A NNALES DE L’I. H. P., SECTION C
J. G AUVIN
R. JANIN
Directional lipschitzian optimal solutions and
directional derivative for the optimal value function
in nonlinear mathematical programming
Annales de l’I. H. P., section C, tome S6 (1989), p. 305-324
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DIRECTIONAL LIPSCHITZIAN OPTIMAL SOLUTIONS
AND DIRECTIONAL DERIVATIVE FOR THE OPTIMAL
VALUE FUNCTION IN NONLINEAR MATHEMATICAL
PROGRAMMING
J. GAUVIN
Département de mathématiques appliquées, École Po4ytechnique, Montreal, Quebec,
Canada H3C 3A7
R. JANIN
Département de mathématiques, Université de Poitiers, 86022 Poitiers Cedex, France
This paper
gives
Lipschitz behaviour
mathematical
in
some
very accurate conditions to have
for the
optimal
programming problem
fixed direction.
with natural
Mathematical
a
perturbations
This result is then used to obtain
the directional derivative for the
Key words:
solutions of
optimal
value function.
programming, Lipschitzian
solutions, optimal
value
function, directional
derivative.
This research
Engineering
was
supported by
the Natural Sciences and
Research Council of Canada under Grant A-9273.
306
1.
INTRODUCTION
We consider in this paper
programming problem
is
a
t
0 , 1,J
&#x3E;
are
of any local
some
The
perturbations
finite sets of indices and
fixed direction for the
for the minimal
near
with natural
s.t.
P(tu):
where
nonlinear mathematical
a
’assumptions
perturbations.
to have the
optimal solution
optimal
Lipschitz
parametric optimization
looking
P(tu)
P(0).
solutions in
optimal
have been studied
We cite the works of Aubin
R
Lipschitz continuity
of program
behaviour of the
are
of program
x(tu)
x*
solution
We
u £
by many
authors.
[1], Cornet-Vial [2] and Robinson
[6] where this property has been obtained under regularity
conditions related somehow with the
regularity condition.
the program
where the programs
perturbations; i.e.,
Lipschitz property
under
is
a
An
the
perturbations
more
v
Theorem
tentative to have this
general regularity conditions.
in this paper is to
of that result.
restricts
have feasible solutions.
P(v)
in Gauvin-Janin [3]
Our purpose
regularity conditions
to be defined in the interior of the
P(0)
domain of feasible
4.3
This
Mangasarian-Fromovitz
give
a
more
example is given which
refined version
shows that
we
may
307
have obtained the most accurate statement for
a
Lipschitz
directional continuity property for the .optimal solutions in
mathematical programming.
obtain
a
nice and
Finally
the result is used to
formula for the directional
simple
derivative of the optimal value function of the mathematical
to
and
programming problem.
This last result
to refine Theorem 3.6
in Gauvin-Tolle [4] where this formula
was
obtained with the
assumption,
Lipschitz continuity property
was
comes
complete
that
among others,
satisfied for the
a
optimal
solutions.
It should be noticed that program
contains the
always
can
equalities
Also
a
be defined
which
can
convex
polyhedron
are
because such
by linear inequalities
or
be included in the above formulation.
mathematical program with general nonlinear
perturbations
linear
can
right-hand
shown
once
some
formulated
as
where the feasible solutions
case
restrained to remain in
set
P(tu)
by
R.T.
be translated to the formulation with
side
perturbations only
as
it has been
(see the Introduction in
Rockafellar
Gauvin-Janin [3]).
2.
ASSUMPTIONS AND PRELIMINARIES
For
~(x~)
an
optimal solution
the set of
multipl iers
x*
of
P(O),
we
denote
Lagrange multipliers; i.e., the
( a , ~ ) , ~ ~ R IuJ
,
such that
by
308
The
corresponding
set of Kuhn-Tucker
multipliers
is denoted
by
with its recession
The first
cone
assumption
is
a
very
general regularity
condition.
H1:
The set
The second
is nonempty.
0 (x*)
assumption is
the direction of
The direction
This
assumption implies
optimal
on
the choice of
u
satisfies
that the
family
{Vf. 1 (x*)I
i E J }
equality constraints is linearly
independent (see (3)
Both
condition
perturbations.
H :
2
related with the
a
of Remark 3.1 in Gauvin-Janin
assumptions implies
that the set
Q
[3]).
1 (x*,u)
of
solutions for the linear program
is nonempty and bounded
even
is unbounded
(see
309
(4)
of Remark 3.1 in Gauvin-Janin
impl ies
that the linear program
subject
to
is f eas ibl e and bounded.
optimal
LP(x*,u):
We denote
by
Y ( x* , u )
the set of
solutions for that program, by
the set of indices
inequality
corresponding
constraints for
some
to
corresponding
multipliers.
we
Finally
tangent subspace
at
denote
x*
possible binding
optimal solution
the subset of indices
the
By duality, this
[3]).
to nonnul
and
by
optimal
by
to the
inequality
related with this last set of indices
constraints
together with
the
equality constraints.
If
we
let
the third and last
sufficient
be the Hessian of the
assumption
is
optimality condition
tangent subspace.
a
Lagrangian
weak second-order
related with the above
310
For any
H3 :
y
E , y ~ 0, there exists
E
n 1 ( x* , u )
such that
This condition is weak in the
À*
hold for all
0
e
1
(x*,u)
By arguments similar
or
optimum
all
À
e
0
1
(x*).
of those in Theorem 2.2
[3], assumption
strict local
that it does not need to
sense
for the
implies
~3
enlarged
that
in
x*
is a
program
n
.
min
f
o
(x) ,
X
R
e
s.t.
x*
Since
program
optimum
of
of
x*
P(0),
of
we
an
optimum
x*
must have that
Therefore
P(0).
where
f 0 (x*)
we
of the
is also
have
a
original
a
X(x*)
neighborhood
for any feasible
f (x)
strict
point
x
P(O).
By
any
is assumed to be
a
local
optimal solution
optimal solution
where
Under
R(tu)
Gauvin-Janin [3]
are
near
x*
(x:) I
x £
H1 and
c
R(tu),
P(tu).
H2, the proof of Theorem
shows that it is
x(t)
we mean
R(tu) nX(x*) }
is the set of feasible solutions of
assumptions
feasible
f
P(tu)
of the restricted program
x(tu)
min {
of
t
e
possible
[0,1: [, for
3.2 in
to construct
some
t
&#x3E;
0
a
0,’
311
such that
x
But since
program
~
lim sup
x*
is the
P(0
( x* ) ,
we
x**
point
have, for any cluster
f0 (x**)
Since
(t) - x* .
f
p
f0 (x (tu) )&#x3E;
of
x(tu)
lim
f
unique optimum
must
neces sari ly
0
fo (x (t) ) ,
_
t10,
as
(x (t) )
we
f
=
p
(x* ) .
for the restricted
have
x** - x*
and
consequently
It should be noticed from above that the
and
H2
implies
if
that the program
feasible
even
feasible
perturbations
v
=
P(tu),
assumptions
t F
is at the boundary of the domain of
0
In that case the condition
on
the direction u
in
H2
implies
that
dom R.
Example ( 3 . 1 ) in Gauvin-Janin [3] illustrates
u
H1
is
[0,
must be
pointing
toward the interior of
that
situation.
3.
LIPSCHITZ DIRECTIONAL CONTINUITY FOR THE OPTIMAL
SOLUTIONS
The next result
for the
on
the
optimal solution
is
Lipschitz directional continuity
a
generalization
and
a
refinement of Theorem 4.3 in Gauvin-Janin [3].
Theorem 1
Let
x*
be
an
optimal solution
of program
P(0)
with
312
assumptions
local
satisfied.
Hand H
H,
123
optimal solution
of
x(tu)
P(tu)
Then,
for any
near
x*,
we
have
Proof.
As
previously noticed, assumption
existence of
optimum
a
neighborhood
X(x*)
for any
and
H
H2’
we
and
is the
unique
by
also have
x(tu)
of the restricted program
which is then feasible for
interval
x*
where
the
.
optimal solution
p(tulx*)
3
P(OJx*);
of the restricted program
assumption
implies
H
some
nontrivial
[O,to[.
Now let suppose the conclusion is false and let take any
t 10, such that
{t},
n
n
sequence
for
some
cluster
point
y
of the bounded set
From Lemma 3.1 and Theorem 3.2
have, f or
a ~
n
0 1 (x*,u),
large enough,
in Gauvin-Janin [3],
f or some 6
and f or any
we
313
where
(x) =
Therefore
Since
we can
n
we
we
0.
for some ~
have,
and
[o,1],
some
i
have assumed that
divided all above
u)-x*1
But it exists
such that
We also have
inequalities
and
equalities by
and take the limits to obtain
a
A
£
n
1 (x*,u)
with
À. 1
&#x3E;
0 ,
i
£
I*(x*,u) ,
314
where
which
K =
I*(x*,u)uJ.
implies
above,
we
then have
that
y £ E.
therefore
From
From
H3’
it exists
n 1 (x* , u)
and
a
0
&#x3E;
such
that
For
n
large enough,
For
n
large enough
left-hand side of
can
we
we
also have
-a/4
The two
inequality (1).
inequalities put together
reduce that
write
inequality
in the
previous
in the left-hand side of (1)
to
which is in contradiction with what
we
have assumed at the
beginning.
The
following example
o
shows that Theorem 1 is
the most accurate statement for
a
perhaps
Lipschitz directional
315
continuity property
for the
optimal solutions in
mathematical programming.
1
Example
min
f. 0 = -x~ 2
s. t.
For
t
(1,2).
x*
0, the optimum is
=
The
mul t ipl iers
with in this
case
0
x*
=
=
0
where
2
1
(x*)
=
( 0 } therefore assumption
is satisfied for any direction u
(u ,u ) .
2
12
optimal multipliers is
=
The linear program
becomes
y~
s.t.
y
~
"2
"2
we
have
=
are
H
for which
I(X*)
5
u
1
2
The set of
316
On this
subspace, the Hessian of the Lagrangian
For the
À*
is
corresponding tangent subspace
The
=
(0,1)
case
£
0
1
-
In that case,
where
(x*,u),
0,
therefore
-
the
optimal
in
0
The
optimal solutions
(x*,u);
and the
is satisfiedwith
3
solution of
=
(
therefore
u -u
H
3
of
is
0,
holds.
we
only have
A*
is not satisfied since
P(tu)
Lipschitz continuity
P(tu)
(0, tu~) )
Lipschitz continuity
For the case where
1
H
have
-
x(tu)
for which the
we
has value
are
in that
does not hold.
case
==
(1,0)
317
4.
DIRECTIONAL DERIVATIVE FOR THE OPTIMAL VALUE FUNCTION
optimal
We now consider the
value function of program
P(tu):
f o (x)I
p(tu) = inf {
where,
previously, R(tu)
as
solutions.
R(tu) }
x ~
is the set of feasible
We need also to consider the local
optimal
value
function for the restricted program
inf {
X(x*)
where
x*
of
is
some
fo (x)J
x e
neighborhood
of
an
optimal solution
P(0) .
The next result is
a
more
refined and
a
more
accurate
statement for the result of Theorem 3.6 in Gauvin-Tolle
[4].
Theorem 2
Let
x*
of program
and
P(0)
(1)
Q 1 (x*) :# ct&#x3E;
(2)
lim sup
Then the
given by
Proof.
optimal
x(tu)
and
and
be
P(tu)
u
optimal solutions respectively
with the
assumptions
’
satisfied
+~ .
value function has
a
directional derivative
318
f or
When
t
some
k.
for some
we
is small enough,
For any
xu (t)
necessarily
A c
o
we
have
1 ( x* ) ,
e [ x* , x (tu) ] .
we
have
Since
o
and
have
~~u~
(P(~) "P~ 0)
)/t ~ ~ ~~x*) ~
This maximum is attained under
’~~ ~’
assumption (1)
as
previously
noticed.
On
or
tile other
hand,
we
have, by Theorem 3.2 in Gollan [5]
Theorem 3.2 in Gauvin-Janin [3],
The result follows from both
inequalities.
a
To
complete finally the result of Theorem
Tolle
[4],
R (tu) ,
t £
closure of
we
can
[,
state the
following
is said to be
R(tu)
theorem.
3.6 in Gauvin-
The
uniformly compact
family
if the
is compact (see also the
319
inf-boundedness condition in Rockafellar [7]). Let
S(0)
optimal solution
be the set of
f or
P(0).
Theorem 3
nonempty and
be
S(0)
Let
on
the nontrivial interval
x*
E
S ( 0 ) , the assumptions
uni f ormly compact
be
R ( tu )
[O,t o [. If, for any optimum
H ,H and H are satisfied,
then the directional derivative of the
optimal
value
function exists and is given by
Proof.
Because
Lemma 2.1 in
a
strict
Gauvin-Tolle [ 4 ] ) .
optimum
of
X (x* )
neighborhood
restricted program
of
and
where
S(0)
x*
are
By
H,
is the
each
S(0)
t = 0+
x*
By
e
[0,t0 [,
for
some
are
also upper
value function
semi-continuous at
t = 0+ ;
therefore continuous at
in
some
For each
nontrivial interval,
x* : S(0)~
optimal solution
x (tu)
we
have
H1
the restricted programs
optimal
t
of the
the number
Therefore the local
For
is
S(0)
unique optimum
must then be finite.
all feasible for t
£
(see
a
By f inite covering,
previously noticed,
as
and
P ( 0 ) ; therefore it exists
P ( 0 ~ x* ) .
optimum points in
H,
uni f ormly compact
is lower semi-continuous at
p (tu)
nonempty,
is
R ( tu )
by
we
t0
&#x3E;
0.
t = 0+.
then have
Theorem 1 that any
of the restricted program
P(tu~x~)
320
is
2
Lipschitz continuous
for each local
at
optimal
t = 0+.
We then
applied Theorem
p (tu ~ x* )
value function
to
finally
When the program
identical for all
is convex, the set
P(O)
x*~S(0);
is
0 (x*)
therefore the above formula
reduces to
which is the classical result of
the Slater
regularity condition
bounded
equivalently
or
In the
for the
case
convex
programming
(0 (x*)
Q (x*) = (0} ).
where the
optimal solution,
nonempty
we
can
still have
under
and
does not hold
Lipschitz property
a
result
on
the
directional derivative for the optimal value function if the
optimal solutions satisfy
for any local
(see Corollary
optimal
4.1
the Holderian
solution x(tu)
of
in Gauvin-Janin [3]).
continuity; i.e.,
P(tu)
In that case,
formula for the directional derivative for the local
value function is
given by
x*
near
I
obtainI
the
optimal
321
where
I
D (x* ) =
v f i (x*) y ~
Q. i £ I (x* ) u { 0 }; vf
x*
is the set of critical directions at
is the subset of
necessary
multipliers satisfying
f x* ) y =
0, i £ J}
and where
the second-order
for that critical direction
optimality condition
y.
This nice formula is not
quite difficult
be useful
as
to evaluate!1
so
simple
and sometime may be
Nevertheless the formula
illustrated by Example 1 where
that for direction
u, with
u 1 -u 2
we
can
have noticed
0, the optimal
solutions
Lipschitzian
are
not
the
optimal
has
a
since
is
but
are
Holderian.
For that
example,
value function
directional derivative with value
u
-u
12.
0, the value given by the formula of Theorem 2
.
322
which
disagree with
apply
for that
case.
(2),
x* = 0
given by
we
we
we
refer to the above valid
have
if and only
and, since
If
have the set of critical directions at
formula
for which
the real value since Theorem 2 does not
Therefore
A .
1
-u
u
1
2
2
0, the formula gives the value
which is in agreement with the real value for the
directional derivative.
Assumption
This
assumption
H3
can
be
equivalently
needed for the
Theorem 1 is contained in the
formulated
by
Lipschitzian property
following
in
less restrictive
323
in Theorem
condition used to obtain the Holderian property
4.1 in
Gauvin-Janin [3]
We can
put together Theorem
Gauvin-Janin [ 3 ]
to obtain the
3 above and
Corollary
4.1 in
and clear
following nice
result for the existence and value for the directional
.
derivative of the optimal value function in mathematical
This last theorem is related with a similar
programming.
result in Rockafellar
[7].
Theorem 4
Let
in
be
S(0)
nonempty and
nontrivial interval
some
x* £ S(0)’
at least
we
one
that
assume
of the
R(tu)
[0,t~[.
H 1 and H2
following
be
uniformly compact
At any
are
optimum
satis f ied with
weak second-order sufficient
optimal ity conditions:
Then the directional
of the
optimal
value
derivative p’(0;u) =
lim
(p(tu)-p(O»jt
function exists and is given by
324
So
far;
we
don’t know any
example
where the
optimal
value function has directional derivative without at least
one
of the condition
or
H3
satisfied.
H4
REFERENCES
[1]
Aubin,
J.P.
"Lipschitz
(1984).
Behaviour of Solutions
to Convex Minimization Problems".
Math.
Oper.
Res.
9,
87-111.
[2]
Cornet,B., Vial, J.P.
(1983).
"Lipschitzian
of Perturbed Nonlinear Programming Problems".
Control and Opt. 24, 1123-1137.
Solutions
SIAM J.
[3]
Gauvin, J., Janin, R. (1987). "Directional Behaviour
of Optimal Solutions in Nonlinear Mathematical
Programming". Math. Oper. Res. (accepted for
publication)
[4]
Gauvin, J., Tolle, J.W. (1977). "Differential
Stability in Nonlinear Programming". SIAM J. Control
and Optimization 15, 294-311.
[5]
Gollan,
B.
Nonlinear
[6]
(1984).
"On the
Programming".
Marginal Function in
Math.
Oper.
Res.
9, 208-221.
"Generalized Equations and
Robinson, S.M. (1982).
their Solutions, Part II:
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Math.
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Prog. Study 19, 200-221.