A NNALES DE L’I. H. P., SECTION C
A. A. AUSLENDER
J. P. C ROUZEIX
Well behaved asymptotical convex functions
Annales de l’I. H. P., section C, tome S6 (1989), p. 101-121
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WELL BEHAVED ASYMPTOTICAL CONVEX FUNCTIONS
A.A. AUSLENDER &#x26; J.P. CROUZEIX
Département de Mathématiques Appliquées, Université Blaise Pascal, BP. 45,
6317D Aubière, France
In this paper,
ABSTRACT :
for CR N
functions
which
lim
satisfy
the
{xn}, {c n}
"For all sequences
have
introduce the class
we
n
Characterizations and
functions and
"Find
lim
a
f(y ) =
In mathematical
m =
{xn}}
is
unbounded,
convergence theorem
{x )}
[8J
converges to
we
shall prove that for
at
the
infinity
the
For
a
are
0
we
proper-closed
convex
large
by
{x n }
for the
class of
sequence
the
What
question
proximal method,
else
convex
is
are
a
set
we
always
one
bounded,
{x:f(x) ~ À}
is bounded.
This
optimal
{xn}}
algori thms,
algorithm
for all B the set
inf-compact?
instance,
such that :
the convergence of
optimal solution,
previous
c nn =
n-oo
given.
constructed
claims : "If the
an
and lim
n
(P), i.e.,
that the sequence
when f is not
sidered in literature.
then
{xn}}
inf-compact (i.e.,
ensures
3f(x )
optimization problem (P)
programming, f or proving
the function f is
E
the functions f
sequence
supposes that the sequences
which in general
then
are
{yn}} of
inf ( f (x) |x ~ RN)
minimizing
convex
E
applications
consider the
we
cnn
N
INTRODUCTION : In this paper all
I
of proper closed
following property.
such that
n
f(xn) = inf(f(x)jx
JF
or
that
is bounded),
happens
when
has not been
con-
the Rockafellar
of solutions of P is nonempty
+~".
have lim
functions which has
minimizing
sequence.
a
In
fact,
good behaviour
102
such
defining
For
decreasing
class, let
a
{f(x}
sequences
the iterative process is
For such
where
x
a
is
e
In
to m.
fact,
that
implies
0 it
as
soon
expected that if
{xn}}
convex
{(p2,p
n
n
function f
As
we
an
are
)} is unbounded
on RN
{cn}
has
measures
the
when f is inf-
functions and
that
one
n
one
orthant
have been considered.
can
speak
In
of
a
shall say that
good asymptotical
behaviour if :
definition,
by.
can
The most used
nonnegative
Rockafellar
3f(x n) (subdifferential of
E
c
!R
at
is closed
we
By
a
so
In
fact,
~.
~
on
case
given by
by
of functions in
in sciences.
-
optimality
f(x )
convex
shall denote this class
entropy defined
Other
the
particular
the class of functions f
important
practice
of
think
enough
to
application
is very
is small
paper
ourselves in the
In
tends to the null vector but the sequence
The class of these functions will be denoted
and
£
Consider for example the function f
"For all sequences (x },
n
general
can
one
case
to m.
the iterate is satisfied.
on
the unbounded case, there
behaviour of the function f.
closed-proper
converging
is bounded, the criteria
This is in
m.
(Vf(x)}
the sequence
case
asymptotical
m.
be
condition
some
the sequence
as
such that
does not tend to
to
necessary and sufficient condition for
converges to
f(xn)
sequences
a
be
can
Unfortunately, in
compact.
In this
Since
positive.
=
when
stopped
expected
are
rule in the dif ferentiable
stopping
is that Vf(x)
which
remark that numerical methods generate
us
E
f at
{f(xn;
f9]:
bad
a
x n) :
we
shall restrict in
f
such that
consider the concept of entropy whir
measure
of entropy is the "x
log x"
by
particular,
the
"Log
x" entropy defined
or
103
the
See for
remark
by
set
same
example,
only
that the first
In section II
will be
allow
we
to
us
study
on
the
II
we
classes J
and
and Lent.
:F1;
one
is
we
shall
only in
JF1.
equivalent characterizations
some
optimization
We shall restrict ourselves
idea initiated
proximal
this time
At
shall prove that these classes of functions
the convergence of classical
inner-product, ~.~
the usual
by Censor
is inf compact while the second
study the
shall
unbounded sequences.
based
one
In section III
given.
[4J
the recent paper
Moreau.
by
to
fundamental
two
the
Through
the associated Euclidian
methods which generate
algorithms
following (.,.)
denotes
norm .
CHARACTERIZATIONS
Let
be
3 f(x)
Proposition
we
2.1I
f
belongs
to
~
x
with
c
&#x3E;
0 :
iff for every sequence (x ,c ,£} such that
n
n
n
have
We have
Proof
{x )} is
f3] there
a
"
the E-subdifferential of f at
-
From
only
minimizing
exis t x
sequence.
and
1
From the
such that
n
n
(2. 1) and (2.2)
prove that if f E
to
we
obtain
E
c
n
lim c n
n-- 00
-
f
and if
theorem of
(2.I) is satisfied then
Br~ndstedt
and Rockafellar
3f(x n),
0 and since f
E j="
it follows:
104
since
But
It follows from
(2.!), (2.2)
characterizing
For
convex
that is to say, which
behaviour
on
the
and
functions which have
satisfy (I. I)
boundary
f’(x;d)
denotes
Set domf =
{x: f(x)
Lecrana 2. l
If
The
proof is
general
context.
Proof
more
briefly.
ri (dom f ) and 03BB
based
on
&#x3E; m
For the sake of
Let y be the
and there exists
a(x-y),
From what
f(y) -
we
À and h
asymptotical
at x
x
to
a &#x3E;
S~,(f)
E
3f(y),
then since
°
and
then
completeness
projection
we
of
x
shall
on
[2J
develop it
SÀ(f) :
II .
0 such that
k(X).
m.
be the relative interior of dom f.
iic e
a(x-y)
E
3f(y).
Then
3f(y).
then :
deduce that
&#x3E;
in direction d.
ideas used in Auslender and Crouzeix
£(x) - À - f(x)-f(y) $. (c,x-y)
Let h -
the
s tudy
{x : f(x) ~ À} for À
S.(f) =
ri(dom f)
d(x~(f))- ~
Clearly fey) - À
to
the directional derivative of f
x ~ S.(f).
Let
good asymptotical behaviour,
be the Euclidean distance from
and let
C
a
i t is useful
of the level set
For this purpose let
whe re
(2.3)
Now let y and h be such that
h ~ 0(a ~ m)
we
have
in
a
105
Taking
x
=
I Ix-ylI
t =
=
we
Theorem 2.2
then
we
then it follows that
+
y
Let f
a
have
x f
S,
(f) for
k(03BB) f(y; 2014-2014))
proper-closed
convex
function
t
&#x3E;
0 and since
finally
and
on RN
satisfying (1.2)
have the equivalences :
1)
2) r(X)
&#x3E;
0
iia
&#x3E;
3) k(À)
&#x3E;
0
iia
&#x3E; m.
1)
Proof
~
r(À) = 0, then
2).
from
m.
Suppose that
(2.4)
which contradicts that f E
and there exis ts
there exists sequences
{xn}}
some
and
À
{cn}}
&#x3E;
m
with
such that
cnE
3f(x )
Let X
E
==&#x3E;
3).
Th i s i s
3)
=~
1) .
Suppose that k(X)
a
(m,A).
sequence
n
tha t f rom
an
inmediate consequen ce f rom the
(xn ,cn}
Then from lemma
be such that
x
J.
2)
there exists
so
j=
that
such
Le t
f E
(2.9):
and X
0 for all À
&#x3E;
&#x3E;
m
2.!,k(A) =
such that
and
&#x3E; m
inequality
and that
f;
then
106
Since f is
convex
and
c
3f(x n)
e
n
which, combined with (2.11) yields
which is
not
What is
decreasing.
support function of
we
-~
&#x3E;
SÀ(f), À
&#x3E;
non
properties
of the
J
&#x3E;m.
hold.
sup{(x,x.) Ix
f) =
F(.,A) is closed
m,
the
are
Let
m.
F(x.,À) ~
ii) for all A
0
investigate
must
following properties obviously
Then the
i)
this
proving
converges to 0.
is that the functions r(.) and k(.)
surprising
more
For
{cn}}
since
possible
to
proper
6 dom f
convex
and
VÀ
positively
homogeneous.
iii) F(0,X) = 0
m.
iv) F(x*,.) is nondecreas ing
Furthermore, set 03C8x *(x,03BB)
Since f is
then ’
convex
v) F(x*, . ) is
For all À
KÀ is
&#x3E; m we
{x*: F(x*,À)
concave
tinuous
and
Then in the
&#x3E;
K
x
(m.-Kc).
function
m;
*
sup
Vx*.
KÀ
+°o).
we
then
and therefore
concave
the barrier
Then
K.
is
cone
convex.
of
S~(f)
i.e.,
’
Because the function
F(x*,.)
.
and the interior of its domain is
on
se ts
x* _is
by
f);
(x,x*) -
=
concave
denote
Vx*.
(m,+oo),
then
F(x*,.) is
con-
have
following
we
furthermore,
Let
us
denote
F(.,~*)
at
x*,
by
denote by K the
F is
a
f ini te
3 ~F(x*,A)
the
common
barrier
convex-concave
cone
of level
func t ion
the subdifferential of the
super-differential
of the
on
convex
concave
107
concave
function
F(x*,.)
at B and define
2.3
Propos i tion
a)
S(x*,A) = 3 x *F(x*,a)
for all x* E
b)
S(x*,A) is non-empty
for all x* E
c)
For A
&#x3E;
if
m
o.(x.ldom
F(x*,A)
&#x3E;
m.
&#x3E;
m.
f) then f(x) = À for all
x
E
S(x*,B).
Proof
Since F(.,~) is the support function of the closed
a)
set
convex
S~(f)
we
have :
+
b)
The subdifferential of
F(x.,À) =
~
S(x*,A).
function is non-empty
convex
a
x
the relative
on
interior of its domain.
c)
p 6
Suppose that x
E
S(x.,À)
and
À.
f(x)
Then
F(x.,1.1) = F(x*,~)
for all
[f(x) ~].
Since
F(x*, . )
is concave,
F(x’,À) =
F(x*,u) = 3*(x*
sup
)dom
f) .
D
1.1
Proposition
Proof
2.4
Assume
and
that x E
a) if F(x.,À)
f),
then
b) if F(x*,~) =
f),
then
Set
continuous
~(1J) = -F(x.,U},
on
(m,-K~).
then ~ is
Consider
4&#x3E;*(1J*) =
sup
~*,
&#x3E;
m.
3.F(x*,~) =
3~F(x*,~) = {A*/x*~
a
the
+
f(x) = À
proper
convex
J (0).
function which is
conjugate function
x*,y&#x3E; :t
À.af(x)}.
of ~ :
tj].
P.y
Then
and
Then
therefore, since f(x) = À and W(X) = - (x,x*), À. belongs
a) and b) follow
straightforwardly.
to
at(1) iff
Q
,
108
Conversely,
Proposition
have
we
2.5
If
3f(x) - {x./x
f (x)
S(x*,f(x)) =
E
x*
x,x*
Now le t
if fI E
optimality conditions
be such that
x E S(x*,f(x));
that is iff x*
=
then
before -I E
as
3f(x).
iff
Furthermore,
-)
E
D
the strong connection between
following function
k(À) = inf
k defined
{-L :
by convention 1 = +~
Proposition
(03BB)
sup
3f(x)
3 F(x*,A)
we
introduce
on
K,Ilx.’1 = I}}
u C
1-1
and
"
(2. 13)
if u " = 0.
2.5
VA
0
&#x3E;
m,
N
-
b)
k is
c)
1~(A) =
non
’properties
c)
p,
on
E
1J
are
a~F(x* , ~) ,
x*
1 },
E
direct consequences of the
of functions
monotonicity
(2.14)
and
concavity
F(x*,.).
E
Ü =
there is
(m,+")~
some
a~F(x~,a)~ .
x* E
ri (K)
such
W~ shall show that for
that ~
~ for all p E
any ~
such
3.F(x*,A).
"
c) will follow.
Let
we
inf sup
x*
p
x*~ K,
that (
Hence
decreasing
a) and b)
Proof
Let
have
we
aÀF(x.,f(x».
In view of
a)
and1 E
3f(x) ==&#x3E; x ~ S(x*,f(x)).
E
f*(x*) - (x,x*) =-f(x),
with
m,
Observe first that from sufficient
Proof
the
&#x3E;
riCK)
and
x; =
x’--
ll (x ~ - x*).
Then from
obtain
lim
F(x’,t)
for all
t
E
Corollary
7.5.11
[9]
109
Set
F(x’.t),
03B8(t) =
there exists
But then there exists
Take x*
Since 8 is
F(x*n,t).
a
and
concave
&#x3E; À such that
t
some
0(t)
n
such that
D
x*.
=
n
2.6
Propos i t ion
S03BB’(f
Let A and ~’ be such that
) C r I (dom f)
that k is
non
decreasing
k(~) ~
f’ (x; c)
Such
lc,
x* =
and
k(A’)
k(03BB) (03BB’)
a) Prove that
and
+00
then
(03BB)
so
t
A*
~
m
then
a
( 1 .2) is satisfied.
when
on
k(X).
Let
x
and
be such that
c
f(x) = X,
~
3f(x)
ri (dom f ) . Take
couple always exis ts since
F(x*,A) =
c
f) and
x
S(x*,X) .
6
Hence,
by proposition 2.4
sup -
e
:
=
~
v
Prove
Let any e
&#x3E;
that
:d 6
u
It follows from 2.6 and 2.13 that
b)
~f(x)] sup[(x*,d)
Sup[ : x* ~
k(~’).
0, by relation (2. 14)
t~(A) ~
Of course,
there is
we
some
assume
that
k(~’)
ri(K) such
x* E
+00.
that ~ ~ x" ~ ~ ~1
and
Next, by propositions 2.3 and 2.4 there
are
some
x
6
S(x*,~’)
and
uo
such that
Now, le t
Then
x
be the Eucl idean
a. and
k(~’ )
and
~
pro jection
there exist
c
6
of
x
+
e.
on
the closed
&#x3E;
i
convex
0 such that
se t
&#x3E;
0
110
Since f is
Utc.
x -
convex
we
have
then
Letting
0
e2014
2. 7
Proposition
set C
of Rn
Proof
deduce
we
Let f be
then
In this
k(À) =
case
k(~) ~ k(~’ ) .
that
dif ferentiable
a
k(À)
for all À
d
inf ( f’ (x;
j
On the other
hand, for al l x*
is dif ferentiable
at
À and
Then it follows
S(x*,A).
2.8
Proposition
&#x3E;
)d
convex
function
that
open
convex
I
E
~f(x)~-1
k(A) =
Let À and ~’ be such that
F(x’,.)
t
VÀF(x*,À) =
easily
an
m.
ri (K) such
E
on
where
x
is any
point
in
k(À).
m
À
À’
~
and
f)
then
so
that
Proof
x*
r
is
a~
Let
= d ~d~
non
.
decreasing
E ~ J1, 7l’ C,
Then
x
when
on
( 1 .2) is satisfied..
and d be such that
E
a F(x*, a)
and
f(x) = a’
and d E
af(x).
consequently, by concavity
Set
of
for all u E
Then from
so
(2.13) it
follows
that
Since from (2.7)
r(X’ ) ~ k(A’)
then from
proposition 2.6 it
follows that
F(x.,.)
111
Proposition
2. 9
and h -
be the inf convolution of f and g :
fVg
Let f and g be
h(x) - inf (f (x-y)
Suppose that
g
E f
Proof
at
iff h E
I)&#x3E;
Obviously,
Set
we
each
x
and then from
closed proper
functions
convex
on
01531’4
g(y) !y E
attained,
and that f is
inf-compact then
y.
m =
inf(h(x)x
inf(g(x)x
E
m2 = inf(f(x)x
E
have :
proposition
Since f, g
+
the infinum is
2) Suppose that g6
From
two
6.6.4
[7J-
T
if
and let
y
(xn ,x*}
n
satisfies
be
a
sequence such that
h(x ) = f(x -v )
+
o(v )
then
it follows from (2.16) that
(2.15)
3) Suppose
we
now
obtain
that h E
F
and le t
{y,y*)}
be
a
sequence such that
E
112
inf-compact, f *
Since f is
neighbourhood V*
a
there exists
conditions
yn
imply
-yn
that
h(xn) =
(2. J5) that
Remark
two
are
h(x) is attained
below, secondly
Corollary
~~~.~i2.
when f =
2.5
~F
to
Let g be
It is not obvious
space Rn,
whole
the function f
g = fV
1 2||.||2,
Given
domain D
we
and
say that
two
closed proper
on
the relation ~ is
by
relation~.
Proposition
Kannai
2.10
iff g
V11./ f E ~
convex
functions defined
convex
belong
not
functions
a
to
(!.0)
on
the
the class
e.7 I,
and
functions which
convex
G, the function
g1
E
F1
the
on R2
same
strictly increasing
and transitive.
a
k,
function
are
g which
such that g -
[6]).
If
having
real valued
reflexive, symmetric
Then there exists in G
that for all g E
[’5J,
corollary
such that
gl(D)
by
(Debreu
and g is bounded from
function then
if there exists
G the class of closed proper
sense
J
where the infimum in
in the introduction by formula
gl rv g2,
continuous function k
Clearly,
cases
Y.
to
g2
[7 ]
6.6.4
it follows from the corollary that g is differentiable
belong
gl
given
optimality
proposition
convex
find examples of
to
large enough
obtain from (2. 17)t
differentiable everywhere which do
Taking
and doe s no t
JF
E
n
there exists
Sufficient
y.n
We obtain then the
2 ) i f g i s bounded f rom below, g
Remark
we
closed proper
a
+
First when f =
x.
follows that:
and from
g(y )
particularly important
for each
It
zn
+
and it follows from
There
0.
and then for
Set
e
Since f and h belong
3h(x~).
e
at
of 0 such that dom
with
zn
is continuous
and g2 ~ g’1 then g2
E
F1.
k
Denote
equivalent
to
g
is minimal in the
o
g,
is
convex
113
Proof
Let
g1 and g
a
functions
3g.(x) =
consider G the class of functions which
us
minimal function in G then there exists
and
k1
LBx. : À
o
3k.
E
increasing
two
kl o g and g2 = k2 g.
It follows that
x* e 3g"(x)}.
such
k2
equivalent
are
to
convex
that
Note
f
if and
only
F1
The class
Examples
functions.
contains,
the class of
of course,
But many other functions
belong
~1.
to
Let
inf-compact
consider,
us
convex
for
instance, the positive semi-definite quadratic function
This function
with
A
decomposed
If
~1.
then
f E
that if
An
as
symmetric positive definite.
necessarily
and
be
can
If
a2 - 0,
then
interesting example
is the
2014~
"Log
0
then
It follows
inf
-
x" entropy defined
on
the
positive
orthant
by
Then -h
belongs
to
"x Log x" entropy.
of the
"Log
reasons
why
x" entropy,
ef
but is not
inf-compact
The fact that "x
the "x
even
Log x" is inf-compact
Log x" entropy has
if the last
one
in contrast with the usual
has
been of ten
some
interest.
seems
be
to
preferred
to
applications including chemistry, image processing,
statistics,
A recent reference is Censor and Lent
entropy is discussed.
Closely
related to the
"Log
the
Entropy arises in
various fields of
...
one
where the
x" entropy
are
"Log x"
the Cobb-
Douglas functions
then
-r
E.
The
Cobb-Douglas functions
are
very much used in economics.
114
class ~
Remark
The
mathematics
dealing
Let
for
differentiable :
[0,
When f is
solution
to
x
a
instance,
A
(P)
To
see
(t
&#x3E;
prope r closed
(2. 18) .
has
that,
0:
function,
convex
If, in addition, the
to
an
optimal solutions
no
report
we
x(t)
such that
But the function
-
functions.
inclusion problem : Find
x
such that
c)
we
to
have
j
{x(tn )}
t
convex
set
then there exists
of
unique
solutions of (P)
optimal
optimal solution
a
(P)
of
when
but f E
dT,
then
we
have the follow-
result.
=
£
in several fields of
( theorem 2, p . 160
When
ing
be use ful
the differential
is non-empty, then x(t) conve rges
t - +0°
to
with the asymptotical behaviour of
consider,
us
appears
+00.
c
n
t
E
-
the
proof given in
meas(A e)
3f(x(t n » and
f(x(t))
=
c n
00.
0.
[I]
page
160.
Then there exists
Since f E
decreases and therefore
I,
Defining
a
then
f(x(t)) --
m
sequence
f(x(t n ))2014
when
m.
115
III
CONVERGENCE OF CLASSICAL ALGORITHMS
As said in
unconstrained
are
suppose in
optimization
(This will be the
bounded.
this section
we
method and
gradient method,
a
the convergence of classical
introduction,
the
and
of the
Convergence
Let
(e}
be
a
provided
approximate proximal
point x.
a
where $
is
{x )}
sequence
that f E
approximate proximal
classical
the
two methods :
on
positive
sequences
inf-compact).
In
approximate proximal
shall prove that convergence
we
sequence of
generated
when f is
particular
shall restrict ourselves
obtained for unbounded sequences,
3.I
in
case
that the
general
in
algorithms
can
be
also
~l.
method
reals
converging
method consists
to
to
Then the
zero.
generate, from
a
starting
the rule :
by
given by :
We shall assume in addition that
which is
equivalent
Remark
Given x , such
n
following
way.
If 03A6 (x )
n
n
"
Proof
a
point
x
n+
1
and is obtained in the
always exists
be such that
x
n
set x
,
n+!
I
=
Suppose that f
otherwise
x
n
~
J" ,
By construction, the sequence
its limit and
that
Let
03A6 n (xn)
Theorem 3.!
to :
assume
!!x . - xn||
then
set x
,
n+!
{x}
{f(x ))
for contradiction that &#x26;
converges to 0.
is
is
&#x3E;
a
=
x .
n
minimizing
non
increasing.
m.
From
sequence.
Denote
(3.3), it
by ~
follows
116
Denote
strongly
we
from what
From
e
converges to 0.
(m,l).
optimality
But then
}
{d
n
modify slightly
of
a
the
along
to
we
f
not
on
inf-compact
e
and
differentiable
f(y n) ?,.. À
regularized proximal
defining correctly
tvf(x )I
Instead of
0},
t
function $
c
f
+
n
the
necessarily
a
1 2||. - !j. 2
x
Indeed, if f
algorithm.
order to obtain convergence results
Vf(.) is uniformly continuous
on
which minimizes
real t
n
we
each level set of
suppose in addition
f, that is
to
say :
where
This
assumption
f (~.) -
e
,
...
is satisfied
trivially
&#x3E; m..
We shall
method.
gradient
half-line{xn -
there does not exist
f!
f ~-
method
rules of the
the descent
that
this half line.
Furthermore, in
that
d
f(y )
3f(y ) such
0, in contradiction with
stepsize
for all
such that
suppose that f is
This is necessary for
is
n
~.
to
converges
proximal-gradient
shall minimize the
we
(f (yn) )
conditions there exists
In this section
f
Since ~ n is
hand,
On the other
There exis ts
converges
Convergence
minimizing
reaches its minimum.
have :
deduce that
we
Let 03BB
we
point where ~ n
deduce
( x - y II
and
y n the
convex,
f rom what
3.2
by
for
example for quadratic functions,
117
Proximal-gradient
3.2.1
from
Starting
n
Theorem 3.2
is
Suppose
minimizing
a
Wi thout loss of
Then
satisfies the
we
From
n
0 stop, else
(xn - tVf(x n)).
then for each
n
f(x ) ~ f (xn)
and
general i ty
we
can
suppose that
0 for al l
n.
equation
(3.7) it follows that
lim
(3.7)
f(x ).
we
If &#x26;
=
~~,
the theorem is
proved, if not 9, is finite
obtain then :
both numbers of
Dividing
so
~ 03A6
7f(x ) "
by
have
n
from
If
{x}
sequence.
Proof
and
~~I
that f 6
t
the sequence
construct
we
x computed.
minimizes on R + the function
where t
{x )
arbitrary point xo
an
rule : suppose
following
the
method with exact minimization
I
(3.6)
we
obtain :
that
and from
Since f
(3.9)
E
~l
I
we
obtain then
it follows that
{xn}
is
a
minimizing
sequence.
and
118
Remark
This
method in
which, instead
does
one
algorithm
be also
can
of
minimizing
the f i rs t s tep of
only
I
an
now,
can
Gradient
3.2.2
Starting
following
t n is
Proposition
Proo f
so
that
proximal
from
rule
Le t
an
on
function 03A6n on RN,
the half line
implementable
(Goldstein-Armi jo)
method with
rule.
introduce
We
for which
Goldstein-Armijo stepsize
arbitrary point x
an
If
given by
3. 3
us
the
Vf(x ) =
0 stop.
implementable
we
we
shall1
rule
construct the sequence
{xn}}
Else :
rule :
.
0 then there exists
If
in
such that
remark f i rs t that
(3.6) is equivalent
to
proximal
method.
minimizing ~n
general
the
rule :
If the supremum in
Passing
in
being
also be obtained.
Suppose x computed.
where
not
implementable step-size
that convergence
the
0) is
t
as
each step the
at
gradient
a
Except for quadratic functions,
{x -
interpreted
(3.!!) is
the limit
we
to :
not
reached then for each i
obtain then
&#x3E;
1,
we
have :
by
see
119
which
yields
to
Theorem 3.4
Proof
From
a
(f(x )i1
The sequence
(3 . 10)
Then the sequence
and
(3.!!)
{f(x ))
contradiction that £
By definition of t
&#x3E;
we
m ~
is
we
non
converges to
m.
obtain
Let L be its limit.
increasing.
Since f(x ) - f(x
) tends
n
n+
-We
to
have :
and
By continui ty
and
by
where
then
the
mean
xn -
,
there exis ts
value theorem,
tn
0
there
E
[1 ,2)
such that
exists n
,
20142014201420142014201420142014x- ,
E
{0, 1 ) such
that
Assume for
0, then from
121
REFERENCES
Cellina, Differential Inclusions, Springer Verlag,BerlinHeidelberg (1981).
[1]
H.P. Aubin and A.
[2]
A. Auslender and J.-P. Crouzeix, "Global
in Math. of Oper. Research.
[3]
A.
regularity theorems",to
appear
Brøndstedt and R.T. Rockafellar, "On the subdifferentiability of
functions", Proceedings
of the American Mathematical
16
Society,
convex
(1965)
pp. 605-611.
[4]
Y.
Censor and A.
constraints",
pp.
[5]
Lent, "Optimization of "Log x" entropy
S.I.A.M. Journal Control &#x26;
over
Optimization,
linear
Vol. 25
equality
(1987)
921-933.
Debreu "Least
(1976
pp.
concave
utility functions",
Journal of Math.
Economics,
Vol.3
121-129.
preferences",
[6]
Kannai "Concavifiability of
Vol.4 (1977) pp. 1-56.
[7]
P.J. Laurent,
[8]
R.T. Rockafellar "Monotone operators and the proximal point algorithm", S.I.A.M.
Journal on Control and Optimization 14 (1976) pp. 877-898.
[9]
R.T.
Approximation
Rockafellar, Convex
N.J.
1970 .
convex
et
optimisation,
Journal of Math.
Hermann
Economics,
(1972).
Analysis, Princeton University
Press -
Princeton,