Directional Lipschitzian Optimal Solutions and Directional Derivative for the Optimal Value Function in Nonlinear Mathematical Programming

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

>

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

obtained with the

was

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

>

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) )> 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

>

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

>

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>

(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

>

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: Application to Nonlinear Math. Programming". Prog. Study 19, 200-221.