P ∗ )
max f (x)
subject to gi (x) ≤ 0 and hj (x) = 0
for all i ∈ {1, 2, . . . , m} and all j ∈ {1, 2, . . . , k} .
Let C := {x : gi (x) ≤ 0, hj (x) = 0 for all i, j} . Let g (x) = (g1 (x) , g2 (x) ,
. . . , gm (x)) and h (x) = (h1 (x) , h2 (x) , . . . , hk (x)). The problem (P ∗ ) reduces to (P ), where the set-valued mapping F : X ⇒ Y = Rm × Rk is defined
by
F (x) := (g (x) , h (x)) + Rm
+ × {0Rk } .
k
∗
∗
Obviously, in that case YF∗ = Rm
+ × R and for any y = (λ, µ) ∈ YF we have
∗
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
CF (y , x) = hλ, g (x)i + hµ, h (x)i .
Close
It can be verified that CF (y ∗ , x) = 0 if and only if λi gi (x) = 0 for all i ∈
{1, 2, . . . , m}.
Quit
Page 22 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
Then I(x) =
sequently
(
HF (x) :=
k
(λ, µ) ∈ Rm
+ × R : λi gi (x) = 0 ∀i = 1, . . . , m , and con-
v ∈ X : ∀ (λ, µ) ∈ I(x)
m
X
0
(v) +
λi gix
i=1
k
X
)
µj h0jx (v) ≤ 0 .
j=1
We deduce from Theorem 4.3 and Theorem 4.5 the following optimality conditions for problem (P ∗ ).
Theorem 4.7. Let x be a solution of (P ∗ ). Suppose that the functions f and hj
are epi-differentiable at x, the functions gi are epi-regular at x and there exist
r > 0 and λ > 0 such that
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
d(g (x) , Rm
− ) ≤ λ d(x, C)
T. Amahroq, N. Gadhi and
Pr H. Riahi
for every x ∈ x + rBX .
k
Then for any v ∈ X such that ∀ (λ, µ) ∈ Rm
+ × R satisfying λi gi (x) = 0
Pm
P
0
and i=1 λi gix
(v) + kj=1 µj h0jx (v) ≤ 0 we have
Title Page
(4.5)
Contents
0
fx (v) ≤ 0.
Remark 4.2. The condition (4.5) ensures the regularity of x.
Theorem 4.8. Suppose that dim (X) < ∞ and that f is Lipschitz at x.
x will be a solution of (P ∗ ) if for any v ∈ X\ {0} such that ∀ (λ, µ) ∈
k
Rm
+ × R , verifying λi gi (x) = 0 and
m
X
i=1
0
λi gix
(v) +
k
X
JJ
J
II
I
Go Back
Close
Quit
µj h0jx (v) ≤ 0,
Page 23 of 26
j=1
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
we have
0
fx (v) < 0.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
References
[1] T. AMAHROQ AND L. THIBAULT, On proto-differentiability and strict
proto-differentiability of multifunctions of feasible points in perturbed optimization problems, Num. Func. Anal. Optim., 16 (1995), 1293–1307.
[2] H. ATTOUCH, Variational Convergence for Functions and Operators,
Pitman Advanced Publishing Program (1984).
[3] J.P.R. CHRISTENSEN, Topology and Borel Structure, Math studies,
Notes de Mathematica.
[4] V.F. DEMYANOV, C. LEMARÉCHAL AND J. ZOWE, Approximation to
a set- valued mapping, Appl. Math. Optim., 14 (1986), 203–214.
[5] P.H. DIEN, Locally Lipschitz set-valued maps and generalized extremal
problems with inclusion constraints, Acta Math. Viet., 8 (1983), 109–122.
[6] J.B. HIRIART-URRUTY, J.J. STRODIOT AND H. NGUYEN, Generalized Heslfan matrix and second order conditions for a problem with C 1,1
data, Appl. Math. Optim., 11 (1984), 43–56.
[7] G. LEBOURG, Valeur moyenne pour gradient généralisée, CR Acad. Sci.
Paris, 218 (1975), 795–797.
[8] R.T. ROCKAFELLAR, First and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc., 307 (1988), 75–108.
[9] R.T. ROCKAFELLAR, Proto-differentiability of set-valued mapping and
its applications in optimization, Ann. Inst. H. Pointcaré: Analyse non
linéaire, 6 (1989), 449-482.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003
[10] R.T. ROCKAFELLAR AND J.B. WETS, Variational Analysis.
[11] L. THIBAULT, On generalized differentials and subdifferentials of Lipschitz vector valued functions, Nonlinear Anal. Th. Meth. Appl., 6 (1982),
1037–1053.
Epi-differentiability and
Optimality Conditions for an
Extremal Problem Under
Inclusion Constraints
T. Amahroq, N. Gadhi and
Pr H. Riahi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 26
J. Ineq. Pure and Appl. Math. 4(2) Art. 41, 2003