Existence
.for
and
Maximizations
Lagrangian
of
Set-Valued
Duality
Functions
H. w. CORLEYI
Communicated by L. Cesari
Abstract. The maximization with respect to a cone of a set-valued
function into possibly infinite dimensions is defined; some existence
results are established; and a Lagrangian duality theory is developed.
Key Words. Set-valued functions, multivalued functions, Pareto optimality, cones, existence, duality, mathematical programming.
I. Introduction
Implicit in the duality theories of Tanino and Sawaragi (Ref. I) and
Corley (Ref. 2) for multiobjective programming was the notion of the
optimization of a set-valued function. The dual problems took this form.
Kawasaki (Ref. 3) subsequently stated a duality theorem in multiobjective
programming where both the primal and dual problems had set-valued
objective functions. His results were obtained for the weak Pareto criterion
in R n under somewhat restrictive assumptions involving the notion of
scalarizability. In this paper, we define the maximization of a set-valued
function with respect to a cone in possibly infinite dimensions and establish
an existence result and Lagrangian duality theory .Optimality
conditions
will be treated elsewhere.
We emphasize here that such problems, rather than being simply
embedded in multiobjective programming, represent a significant generalization in mathematical optimization theory because of the wide application
of set-valued functions. For example, Klein and Thompson (Ref. 4) survey
their use in economics, in addition to presenting their theQry. Zangwill (Ref.
5) uses them to present a unified treatment of convergence of nonlinear
programming algorithms, while Hogan (Ref. 6) studies their properties from
I Professor,Departmentof Industrial Engineering,The University of Texasat Arlington,
Arlington, Texas.
this viewpoint. Generalized equations (Ref. 7) and differential inclusions
(Ref. 8) are other applications. Frequently occurring examples of set-valued
functions include inverses of functions, cones of tangents, and subgradients.
It is therefore anticipated that an optimization theory for set-valued functions will provide a useful analytical tool.
2. Maximizations
of Set-Valued Functions
Let X, Y; Z be real topological vector spaces, and let F: X ~ 2 Y,
a: X ~ 2z be relations. In optimization literature, relations are often called
set-valued functions, multifunctions, or point-to-set maps. We refer to them
as set-valued functions. The domain of F: X ~ 2 y is given by
D(F) ={xe
X: F(x) ,e (2:;}.
A set C in y is a cone if
Aye C,
for all ye C, and A ~O.
A pointed cone C is one for which C n -C = {6}. C is said to be acute if
the closure C is pointed. A convex cone C is one for which
Alyl+A2y2e
C,
for all YI,Y2e C, and A1, A2~O.
The following notions of optimality are used here. Let C be a pointed
cone in Y and B C 1': For YI , Y2e Y; write
YI~sY2,
ify2-YleC.
If Y2-Yle C\{6}, write YI <sY2; if Y2-Yle CO (the interior of C), write
YI <wY2. The point Yoe B is a strong maximal element of B with respect
to C, denoted Yoe max B [or ye max(B; C) when more specificity is
required], if there exists no yeB for which Yo <sY. Similarly, YoeB is a
weak maximal element of B with respect to C, denoted Yoe wmax B [ or
ye wmax(B; C)], if there exists no ye B for which Yo <wy. The set sup B
of strong supremal elements of B with respect to C is defined as sup B =
max B, and the set wsup B of weak supremal elements as wsup B = wmax B.
The notions of min B, wmin B, inf B, and winf B are similarly defined. For
example, Yoe min B [ or Yoe min(B, C) ], if there exists no ye B for which
y <syo.
We are concerned with the following problems. Let Y be ordered by
the pointed convex cone C C Y; Z by the pointed convex cone D C Z, and
let ACX, F: X~2Y, a: X~2z. Also, denote
F(A) = U F(x),
xeA
a-(u)={x:
a(x)
n U,e(2:;}.
.
Then, the basic problems are to
maximize
xEA
F(x),
(I)
i.e., to find all xoe A for which Yoe max F(A)
wmaximize
xEA
F(x),
and Yoe F(xo), and to
(2)
i.e., to find all xoe A for which Yoe wmax F(A) and Yoe F(xo). Special cases
of (I) and (2) are the strong and weak primal problems, respectively. The
strong primal problem is to
maximize F(x),
xEA
(3)
s.t. G(x) n D ¥ (ZJ,
i.e., to find all xoe An G-(D) for which Yoe max F[An G-(D)] and Yoe
F(xo). Note that the constraint specializes to G(x) ...O when G is real-valued
and D is the cone of nonnegative reals.
The weak primal problem is to
wmaximize F(x),
xEA
(4)
s.t. G(x) n D ¥ (ZJ,
i.e., to find all xoeAnG-(D)
for which YoewmaxF[AnG-(D)]
and
Yoe F(xo). Any Xo solving one of the above problems is called a weak or
strong maximal point for the problem as the case may be.
To define the dual problems associated with (3) and (4), let B(Z, Y)
denote the set of all bounded, continuous linear functions s: Z-+ y: A
function s e B(Z, Y) is said to be nonnegative with respect to the cones C
and D, written s e B+(Z, Y), if s(D) C C The strong dual problem associated
with (3) is to
minimize
[
sED
~
(Z.Y)
SUP(F+SG)(A)
] .(5)
-
Letting
<I>(s)= sup(F+ sG)(A),
(5) is the problem of determining the strong maximal elements of
USED+(Z.Y) <I>(s) with respect to the cone -C. Similarly, the weak dual
problem associated with (4) is to
wminimize
[
~
sED (z. Y)
WSUP(F+SG)(A)
],
(6)
i.e., to determine the weak minimal elements of U.e8+(Z, y)A(s), where
A(s) = wsup(F+
sG)(A).
Existence results are stated for problems (1) and (2) in Section 3. In
Section 4, duality relationships are established between the primal and dual
problems.
3. Existence Results
The definitions of upper semicontinuity
and semicompactness are first
given.
Definition 3.1.
The set-valued function F: X ~ 2 y is said to be upper
semicontinuous (usc) if
F+( U) = {x: F(x) C U}
is open for each open uc
y:
Definition 3.2. Let C be a cone in Y and B C 1': B is said to be
C-semicompact if every open cover of complements of the form {(t+
Ya)c: Ya E B, a E A} has a finite subcover.
Existence results will be stated for set-valued functions which are usc
and C-semicompact-valued. The following three examples illustrate such
functions, after which two lemmas are given. Lemma 3.2 is proved in Ref. 9.
Example 3.1.
Let g, h : R I ~ R I, where g(x) is lower semicontinuous
in the usual sense and h(x) is upper semicontinuous in the usual sense. Then,
F(x)={y-:
g(x)~y~h(x)}
is usc and compact-valued (and hence R~-semicompact-valued),
Example 3.2. Let I: Rm ~ RI be locally Lipschitz and d E Rm. Then,
as in Clarke (Ref. 2), let
IO(x; d) = lim sup{[I(x
h-+O
tJ,O
+ td + h) -I(x
+ h )]1 t},
and let
al(x) = {y E Rm: (y, d)~IO(x;
d), for all d E Rm}
be the generalized gradient of I at x. Then, al is usc and compact-valued
(and hence R'.;'-semicompact-valued).
Example 3.3.
Let X = (0,00) with the subspace topology, y = R2 with
the usual topology, and
F(x)={(al,aJ:
ai+a~~x;
Then, F is R~-semicompact-valued
al,a2>0}.
(but not compact-valued)
and usc.
Lemma 3.1. Let A be compact in X, and let F be C-semicompactvalued as well as usc. Then, F(A) is C-semicompact in y:
Proof.
Let { O.. : a E A} be an open cover of F(A)
O.. = ( C + Y..)C,
of the form
for Y.. E y:
For each x E A, {0..: a E A} covers F(x), which is C-semicompact, so there
is a finite sub~over. Let Nx be a finite index set such that
F(x)COx=
U O..,
..ENx
an open set. Since F is usc, Vx = F+( Ox) is open for each x E A. Moreover,
{ Vx: x E A} is an open cover of A and thus has a finite subcover, say
Vj=F+(Ox,),
i=I,...,n.
Then,
N=
n
U NXi
;=1
is a finite subset of A and
U F(x)C U
XEA
-..EN
Hence, F(A)
O...
is C-semicompact.
O
Lemma 3.2. Let C be an acute convex cone in Y and B a nonempty,
C-semicompact set in y: Then, max(B; C) ,c 0.
An immediate consequence of Lemmas 3.1 and 3.2 is the following
theorem.
Theorem 3.1. Let A be compact in X, let C be an acute convex cone
in Y, and let F: X-+ 2 y be C-semicompact-valued and usc. Then, there
exists a strong maximal point for (1).
The existence of weak maximal points is a corollary to Theorem 3.1
as a consequence of the following two properties and the fact that the cone
COu{8}CC.
Property 3.1.
If C is a pointed cone in Y and H C Yo then
wmax(H; C) = max(H; Cou {8})
for the pointed cone COu{8}.
Property 3.2. If H C Y and C1, C2 are two pointed cones in Y for
which C1 ("'IC2, then
max(H; C2)Cmax(H;
C1).
Corollary 3.1. Under the assumptions of Theorem 3.1, there exists a
weak maximal point for (2).
The next example illustrates that Theorem 3.1 is not valid even if A is
compact, F is usc, and each F(x) is closed and bounded.
Example 3.4. Let X = R1, Y= Co, the Banach space of all sequences
y = (an} of real numbers converging to 0 in sup norm. Let C be the set of
all nonnegative sequences in Co, a pointed, closed, and convex cone. Let
A=[!, 1], and define F:X-+2Y
by
F(x)=Hx={Y:
IIYII~lxl},
a closed and bounded set. Then,
F(A)=H1={y:
Ilyll~I}.
To show that max(F(A); C) = (Zj, fix y = (an} in HI and note that
lakl < 1,
for some k.
Let Yk be the sequence which is 0 except in the kth place, where it has
value 1-lakl. Thus, both y + Yk E HI and y + Yk E C + y. Since y is arbitrary,
max(F(A);
C) = (Zj.
4. Duality Relationships
Generalizations of the Lagrangian duality theorems of mathematical
programming are established in this section. Concavity for set-valued functions is first defined. This definition is the obvious counterpart of convexity
as defined in Ref. 10 and elsewhere.
Definition 4.1. Let A C X be convex, let C be a convex cone in Y;
and let F: X-+ 2 y. Then, F is C-concave on A if, for any XI, X2E A and
AE[o,I],
F[Axl + (1- A )X2] -C
:) AF(xl) + (1- A )F(xJ.
A readily established consequence of Definition
following
property.
(7)
4.1 is given in the
Property 4.1. If F is C-concave on the convex set A, then the sets
F(A) -C and A n F-( C) are convex.
Some other easily proved properties to be used subsequently are stated
below.
Property 4.2.
Then,
Let C be a pointed convex cone in Y; and let B C Y:
max B=max(B-C)
Property 4.3.
and
wmax BCwmax(B-C).
Let C be a pointed cone in Y; B c Y; and Yoe B. Then,
Yoe max B,
if and only if B n [ C + Yo] = {Yo}.
Yoe wmax B,
ifand onlyifBn[Co+
Yo] ~(25.
Property -4.4. Let C be a pointed convex cone in Y: If Yl e CO and
Y2e C, then Yl+ Y2e Co.
We now state a theorem to be used in proving duality relationships
between the primal and dual problems. Let Y* denote the topological dual
of Y; and let
C+ = {le Y*: I(y) ~ 0, for all ye C},
the nonnegative dual cone of the cone C in Y:
Theorem 4.1. Let C be a pointed convex cone in y with Co;6 (25,D
a pointed convex cone in Z with Do;6 (25,and A a convex set in X. Suppose
that F is C-concave on A, G is D-concave on A, A n G-(DO) ;6 (25,and Xo
is a weak maximal point for (4). Then, there exists soe B+(Z, Y) such that
Xo is a weak maximal point for the problem
wmaximize F(x)+soG(x),
xEA
and there exists zoe D n G(xo) for which so(Zo)= 8.
Proof.
Let
E =AnG-(D),
which is convex from Property 4.1. Also from Property 4.1, F(E) -C is
convex. Let Yoe wmax F(xo), which must be nonempty, since Xo solves (4).
From Property 4.2,
Yoe wmax[F(E)
-C].
Property 4.3 then requires that
[F(E)-
C]n[Co+
Yo] =(25
Property 4.1. If F is C-concave on the convex set A, then the sets
F(A) -C and A n F-( C) are convex.
Some other easily proved properties to be used subsequently are stated
below.
Property 4.2.
Then,
Let C be a pointed convex cone in Y; and let B C Y:
max B=max(B-C)
Property 4.3.
and
wmax BCwmax(B-C).
Let C be a pointed cone in Y; B c Y; and Yoe B. Then,
Yoe max B,
if and only if B n [ C + Yo] = {Yo}.
Yoe wmax B,
ifand onlyifBn[Co+
Yo] ~(25.
Property -4.4. Let C be a pointed convex cone in Y: If Yl e CO and
Y2e C, then Yl+ Y2e Co.
We now state a theorem to be used in proving duality relationships
between the primal and dual problems. Let Y* denote the topological dual
of Y; and let
C+ = {le Y*: I(y) ~ 0, for all ye C},
the nonnegative dual cone of the cone C in Y:
Theorem 4.1. Let C be a pointed convex cone in y with Co;6 (25,D
a pointed convex cone in Z with Do;6 (25,and A a convex set in X. Suppose
that F is C-concave on A, G is D-concave on A, A n G-(DO) ;6 (25,and Xo
is a weak maximal point for (4). Then, there exists soe B+(Z, Y) such that
Xo is a weak maximal point for the problem
wmaximize F(x)+soG(x),
xEA
and there exists zoe D n G(xo) for which so(Zo)= 8.
Proof.
Let
E =AnG-(D),
which is convex from Property 4.1. Also from Property 4.1, F(E) -C is
convex. Let Yoe wmax F(xo), which must be nonempty, since Xo solves (4).
From Property 4.2,
Yoe wmax[F(E)
-C].
Property 4.3 then requires that
[F(E)-
C]n[Co+
Yo] =(25
and so all suprema are attained in (II).
Let
ZoE D("1 G(xo) ~ g.
Then,
(l(yo), Zo)E A ("I <I>,
so that, from (10), with {3 = I,
l(yo) + uo(Zo):s;l(yo).
Thus, uo(Zo) = 0, since u(zo) ~ 0.
We have now established that there exist UoE D+ and ZoE D("1 G(xo)
with uo(Zo) = ° for which
l(y) + uo(z) :s;l(yo) + uo(zo),
for all x E A, y E F(x), z E D ("I G(x).
(12)
Since CO~ 9 and l(y) > ° for all y E CO, choose Yl E CO such that l(yl) = I.
Define So:Z-+ y by
so(z) = UO(Z)Yl.
It follows that
SoE B+(Z, Y)
and
so(zo) = 6.
Moreover,
y+so(z) -Yo-so(zo)
t CO,
Otherwise l(y)+uo(z»l(yo)+uo(zo),
is now complete.
for all x E A, Y E F(x),
in contradiction
ZE D("1 G(x).
to (12). The proof
D
The duality relationships are next established. Theorem 4.2 bel~w is a
generalization of the weak duality theorem of mathematical programming
which states that the value of the primal objective function at any feasible
point is never larger than the value of the dual objective function at any
feasible point.
Theorem 4.2. Suppose that C is a pointed convex cone in ~ Xo is
feasible to (4) [i.e., xoEA("1G-(D)],
and So is feasible to (6) [i.e., SoE
B+(Z, Y)]. Then, for every YoE F(xo) and VoEO(So), it is not the case that
Yo <MIVo. A similar relationship holds for (3) and (5).
Proof. The result is proved only for (4) and (6). Suppose to the
contrary that there exist Yoe F(xo) and voe .n(so) such that
Yo- voe Co.
(13)
However, since
voe wsup(F+
soG)(A),
there does not exist xI e A with YI e F(xl)
and Zl e G(xl)
for which
YI + SO(ZI)-voe Co.
But xoe A, so, in particular,
for zoe Dn G(xo) ,e 125,we have that
Yo+ so(zo) -vot: Co.
(14)
Upon adding so(Zo)e C to (13) and using Property 4.4, a contradiction
(14) is obtained to establish the result.
to
O
D need not be a cone, A does not have to be a convex set, and no
concavity restrictions are placed on F or G in Theorem 4.2. A stronger
result requires these assumptions. Theorem 4.3 below may be very loosely
interpreted in the context of mathematical programming as follows. A
solution to the primal implies the existence of a solution to the dual, and
the values of the two objective functions are equal. Theorem 4.4 is somewhat
similar; it should be obvious that its conclusion is equality for real-valued
functions.
Theorem 4.3. Suppose that C is pointed convex cone in y with
Co,e 125,D is a pointed convex cone in Z with Do,e 125,A is a convex set,
F is C-concave on A, G is D-concave on A, and A n G-(DO) ,e 125.If Xo is
a weak maximal point for (4), then there exists Yoe F(xo) such that
Yoe wmin
[
U
SE B+(Z' Y)
Proof.
.n(S)] .(15)
From Theorem 4.1, there exists soe B+(Z, Y) for which
Yoe .n(so),
for some Yoe F(xo).
To verify (15), assume the contrary. Then, there exists SI e B+(Z, Y) and
YI e .n(SI) for which
o
YO-YleC.
(16)
For zoe Dn G(xo), add sl(zo)e C to (16) and use Property 4.4 to get
Yo=sl(zo)-YleCo.
But (17) contradicts the assumption that YI e .n(SI), so (15) follows.
(17)
D
Corollary 4.1.
wmin
Let the assumptions of Theorem 4.3 be satisfied. If
[sEB (z, Y)
~
O(S)
] ~ (25,
(4) has no weak maximal points and (3) has no strong maximal points.
Proof. Apply Theorem 4.3. Then, (4) has no weak maximal points;
and, by Properties 3.1 and 3.2, (6) has no strong maximal points.
D
Theorem 4.4. Suppose that the assumptions of Theorem 4.3 are
satisfied. Let Xo be a weak maximal point for (4) and
voe wmin
[sEB (z. Y)
~
O(S)
].
Then, for every ye F(xo), it is not the case that y <w Vo. Moreover, there
exists Yoe F(xo) for which it is not the case that Vo<wYo.
Proof.
Since
voe wmin
U
O(s),
SEB+(Z'Y)
there exists SI e B+(Z, Y) for which voe O(Sl). It follows immediately from
Theorem 4.2 that
Vo-y t Co,
for every ye F(xo).
On the other hand, from Theorem 4.1, there exist soe B+(Z, Y) and Yoe
F(xo) such that Yoe O(so). But
voewmin
U
O(s),
sEB+(z,Y)
so
Yo- vot Co.
D
s. Remarks
Optimization theory at this level of abstraction begins to diverge from
results in mathematical programming for the real-valued case. For example,
both
G(x)CD
(18)
G(x) n D ~ (25
(19)
and
generalize the mathematical programming inequality constraints gj ( x) ~ 0,
j = 1, ..., m. However, to prove the analog of Theorem 4.1 for the constraint
(18) would require, in addition to (7), a second type of concavity characterized by
F[Axl +(I-A)x2]
CAF(xl)+(I-A)F(X2)
+ C.
Moreover, the properties of both (18) and (19) are apparently needed to
generalize the usual saddle-point optimality conditions. A natural such
generalization would be as follows. Let
L(x, s) = F(x)+sG(x),
and define All ~~~ A2 for sets AI'A2 to mean, as in Ref. 3, that there
exists xI E AI not dominated (weakly or strongly as the case may be) by
any X2E A2. A saddle-point (xo, so) could then be defined as satisfying
L(x, so) ~... ~ L(xo, s) ~~~ L(x, so),
for all x E A and s E B+(Z, Y).
However, neither (18) nor (19) as the primal constraint yields saddle-point
optimality conditions, though ( 19) preserves enough properties via Theorem
4.1 to produce the duality relations given in Theorems 4.3 and 4.4. It should
also be remarked that, in infinite dimensions, the duality relations are stated
for weak optimality due to the requirements of the standard separation
theorems used in the proofs. It is conceivable that, in finite dimensions,
related results could be stated in terms of strong optimality by somehow
sharpening these separation theorems.
References
1. TANINO,T., and SAWARAGI,Y., Duality Theoryin MultiobjectiveProgramming,
Journal of Optimization Theory and Applications, Vol. 27, pp. 509-529, 1979.
2. CORLEY, H. W., Duality Theoryfor Maximizations with Respectto Cones,Journal
of Mathematical Analysis and Applications, Vol. 84, pp. 560-568, 1981.
3. KAwASAKI, H., A Duality Theory in Multiobjective Nonlinear Programming,
Mathematics of Operations Research,Vol. 7, pp. 95-110,1982.
4. KLEIN, K., and THOMPSON, A. C., Theory of Correspondences,
John Wiley,
New York, New York, 1984.
5. ZANGWILL, W. I., Nonlinear Programming:A Unified Approach,Prentice-Hall,
Englewood Cliffs, New Jersey, 1969.
6. -HOGAN, W. W., Point-to-SetMaps in Mathematical Programming,SlAM Review,
Vol. 15, pp. 591-603, 1973.
7. ROBINSON, S. M., Generalized Equations and Their Solutions, Part I: Basic
Theory, Mathematical Programming Study, Vol. 10, pp. 128-141, 1979.
8. CLARKE, F. H., Optimization and NonsmoothAnalysis, John Wiley, New York,
New York, 1983.
9. CORLEY, H. W., An Existence Resultfor Maximizations with Respect to Cones,
Journal of Optimization Theory and Applications, Vol. 31, pp. 277-281, 1980.
10. BORWEIN, J., Multivalued Convexity and Optimization: A Unified Approach to
Inequality and Equality Constraints, Mathematical Programming, Vol. 13, pp.
183-199,1977.
11. Eow AROS, R., Functional Analysis: Theory and Applications, Holt, Rinehart,
and Winston, New York, New York, 1965.
12. LUENBERGER, D. G., Optimization by Vector Space Methods, John Wiley, New
York, New York, 1969.