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Optimizationofapproximateintegrationof
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Ukrainian Mathematical Journal, Vol. 63, No. 2, July, 2011 (Ukrainian Original Vol. 63, No. 2, February, 2011)
OPTIMIZATION OF APPROXIMATE INTEGRATION OF SET-VALUED
FUNCTIONS MONOTONE WITH RESPECT TO INCLUSION
V. F. Babenko 1, 2 and V. V. Babenko 1
UDC 517.5
We obtain the best quadrature formula on the class of convex-valued functions defined on the
segment [0, 1] and monotone with respect to inclusion.
Introduction
The theory of set-valued mappings has been extensively developed in recent decades due to the needs of the
theory of optimization, the theory of games, mathematical economics, and other branches of mathematics. For a
survey of results obtained in this direction, see [1 – 3] and references therein. In recent years, the problems of approximation of set-valued mappings have attracted the interest of mathematicians (see, e.g., [4 – 8]). An important branch of approximation theory and numerical analysis is the theory of quadrature formulas (see, e.g., [9]).
At the same time, we do not know works related to the optimization of approximate integration of set-valued
functions. The present paper is devoted to the investigation of exactly these problems.
There are many different approaches to the definition of integrals of set-valued functions (see [10]). One of
the most elementary approaches is the approach of Hukuhara [11], who proposed to consider a generalization of
the Riemann integral for functions with values in the space K(R d ) of compact convex subsets of the space
R d (below, for completeness of presentation, we give the definition of this integral and describe its elementary
properties).
We consider the problem of optimization of approximate calculation of integrals in the sense of Hukuhara
on the class of functions f : [0, 1] → K(R d ) monotone with respect to inclusion and taking given values
f (0 ) = A and f (1) = B at the endpoints of the segment [0, 1]. The results obtained for this class generalize
the known results of Kiefer for numerical function [12].
We now briefly describe the structure of the paper.
In the first section, we give necessary definitions and facts related to the space of compact convex sets. In
the second section, we give necessary definitions and facts related to the integration of set-valued (convexvalued) functions. In the third section, we solve the problem of the best quadrature formula on the class of functions f : [0, 1] → K(R d ) monotone with respect to inclusion.
1. Space of Convex Sets in R d
Let K = K(R d ) denote a collection of nonempty, compact, convex subsets of the space R d . In the
collection K, we introduce the following operations:
1 Dnepropetrovsk National University, Dnepropetrovsk, Ukraine.
2 Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 2, pp. 147–155, February, 2011. Original article submitted August
6, 2010.
0041–5995/11/6302–0177
© 2011
Springer Science+Business Media, Inc.
177
178
V. F. BABENKO
AND
V. V. BABENKO
Let A , B ∈K and α ≥ 0 . Then
A + B =:
{x + y :
x ∈ A, y ∈ B } ,
αA =:
{ αx :
x ∈ A} .
The set A + B is called the Minkowski sum of the sets A and B.
In the investigation of approximation of convex bodies by polyhedra, one uses different metrics in K =
K(R d ) (see, e.g., [13]). We use the Hausdorff metric δ H ( A, B) , which is defined as follows:
⎫
⎧
δ H ( A, B) = max ⎨ sup inf x − y , sup inf x − y ⎬ ,
x ∈B y ∈ A
⎭
⎩ x ∈ A y ∈B
where ⋅ is the Euclidean norm in R d .
The support function of a convex set A ∈K is introduced as a function
h A (u ) = sup ⟨ x, u ⟩ ,
x ∈A
u ∈ S d −1 ,
defined on the unit sphere S d −1 of the space R d . In terms of support functions, the metric δ H can be represented in the form
δ H (C , D) =
sup
u ∈S d −1
hC (u ) − hD (u ) .
(1)
Note that the Minkowski sum and the operation of multiplication by nonnegative numbers are continuous
with respect to the metric δ H , and the metric space ⟨K, δ H ⟩ is complete.
The following two properties of the metric δ H play an important role in our investigation:
(i) for any A, B, C , D ∈K , one has
δ H ( A + B, C + D) ≤ δ H ( A, C ) + δ H ( B, D) ;
(2)
(ii) for any A, B ∈K and α ≥ 0 , one has
δ H (αA, αB) = αδ H ( A, B) .
(3)
Property (3) is obvious. Let us verify property (2). For arbitrary convex sets A, B, C, and D, using
representation (1), we get
δ H ( A + B, C + D)
⎧
= max ⎨ sup ( h A + B (u ) − hC + D (u ) ) ,
⎩ u ∈S d −1
sup
u ∈S d −1
⎫
( hC + D (u) − h A+ B (u) ) ⎬
⎭
⎧
⎫
= max ⎨ sup ( h A (u ) − hC (u ) + hB (u ) − hD (u ) ) , sup ( hC (u ) − h A (u ) + hD (u ) − hB (u ) ) ⎬ .
u ∈S d −1
⎩ u ∈S d −1
⎭
OPTIMIZATION
OF
APPROXIMATE INTEGRATION OF SET-V ALUED F UNCTIONS MONOTONE
WITH
RESPECT TO INCLUSION
179
Taking into account that
sup
( h A (u) − hC (u ) + hB (u ) − hD (u ) )
≤
sup
( hC (u) − h A (u ) + hD (u ) − hB (u ) )
≤
u ∈S d −1
sup
( h A (u) − hC (u) ) +
sup
( hC (u) − h A (u) ) +
u ∈S d −1
sup
( hB (u) − hD (u) )
sup
( hD (u) − hB (u) ) ,
u ∈S d −1
and
u ∈S d −1
u ∈S d −1
u ∈S d −1
we obtain
⎧
δ H ( A + B, C + D) ≤ max ⎨ sup ( h A (u ) − hC (u ) ) + sup ( hB (u ) − hD (u ) ) ,
u ∈S d −1
⎩ u ∈S d −1
sup
u ∈S d −1
( hC (u) − h A (u) ) +
⎧
≤ max ⎨ sup ( h A (u ) − hC (u ) ) ,
⎩ u ∈S d −1
⎫
sup
( hD (u) − hB (u) ) ⎬
sup
( hC (u) − h A (u) ) ⎬
u ∈S d −1
u ∈S d −1
⎭
⎧
+ max ⎨ sup ( hB (u ) − hD (u ) ) ,
⎩ u ∈S d −1
⎫
⎭
sup
u ∈S d −1
⎫
( hD (u) − hB (u) ) ⎬
⎭
= δ H ( A, C ) + δ H ( B, D) .
In conclusion of this section, we note the following: If we extend the Minkowski sum to an arbitrary finite
number of sets Ak , k = 1, … , n, by setting
n −1
n
∑ Ak
k =1
:=
∑ Ak
k =1
+ An ,
then, by induction, using property (2) of the metric δ H , we get
⎛ n
δ H ⎜ ∑ Ak ,
⎝ k =1
⎞
∑ Bk ⎟⎠ ≤
k =1
n
n
∑ δ H ( Ak , Bk ) .
(4)
k =1
2. Integration of Set-Valued Functions
Recall that a partition P of a segment [ a, b ] , a < b , is defined as a finite system of points x 0 , … , x n of
this segment such that a = x 0 < x1 < … < x n = b . The number
180
V. F. BABENKO
AND
V. V. BABENKO
λ( P ) : = max x i − x i −1
i = 1, n
is called the parameter of the partition P. Choosing a point ξ i ∈ [ x i −1, x i ] , i = 1, … , n, in each segment
[ x i −1, x i ] of the partition P, we obtain a partition ( P, ξ ) of the segment [ a, b ] with marked points
( ξ = ( ξ1 , … , ξ n ) ) .
Let a function f : [ a, b ] → K be given. We associate every partition ( P, ξ ) with marked points with the
integral sum
σ ( f ; ( P, ξ ) ) =
n
∑ f (ξ i ) Δ x i ,
i =1
where Δx i = x i − x i −1 .
Definition 1. If there exists an element I ∈K such that, for any ε > 0 , one can find δ > 0 such
that, for any partition ( P, ξ ) with marked points and parameter λ( P ) < δ , one has
⎛
δH ⎜ I,
⎝
⎞
n
∑ f (ξ i ) Δ xi ⎟⎠
i =1
< ε,
then the function f : [ a, b ] → K is called integrable on the segment [ a, b ] , and the element I is called
its integral. In this case, one writes
b
I =
∫ f ( x ) dx .
a
Denote the collection of all integrable functions f : [ a, b ] → K by R ( [ a, b ], K ) .
By analogy with the case of numerical functions, one can prove the following statements:
Proposition 1. Every function f : [ a, b ] → K continuous on [ a, b ] is integrable on [ a, b ] .
Proposition 2. Every function f : [ a, b ] → K monotone in the sense that
a ≤ x1 < x 2 ≤ b ⇒
f ( x1 ) ⊂ f ( x 2 )
(5)
is integrable on [ a, b ] .
Proposition 3. If f and g belong to R ( [ a, b ], K ) , then their linear combination αf + βg with
nonnegative coefficients is also integrable on [ a, b ] , and, furthermore,
b
b
b
a
a
a
∫ (αf + βg)( x ) dx = α ∫ f ( x ) dx + β ∫ g( x ) dx .
OPTIMIZATION
OF
APPROXIMATE INTEGRATION OF SET-V ALUED F UNCTIONS MONOTONE
WITH
RESPECT TO INCLUSION
181
Proposition 4. If a < b < c and f ∈ R ( [ a, c], K ) , then
f
[ a, b ]
∈ R ( [ a, b ], K ) ,
f
∈ R ( [ b, c ], K ) ,
[ b, c ]
and the following equality is true:
c
∫ f ( x ) dx
=
a
b
c
a
b
∫ f ( x ) dx + ∫ f ( x ) dx .
Proposition 5. If a function f is monotone on [ a, b ] in the sense of (5), then the following inclusions
are true:
b
f (a )(b − a ) ⊂
∫ f ( x ) dx
⊂ f (b )(b − a ) .
a
Using the definition of integral and property (4) of the Hausdorff metric, one can easily prove the following
statement:
Proposition 6. If f and g belong to R ( [ a, b ], K ) , then
⎛
δ ⎜
⎝
H
b
∫
a
⎞
f ( x ) dx, ∫ g( x ) dx ⎟ ≤
⎠
a
b
b
H
∫ δ ( f ( x ), g( x ) ) dx .
a
3. Optimization of Quadrature Formulas on Classes of Monotone Set-Valued Functions
Let M A, B ( A ⊂ B) be the class of functions f : [ 0; 1] → K(R d ) monotone in the sense of (5) and such
that f (0 ) = A and f (1) = B , where A and B are given sets.
Consider the problem of the best quadrature formula of the form
q( f ) = C +
n −1
∑ ck f ( x k )
(6)
k =1
on the class M A, B , where C ∈K(R d ) , c1, … , cn −1 ≥ 0 , and 0 ≤ x1 < … < x n −1 ≤ 1 . Denote the collection of
all these formulas by Q. The problem is formulated as follows:
Denote
⎛
Rn −1 ( M A, B ) = inf sup δ ⎜
q ∈Q f ∈ M A , B
⎝
H
1
∫
0
⎞
f ( x )dx , q( f ) ⎟ .
⎠
(7)
It is necessary to determine quantity (7) and find the quadrature formula of the form (6) that realizes the greatest
lower bound on the right-hand side of (7). This formula is called the best quadrature formula on the class M A, B .
182
V. F. BABENKO
AND
V. V. BABENKO
The following theorem is true:
Theorem 1. Among all quadrature formulas of the form (6), the best formula on the class M A, B is the
following:
A+ B
1 n −1 ⎛ k ⎞
+
∑ f⎜ ⎟;
n k =1 ⎝ n ⎠
2n
qn−1 ( f ) =
in this case, one has
Rn −1 ( M A, B ) =
⎛
sup δ H ⎜
f ∈ M A, B
⎝
⎞
1
∫ f ( x )dx, qn −1( f ) ⎟⎠
=
0
1 H
δ ( A, B) .
2n
Proof. Using the additivity of an integral (Proposition 4), the monotonicity of the function f, and the
monotonicity of an integral (Proposition 5), we get
n −1 ( k +1)/ n
1
∫
f ( x )dx =
∑
k=0
0
∫
1 n −1 ⎛ k ⎞
B
f⎜ ⎟ +
.
∑
⎝
⎠
n k =1
n
n
f ( x )dx ⊂
k /n
By analogy, we obtain
A
1 n −1 ⎛ k ⎞
+
∑f⎜ ⎟ ⊂
n
n k =1 ⎝ n ⎠
n −1 ( k +1)/ n
∑
k=0
∫
1
f ( x )dx =
k /n
∫ f ( x )dx .
0
Thus,
A
1 n −1
+
∑
n
n k =1
⎛k⎞
f⎜ ⎟ ⊂
⎝ n⎠
1
∫
f ( x )dx ⊂
0
B
1 n −1
+
∑
n
n k =1
⎛k⎞
f⎜ ⎟.
⎝ n⎠
(8)
The following inclusions are true:
A
1 n −1
+
∑
n
n k =1
A+ B
1 n −1
⎛k⎞
f⎜ ⎟ ⊂
+
∑
⎝ n⎠
n k =1
2n
B
1 n −1
⎛k⎞
f⎜ ⎟ ⊂
+
∑
⎝ n⎠
n
n k =1
⎛k⎞
f⎜ ⎟.
⎝ n⎠
(9)
Let us estimate
⎛
δH ⎜
⎝
1
∫
0
f ( x )dx ,
A + B 1 n −1
+ ∑
2n
n k =1
⎛ k⎞⎞
f ⎜ ⎟⎟.
⎝ n⎠⎠
We prove that if
X ⊂ Y ⊂ Z,
(10)
OPTIMIZATION
OF
APPROXIMATE INTEGRATION OF SET-V ALUED F UNCTIONS MONOTONE
WITH
RESPECT TO INCLUSION
183
then
1 H
X+Z⎞
⎛
δ (X, Z ) ,
δH ⎜ Y,
⎟⎠ ≤
⎝
2
2
(11)
whence, using (8) and (9), we obtain
⎛
δ ⎜
⎝
H
1
∫
f ( x )dx,
0
A+ B
+
2n
n −1
k ⎞
k =1
⎠
∑ f ⎛⎜⎝ n ⎞⎟⎠ ⎟
≤
1 H
δ ( A, B) .
2n
Let us prove relation (11).
For X , Y ∈K , we denote
e( X , Y ) = sup inf x − y .
x ∈ X y ∈Y
Consider
X+Z⎞
X+Z⎞
⎛ ⎛
⎛
⎛ X+Z
⎞⎞
δH ⎜ Y,
,Y⎟⎟.
⎟⎠ = max ⎜⎝ e ⎜⎝ Y ,
⎟⎠ , e ⎜⎝
⎝
⎠⎠
2
2
2
By virtue of relation (10) and property (2) of the Hausdorff metric, we have
1 H
X+Z⎞
X+Z⎞
X+Z⎞
⎛
⎛
H ⎛
δ (X, Z ) .
e ⎜ Y,
⎟⎠ ≤ e ⎜⎝ Z ,
⎟⎠ ≤ δ ⎜⎝ Z ,
⎟⎠ ≤
⎝
2
2
2
2
Further, we get
1 H
⎛ X+Z
⎞
⎛ X+Z
⎞
⎛ X+Z
⎞
, X⎟ ≤
δ (X, Z ) .
e⎜
, Y ⎟ ≤ e⎜
, X ⎟ ≤ δH ⎜
⎝ 2
⎠
⎝ 2
⎠
⎝ 2
⎠
2
Thus,
1 H
X+Z⎞
X+Z⎞ ⎛ X+Z
⎛ ⎛
⎛
⎞⎞
δ (X, Z ) .
δH ⎜ Y,
,Y⎟⎟ ≤
⎟⎠ = max ⎜⎝ e ⎜⎝ Y ,
⎟⎠ , e ⎜⎝
⎝
⎠
⎠
2
2
2
2
Relation (11) [and, hence, relation (12)] is proved.
We now show that, for any quadrature formula of the form (6), one has
⎛
sup δ ⎜
f ∈ M A, B
⎝
H
which yields the statement of the theorem.
1
∫
0
⎞
1 H
f ( x )dx, q( f ) ⎟ ≥
δ ( A, B) ,
2n
⎠
(12)
184
V. F. BABENKO
AND
V. V. BABENKO
For an arbitrary collection of points 0 = x 0 ≤ x1 < … < x n −1 ≤ x n = 1 , one can find k = 0, 1, … , n − 1
such that x k +1 − x k ≥ 1 / n . We set
⎧⎪ A,
f1 ( x ) = ⎨
⎩⎪ B,
x ≤ xk ,
x > xk ,
and
x < x k +1,
⎧⎪ A,
f2 ( x ) = ⎨
⎩⎪ B,
x ≥ x k +1.
Then
1
∫ f1( x )dx
= Ax k + B(1 − x k ) ,
0
1
∫ f2 ( x )dx
= Ax k +1 + B(1 − x k +1 ) ,
0
and
q( f1 ) = q( f2 ) .
Therefore,
⎛
sup δ H ⎜
f ∈ M A, B
⎝
1
∫
0
⎧⎪
⎞
⎛
f ( x )dx , q( f ) ⎟ ≥ max ⎨ δ H ⎜
⎠
⎝
⎪⎩
≥
1
2
⎧⎪
⎛
H
⎨δ ⎜
⎝
⎪⎩
1 H
δ
≥
2
=
⎛
⎜
⎜⎝
1
∫
0
1
∫
0
⎞
⎛
f1 ( x )dx, q( f1 ) ⎟ , δ H ⎜
⎠
⎝
⎞
⎛
f1 ( x )dx , q( f1 ) ⎟ + δ H ⎜
⎠
⎝
1
1
⎞
0
0
⎠
1
⎞ ⎫⎪
0
⎪⎭
∫ f2 ( x )dx, q( f2 ) ⎟⎠ ⎬
1
⎞ ⎫⎪
0
⎪⎭
∫ f2 ( x )dx, q( f2 ) ⎟⎠ ⎬
∫ f1( x )dx , ∫ f2 ( x )dx ⎟⎟
1 H
δ ( Ax k + B(1 − x k ), Ax k +1 + B(1 − x k +1 ) ) .
2
We now consider e( Ax k + B(1 − x k ), Ax k +1 + B(1 − x k +1 )) . Using the duality theorem (see, e.g., [14],
Sec. 2.3), we get
OPTIMIZATION
OF
APPROXIMATE INTEGRATION OF SET-V ALUED F UNCTIONS MONOTONE
WITH
RESPECT TO INCLUSION
185
e( Ax k + B(1 − x k ), Ax k +1 + B(1 − x k +1 ))
=
sup
z ∈ Ax k + B (1− x k )
⎧
sup ⎨ f ( z ) −
f ≤1 ⎩
sup
ω ∈ Ax k +1 + B (1− x k +1 )
⎫
f (ω ) ⎬
⎭
=
⎫
⎧
sup ⎨ sup ( x k f ( x ) + (1 − x k ) f ( y )) − sup ( x k +1 f (u ) + (1 − x k +1 ) f ( v)) ⎬
u ∈ A, v ∈ B
f ≤ 1 ⎩ x ∈ A, y ∈ B
⎭
=
sup
( x k h A ( f ) + (1 − x k )hB ( f ) −
=
sup
( ( x k − x k +1 )h A ( f ) + ( x k +1 − x k )hB ( f ) )
f ≤1
f ≤1
= ( x k +1 − x k ) sup
f ≤1
x k +1h A ( f ) − (1 − x k +1 )hB ( f ) )
( hB ( f ) − h A ( f ) )
= ( x k +1 − x k ) e ( B, A) .
By analogy, we obtain
e ( Ax k +1 + B(1 − x k +1 ), Ax k + B(1 − x k ) )
=
sup
z ∈ Ax k +1 + B (1− x k +1 )
⎧
sup ⎨ f ( z ) −
f ≤1 ⎩
sup
ω ∈ Ax k + B (1− x k )
⎫
f (ω ) ⎬
⎭
=
⎫
⎧
sup ⎨ sup ( x k +1 f ( x ) + (1 − x k +1 ) f ( y )) − sup ( x k f (u ) + (1 − x k ) f ( v)) ⎬
u ∈ A, v ∈ B
f ≤ 1 ⎩ x ∈ A, y ∈ B
⎭
=
sup
( x k +1h A ( f ) + (1 − x k +1 )hB ( f ) −
=
sup
( ( x k +1 − x k )h A ( f ) − ( x k +1 − x k )hB ( f ) )
f ≤1
f ≤1
= ( x k +1 − x k ) sup
f ≤1
( h A ( f ) − hB ( f ) )
x k h A ( f ) − (1 − x k )hB ( f ) )
= ( x k +1 − x k ) e ( A, B) .
Since B is wider than A, we conclude that e ( A, B) = 0.
Thus, for any quadrature formula q ∈ Q , we have
⎛
sup δ ⎜
f ∈ M A, B
⎝
H
The theorem is proved.
1
∫
0
⎞
1
1 H
f ( x )dx , q( f ) ⎟ ≥
( x k +1 − x k ) δ H ( A, B) ≥
δ ( A, B) .
2
2n
⎠
186
V. F. BABENKO
AND
V. V. BABENKO
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