or-CONCAVE FUNCTIONS AND MEASURES AND THEIR
APPLICATIONS
UDC 519.95
V. I. Norkin and N. V. Roenko
Properties of functions arising in probability optimization problems are studied. Probability distribution
functions are shown to have certain concavity properties under natural conditions. A calculus of these functions
is constructed.
INTRODUCTION
Many applied optimization problems contain functions that are not concave (convex), but have some other useful
properties. Various generalizations (weaker forms) of concavity (convexity) of functions have been proposed. One of these
generalizations is the notion of a quasiconcave (quasiconvex) function (see [1, 2]).
A quasiconcave function has a number of special properties: its Lebesgue sets are convex and, if they are bounded,
then the function is bell-shaped; the set of its global maxima in any convex domain is convex. However, quasiconcave functions
may be discontinuous inside the domain of definition and their stationary points and regions are not necessarily global optima.
In this respect, they differ from concave functions.
In this paper, we identify and analyze a subclass of quasiconcave functions - the so-called a-concave functions, where
a is a numerical parameter. The family of a-concave functions (a E [ - oo, + oo]) is ordered by the parameter a. If some
function is al-concave and a I > a2, then it is also a2-concave. The parameter value a > - oo corresponds to nonnegative
quasiconcave functions.
A positive function f is a-concave ( - Qo < a < + oo) if t~ is concave (a > 0), In f is concave (a = 0), and t~ is
convex (a < 0).
a-Concave functions arise, for instance, in evaluating integral functionals of set-valued mappings, in computing the
probability that a random variable dependent on parameters exceeds a given limit, and also in fuzzy optimization problems.
Optimization problems with a-concave functions are reducible to convex programming problems by raising the aconcave functions in an appropriate power. This requires an estimate of the concavity parameter a, and the theory accordingly
focuses on development of techniques for computation or estimation of a. Sometimes we may not want to transform the original
problem to a convex problem. For instance, it is undesirable to complicate the a-concave stochastic programming problem by
raising the expectations to the power a. The theory therefore also studies the properties of a-concave functions proper.
c~-Concave functions (a > - ~ ) inherit the properties of concave functions to a greater extent than quasiconcave
functions do. They are continuous in the interior points of their domain of definition; they are differentiable in a certain sense,
have a unique extremum (a maximum), and can be maximized by generalized gradient method. In this paper, we construct a
calculus of a-concave functions, which includes assertions of a-concavity of the sum, the product, the maximum, the minimum,
and the integral of a-concave functions.
We also study c~-concave measures (a-concave functions of bounded convex sets in Rm). An example of an [l/m]concave measure is the volume of bounded measurable sets in Rm. The main classical probability distributions in R m are shown
to be a-concave functions and the measures generated by these probability distributions are a'-concave. Conversely, if a
probability measure is et-concave, then the probability that some random variable dependent on parameters exceeds a given
level is an a-concave function of the parameters under certain conditions. These properties open opportunities for effective
solution of stochastic programming problems that contain probability distribution functions.
Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 77-88, November-December, 1991. Original article
submitted February 8, 1990.
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1060-0396/91/2706-0860512.50 ©1992 Plenum Publishing Corporation
P R O P E R T I E S O F or-CONCAVE F U N C T I O N S
Defin#ion 1. The function F(x) defined on the convex set X C R n is called quasiconcave if for any Xo, x I ~ X and
X E (0, 1) we have the inequality F((1 - X)xo + XXl) _> min(F(xo), F(xl)).
Definition 2. The nonnegative function F(x) defined on the convex set X C R n is called log-concave (or logarithmic
concave) if the function In F(x) is concave on X, i.e., for any Xo, x I E X and X E (0, 1) we have In F((1 - X)xo + Xxl) _
(1 - X)ln F(xo) + kin F(Xl) , which is equivalent to F ( ( 1 - X)xo + XXl) _> F(xo)l-hF(Xl)X.
Here and in what follows we assume that In 0 = - no, 0"(+ oo) = 0, 0° = 1, 00 -[c~[ = 0, 0 - ] ~ l = + ~ , + ooo =
1, - o o + oo = 0, etc.
Definition 3 [3]. The function f: X --, R 1 defined on the convex set X C R n is called g-concave if there exists a strictly
monotone (strictly monotone increasing [4]) function g: R 1 --, R 1 such that the function g(f(x)) is concave on X.
Log-concave functions are obviously quasiconcave and g-concave.
Definition 4. The nonnegative function F(x) defined on the convex set X C R n is called a-concave (a is a numerical
parameter, a E ( - oo, + oo)) if for any Xo, x I E X and X E (0, 1) we have
I
/?((1 - - ~ ) x, +
~X1)
((1 - - ~ ) F ( r o ) ~ +
)~F (xl)C*)I/c*, cz :# O, 4- oo,
{ F(xo) l-x, F(xl) ~,
o: = O,
I min(F(xo) ' F(.h)),
([ max (F (Xo), F (x0),
cz = - - oo,
o~ = 4- ~ , , .
a-Concave functions are obviously quasiconcave and 0-concave functions are log-concave.
It follows from these definitions that F(x) is an a-concave function on X ( - oo < a < + 0o) if and only if F°L(x) is
concave (a > 0), In F(x) is concave (a = 0), and F~(x) is convex (a < 0) on X. We are thus dealing with a subclass of gconcave functions where g = (sign a)t ~ or g = In f.
For instance, the function fl(x) = max(0, x), x E R l, is log-concave and the function f2(x) = x - l , x E R 1, is ( - 1 ) concave. The indicator function of a convex set in R n is log-concave in R n.
For X = {x E R n, f(x) > 0}, a-concave (a-unimodal) functions have been considered by Borrell [51, Das Gupta
and Roenko [7].
We have the following simple lemma.
L E M M A 1. The generalized mean function
[ ((1--~,)A ~ 4- )~B~')v~,
~)~,A,B (0~) =
[ AI-~,B x,
{
[61,
a =/= O, 4- oo,
(Z = O,
[ min(A,B),
a------oo,
It m a x ( A , B ) ,
a=4-oc
is monotone increasing in a E ( - ao, + oo) for fixed A, B E [0, + oo] and X E (0, 1) and is also continuous in a E [ - oo,
+oo] for fixed A, B E (0, + oo)and X E (0, 1).
Proof For A, B E {0, +o~} the first assertion is verified directly. For A, B E (0, + o o ) and rational X E (0, 1),
it follows from Theorem 16 [8]. For other X, it is true by continuity of the function k --, ~,A,B(a). The second assertion is
also directly verified.
By L e m m a 1, if F(x) is a v c o n c a v e and a x _> a 2 _> - oo, then F(x) is az-concave.
Definition 5. The degree of concavity of a nonnegative quasiconcave function F is the number C F = {sup a lF(x) is
an a-concave function}.
Let us establish some properties of (x-concave functions.
T H E O R E M 1. The a-concave function F(x) ( - o o < a < oo) defined on the convex set X is continuous, locally
Lipschitz, and directionally differentiable at the points of the relative interior of the sets X (a > 0) and X' = {x E X I F(x) >
o} (a __ o).
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Proof. Represent
F(x) = I(F (x}~)'m, ~=/=0,
|exp (In F(x)),
~z = 0.
(1)
The function yl(x) = F(x) ~ (c~ > 0) is finite and concave on X, yz(x) = In F(x) is finite and concave on the set X' = {x E
X IF(x) > 0}, and y3(x) = F(x) ~ (c~ < 0) is finite and concave on X' = {x E X IF(x) > 0}. They are therefore continuous,
locally Lipschitz, and directionally differentiable at the points of the relative interior of X (c~ > 0) and X' (~ _< 0) [9]. The
same properties are preserved for the function F(x), which is a superposition of the functions y~(x) and the smooth functions
yl/~ and exp y.
COROLLARY
1 . If an a-concave function is positive in some neighborhood of the closed set X C Rn, then it
achieves its supremum on X.
Definition 6 [10]. The function f: Rn ~ R l is called generalized differentiable at the point x E R n if in some
neighborhood of the point x an upper semicontinuous (in x) set-valued mapping Gf is defined, Gf: y --, Gf(y) C R n, such that
its values Gf(y), y E R n, are nonempty bounded convex sets and in the neighborhood of x we have the decomposition f(y) =
fix) + (g, y - x) + o(x, y, g), where g E Gf(y), (,) is the scalar product, and the function o(x, y, g) satisfies the condition
lira o(x k,y ~,gk) = 0
~-,+~ IIV ~ - - ~ I1
for any sequences yk ~ x (yk ;e x), gk E Gf(yk). A function is called generalized differentiable in a domain if it is generalized
differentiable at every point of the domain. The vectors g E Gf(yk) are called the pseudogradients (generalized gradients) of
the function f at the point y.
Continuously differentiable convex and concave functions are generalized differentiable, and their gradients and
subgradients are pseudogradients. The class of generalized differentiable functions is closed under the finite operations of taking
the maximum and the minimum, superposition, and the operation of taking the expectation. A calculus of pseudogradients is
available for compound generalized differentiable functions. Generalized gradient methods, including stochastic methods, have
been developed for minimization of generalized differentiable functions.
THEOREM 2. The function F(x) c~-concave on X for c~ E ( - o~, + oo) is generalized differentiable in the interior
points of the set X (c~ > 13) or X' = {x E X IF(x) > 0} (c~ _< 0).
Proof The theorem is true because F(x) is representable in the interior points of X or X' as the superposition (1) of
convex (or concave) and smooth functions. Finite concave, convex, and smooth functions are generalized differentiable, and
their superposition is also a generalized differentiable function [10].
Thus, c~-concave functions can be optimized by the generalized gradient methods developed in [10].
Definition Z The point x* is called a stationary point for the c~-concave function F(x) on the convex set X C R n if F(x)
is continuous in the neighborhood of x* and 0 E 0F(x*) + Nx(x*), where Nx(x ) is the normal cone to the set X at the point
x E X and 0F(x) is the Clarke subdifferential [11] of the function F at the point x (it is also the pseudogradient set GF(X) for
the generalized differentiable function F(x) [10]).
THEOREM 3. Assume that the ~x-concave function F(x) is positive in some neighborhood of the convex set X C R n.
Then all the stationary points of F(x) on X, and in particular all the local maxima of F(x) on X, are global maxima and the
set of global maxima is convex.
Proof Let x* be a stationary point of the Lipschitzian function F(x) on X. Represent
OF(x)= IF(x)Z-%O1F(x)~ , c~:#O,
t F (x).O ln F (x), c~=O,
where OF is the Clarke subdifferential of the function F(x). Then the condition of stationarity of F(x) at the point x* E X (see
[11]) 0 E 0F(x*) > Nx(x* ) is equivalent to the following conditions because F(x*) > 0 : 0 E 0(1/a)F(x*) + Nx(x*), a # 0,
0 E 0 In F(x*) + Nx(x*), c¢ = 0. Thus, if x* is a stationary point on X for F(x) > 0, then it is also a stationary point for
the functions (1/~)F(x) c~(c~ ;e 0)and In F(x) (a = 0) that are concave on X. For a concave function, stationarity of x* implies
that x* is a global optimum (maximum) point. By monotonicity of the functions c@/~ and exp y, which transform the functions
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y(x) = (l/ot)F(x)" and y(x) = In F(x) to F(x), we thus conclude from the above that x* is a global optimum (maximum) point
for F(x) on X.
Now let x* be a local maximum of F(x) on X. Since F(x) is Lipschitzian in the neighborhood of X, x* is a stationary
point and therefore a global optimum point. The set of global maxima is convex by quasiconcavity of F(x) on X.
A proposition related to Theorem 3 has been proved in [12].
C A L C U L U S O F o~-CONCAVE F U N C T I O N S
The calculus of or-concave functions is constructed in what follows using two lemmas, which are a consequence of the
Minkowski and H61der inequalities [8].
L E M M A 2. Let a G (0, 1) be a real number. Then the function (x~, x 2) --, (xl ~ + x2°~)1/~ is concave for (x 1, x z)
[0, +~)2.
L E M M A 3. Let cq ..... c~k > 0, Ei=~k c~i = 1. Then the function (x l ..... Xk) --) IIi=l k xi ai is concave for (x 1..... xk)
[0, +oo)k. I f o q ' > O , a 2' > 0, oq' + a 2 ' = l, then the function (x ~, x2) -" x ~°t'l x 2'~'"- is concave for (XI, X2) E [0, .jr_00)2.
The following theorem provides a bound on the degree of concavity of the product of o~-concave functions.
T H E O R E M 4 [13]. If F~ is at-concave and F2 is o~2-concave on X and moreover either oq > 0, o~2 > 0 or a l a 2 >
0 and a~-~ + o~2-~ > 0, then Ft(x)-F2(x ) is an ~ - c o n c a v e function, where o~o = (c~ -~ + c~2-~) -~ > 0.
Proof. Consider the case oq > 0 and c~2 > 0. By assumption,
F~ (xx) ~ ((1 - - ~) Y~ (x0) c~' + )~F~ (z~)~') I/oh,
F~ (xx) ~> ((1 - - ~) F~. (Xo) ~'- ÷ ;~F~ (x~)~')'/%
where x x = (1 - X)xo + Xx 1, X ~ (0, i), Xo, x~ ff X. Hence
F 1 (xz). F 2 (xz) ~- ((1 - - ~,) ~'t (Xo)cz' 2_ $.F1 (.vl)a,) l/~zt x
(2)
×((I - - ;~) F,, (Xo) ~'~ + ~.F~ (.~:~)~'9~/~"-.
Now apply L e m m a 3, which asserts that the function f(y) = Yl a'l Y2a'~-, where y = lYx/ , ct I ' + ot 2' = l ,
~ ~t2 J
isconcave.
We thus have the inequality
(3)
Y~l "Y~ ~ (1 - - L) ty01 "V02 + ~b'ti ~'yI2,
where
~ ( o , 1).
Let
Yol = F1 (x0) ~' , Yu = F~ (x~) ~'', Vo2 = F~ (xo) ~, Y12 = Y.2 (Xl) ~'',
Substituting these values in inequality (3) and raising it to the power a.o -1 = (a I + ~2)/(oqot~, we obtain ((1 - ~)Fl(x0)~'+
hF~ (xl)~') I/~' • ((1 - - ~,) × F2 (x0)~'+~,F2 (X1)O~2)1/0~'~9 ( ( 1 - - L) (F~ (x0)" F, (xo))~°q-& ( F 1 (xt). F~ (xl))~°) %~. Combining inequalities (2) and (3), we obtain the sought proposition. For ~x1-1 + o~2-1 > 0 and oqo~2 > 0, the theorem is proved similarly
using the second part of L e m m a 3.
C O R O L L A R Y 2 [7]. Let Fi(x), i --- 1 ..... m, be nonnegative concave functions defined on the convex set X C R n.
Then F(x) = IIi=l m Fi(x ) is a (1/m)-concave function on X.
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L E M M A 4; I f F 1 and F 2 are log-concave functions, then FiCq.F2 c~2, a 1 ~ 0, a 2 >__ 0, is also a log-concave function.
Proof The function In FlCZl-F2 ~2 = a I In F 1 + a 2 In F 2 is concave as the sum of concave functions.
L E M M A 5 [13]. Let f be a nonnegative concave function defined on the convex set X and so an a-concave and
monotone function on Y; also fiX) C Y. Then the composition SOof is an a-concave function.
Proof Let Xo, x 1 E X, x x = (1 - X ) x o + ~kXl, )k ~ (0, 1). T h e n f ( x ~ ) ~ ( 1 - - K ) : ( X o ) q-K:(xx), ~ ( f ( x ~ ) ) ~ ×
((1 -- ),) f (Xo) -{- X/(x~))~ [(1 -- X) q~'- (f (Xo)) ÷ ~.q:~ (:(x~))l '/~
L E M M A 6. Let fl be an al-concave function and t'2 an az-concave function on X C R n, where a 1 ~ 1, a 2 ;> 1.
Then fl + f2 is an a-concave function with a = min(a~, a2).
Proof fl and f2 are a-concave functions, and therefore for x0 ~ X, x~ E X, X E (0, 1), and x x = (1 - )Qx0 + Xx~
we have [1 (x~) I.. (( 1 - - ;~) [: (xo) C¢÷ ~,:~ (x:)~):/~, [~ (x x) ~ (( 1 - - ~)/~,~(Xo)~ + ~,:,_(X1)~) I/~z "
Adding these inequalities and using the Minkowski inequality, which holds for a ~ 1, we obtain [~ (xa) -i- f~ (xz)
[(1 -- ~) ([1 (Xo) -+- T., (xo)) ~ -+- ~. ([~ (x0)%? f., (xa))~] ~/~.
L E M M A 7. Let f(x, y) be a-concave in x E X for any y ~ Y, Y is a set. Then the function SO(x) = infy {fix,
y) ly E Y} is a-concave on X.
Proof The function SO(x) is clearly nonnegative and finite. Take xo ~ X, xl ~ X, ~, E (0, 1), and x x = (1 - ),)xo +
Xx 1. Find a sequence of points Yk such that so(xx) = infyE¥ f(x x, y) : limk._,= f(xx, Yk)"
By c~-concavity of the function f(', Yk), we have the inequalities
[
] ((l -- ~)f(Xo,
~h) c¢ "7- ~f(Xl, Yh)°') 1/~,
~4=0,
:(xx ' y~)>~ { f(Xo, ty~)~-~.f(x~,y~) ~, ~ = o,
] rain (: (Xo, Yh), : (&, Yh)), ~ = -- co,
tI max (/~(Xo, b'i0, : (x~, Yi0), c¢ = -5 0o.
Clearly, f(xo, y) >__ so(Xo), f(xl, y) >_ so(xl). Now by monotonicity in A, B of the functions ((1 - X)A~ + ~kBt~)l/% A l-xBx,
min(A, B), and max(A, B), we have the inequalities
[ ((1 --z)q~(xo) = +
: (x~, yh) ~> { q~(xo)
q~(x0 ~,
~o~(xf) 1/~, ~:/=0,
~ = 0,
min (q~(Xo), q~(x0) ,
a ---- - - co,
max (q~ (xo), q~(x0) ,
~ = q- co.
Passing to the limit in k, we obtain the sought assertion.
In what follows, we will need the concept of a convex set-valued mapping [14].
Definition 8. The graph of the set-valued mapping H: X ~ 2 Y, where X and Y are sets, is the set gf H = {(x, y) E
X × YlY E H(x)}. The domain of definition of the mapping H is the set dom H = {xlH(x) ~ ~ } .
Each set 1" in X x Y defines a set-valued mapping from X to 2 Y with the graph P by the formula H(x) = {yl(x,
y) E F}.
Definition 9. The set-valued mapping H(x) is called convex (closed) if gf H is a convex (closed) set.
We have the following convexity criteria for H.
L E M M A 8. The set-valued mapping H: X --~ 2 Y is convex if and only if for any xo E dora H, x I (E dom H, and
), E (0, 1), we have (1 - X)H(xo) + XH(x l) C H((1 - X)xo + Xxl).
L E M M A 9. For the set-valued mapping H: X ~ 2 ¥, X C R n, Y C R m, to be convex it is necessary (when H is
closed) and sufficient that there exists a function f(x, y), x C R n, y C R m, quasiconcave in all its variables and such that
H(x) = {y C R mlf(x, y) > 0}.
L E M M A 10 [12]. Let H: X --, 2 Y be a convex set-valued mapping, let f(x, y) be a-concave on the graph of H, and
let X C R n and Y C R m be convex sets. Then the function so(x) = SUpy {fix, y) ly E H(x)} is a-concave on dom H.
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Proof L e t a ~ O, +o~. Take any xo, x I E d o m H a n d y o E H(xo), Yl E H(xl), X E (0, 1). S e t x x = (1 - X)xo +
Xxl, YX = (1 - X)yo + Xy 1. By convexity of H, Yx G H(xx). By a-concavity of the function f(x, y),
(( 1 - - ~.) ~ ( r o, To) + Xfc, ( r~, t2~) ~/c~~ f (xz, tj~ ) ~ q) (x~ ).
(4)
Take the sequences yl k and y2 t such that f(xo, yo k) --, ~O(Xo)and f(xl, y / ) ----~,(Xl) as k, 1 ~ + o0. Substitute yok and
y / i n inequality (4) and pass to the limit first in k and then in l; by continuity of the function ((1 - X)A c~ + XB~) 1/a in
A E [0, + o o ] a n d B E [0, + ~ ] , w e o b t a i n
q) (.r~) ~> [( 1 - - ~) ~ re0) + )~q0~ (x01 ~/~.
The proof for a = 0,
+ ~
is similar. For convex functions f(x, y), analogs of L e m m a 10 have been proved in [15,
16].
The following assertions show that the property of a-concavity of functions is preserved under integration in certain
cases.
T H E O R E M 5. Let H: R n ~ 2 Rm be a convex set-valued mapping such that the minimal affine subspace containing
the graph of H is (n + m)-dimensional. Assume that the function f: R n × R m --, R t is positive and a-concave on the graph
of H, where - m - 1
_ a _< + oo. Then the function
F(x)=
.I f(x,y)dy,
xE p,n, ~JE R'~
H(x~
is a'-concave on the set D H, which is the projection of the interior o f g f H on the space R n, and a ' = a/(1 + a m ) ( a ' = - o .
for a = - l/m).
For H(x) -- R n this theorem was established by Borrell [5] ( - ( n + m ) - I _ a _< + 0o) and by Brascamp and Leib
[17]. For a convex set-valued mapping H, the theorem was proved by Das Gupta [6] and Prekopa [18, 19] (ct = 0).
a - C O N C A V E M E A S U R E S IN R n
Measures, as well as functions, may have certain concavity properties. This was first noted for the Lebesgue volume
of bounded convex sets in R n ( B r u n n - M i n k o w s l d - L y u s t e r n i k theorem).
T H E O R E M 6. Let Ao, A l be nonempty Borel sets in R n, V(Ao) , V(A1) their Lebesgue volumes. Then V(Ax)I/n _>
(1 - X)V(Ao)Vn + XV(A1)~/n , where A x = (1 - X)Ao + XA 1 = {(I - X)a o + Xa I la0 E Ao, a 1 E A1} , X E (0, 1), i.e.,
the Lebesgue volume is a (1/n)-concave function of bounded convex sets in R n.
This theorem was established by Brunn [20] and Minkowski [21] for convex sets Ao, A 1 and subsequently generalized
by Lyusternik [22] to measurable sets Ao, A s. Further generalizations were developed by Brascamp and Leib [17] and by Das
Gupta [23].
We introduce a-concave measures similarly to a-concave functions.
Definition 10. The nonnegative measure P defined on the a-algebra of Lebesgue-measurable subsets of the convex set
f/ C R n is called a-concave if for any measurable sets Ao, A t C ~ and any number X C (0, 1) we have
((1 --)gP~(Ao) + ~P=(A1)) v=,
f (Ax)
~ --/=0,
P(Ao)t-~P(AO ~, c~ = O,
rain (P (Ao), P (A1)),
max (P (Ao), P (AO),
~z = - - oo,
~ = + oo,
where A x = (1 - X)Ao + XAl, P(Ax) is the lower measure of the set A x.
Here we use the lower measure P, because the set A x is not necessarily measurable for Lebesgue-measurable sets A o
and A 1.
a-Concave measures are an important particular case of g-concave measures (see [4]).
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Definition 11. The probability measure P defined on the a-algebra of Lebesgue-measurable subsets of the convex set
(5 R m is called g-concave if for each pair of convex measurable sets Ao, A 1 C ~ and any number k E (0, 1) we have
g(P{(1 - X)Ao + XA1} ) _> (l - X)g(P(Ao) ) + Xg(P(A1) ).
The relationship between a-concave functions and a-concave measures as defined above is established by the following
theorem.
T H E O R E M 7. Let f / b e an open convex set in R n and P a positive measure on ft. Assume that L is the least affine
subspace that contains ~ and let no be the dimension of L. Then the measure P is a-concave ( - oo < a < i/n o) if and only
if it has an a'-concave density function f on f~ with respect to the Lebesgue measure on L, where a ' = a/(1 - ano)
( - 1 / n o < a ' < +0o).
In the forward direction Theorem 7 was proved in [5] (see also [24]). In the reverse direction, some particular cases
were proved in [25] (n o = 1 , 0 < a < + o o ) , [ 2 6 ] (no _> 1 , 0 < a < + o o ) , [ 2 7 ] ( n o = 1, a = 0 , X = 1/2), [28] (n o =
1, a = 0, X E (0, 1)), [18] (n o >_ 1, a = 0, X E (0, 1)), [24] ( - 1/no < a -< 0); the general proof was given in [5, 17,
231.
C O R O L L A R Y 3. Assume that the function f(0) is defined and nonnegative on a convex set O C R n, a-concave on
the interior of 0 ( - 1/n < a < ~ ) , and Lebesgue-integrable on @. Then the measure P on O defined by the formula P(K) =
S K fr0)d0, K C O, is (a/(1 + an))-concave on O.
The corollary holds because the boundary of a convex set in R n is of measure zero (see [29]).
EXAMPLES OF a-CONCAVE FUNCTIONS AND MEASURES
The main classical probability distributions have a-concave density functions and are generated by a'-concave
probability measures [13].
1. Consider the density function of the nondegenerate multivariate normal distribution in Rn:
1
exp( - 1
- - m ) ) r B- ~ ( x - t n ) )
P1 (x) = _[/(2n) ~ det B
~- (x
Here B - l is an n × n positive definite matrix, m is an n-dimensional vector. The function In Pl(x) is concave, and therefore
the density function Pl(x) generates a tog-concave measure in R n [18].
2. Consider the uniform probability distribution in R n with the density function
P2 (x) = j1/v
x E 6,
l o, xCC,
where G is a convex set in R n, V(G) is the Lebesgue volume of G. The function Pz(x) is quasiconcave, but it is not a-concave
(a > 0) in R n. But on the set G the function P2(x) = const is pseudoconcave, and therefore by Corollary 3 the measure
generated by this function on the set of measurable subsets of G is 1/n-concave.
3. Consider the beta-distribution. The random variable ~ is beta-distributed with the parameters a , / 3 (a > 0,/3 >
0) if its density function has the form
(x) =
r ( a + 13) x 21(1
r
r (13)
0,
xE[0, 11,
x ~ [ 0 , 11.
Here and in what follows, l-'(t) is the gamma-function.
If o~ _> 1, /3 _> 1, then by Theorem 4 the function P3(x) is (a + /3 - 2)-2-concave on [0, 1] and the measure
generated by this function is (a + /3 - 1)-l-concave by Corollary 3.
866
4. The multivariate analog of the beta-distribution is the Dirichlet distribution. The random vector ~ = {~1 .... ,~n} is
Dirichlet-distributed with the parameters ot = {cq ..... %~}, ai > 0, /3 = 1 ..... n, if its density function has the form
.F (~1 '-I- (Y"2 +
"'" '~- ~n)Y~'
F (~1) ..- I' (~z,,)
i f x l ~ 0 . . . . . xn-11~0,
& (x) =
0
"'" "n--lv~n--I ,-/'l --
l~xa
...
X1 --
... --
Ifn_l) ~n,
xn-1~O,
otherwise.
By Theorem 4 we obtain that on the (k - 1)-dimensional simplex {x [ ~i= t n xi = 1, x i ~ 0, i = 1 .... ,n} the density function
P4(x) is (c~1 + ... + C~n)-l-concave , and the corresponding measure is (a I + ... + a n + n ) - L c o n c a v e .
5. Consider Student's distribution. The random vector ~ = (~l ..... ~n) follows the/-dimensional Student's distribution
(t-distribution) with n degrees of freedom, location vector m, and accuracy matrix T if its density function has the form
1
1 "F --~ (x - - m) T T (x - -
P~(x)=
m)J]-ct+m/2
where T is a symmetric positive definite matrix.
The density function Ps(x) is ( - 1 / ( 1 + n))-concave and the corresponding measure is ( - 1 / n ) - c o n c a v e .
6. Pareto distribution is often used in econometrics. The random vector ~ = (~l .... ,(n) follows the multivariate Pareto
distribution with the parameters o~, 01 ..... On > 0 if its density function has the form P6(x) = const(]]i=l n xi/0 i - n +
1)-(u+n), xi _> 0i ' i = 1 .... ,n. Clearly P6(x) is a ( - 1/(u + n))-concave function and the corresponding measure on the set
{x[x i _> 0i, i = 1 ..... n} is ( - c ~ - l ) - c o n c a v e .
7. The random vector ( = ((t .... ,(n) follows the multivariate F-distribution with the parameters no,n I .... ,n t > 0, gi =0 l
n i = n, if its density function is
l
i=1
t ~i/~-~ is
The function I-] *i
i=1
(1
2
~
l
j[no+
i = 1 . . . . . l.
i=I
ni -- 1
)-1
-concave and the function (n o + ~i=l / nixi] -n/2 is (--2/n)-concave.
i=l
Therefore, by Theorem 4, PT(X) is [ - ( n o / 2 + /)-1J-concave on the positive orthant and the corresponding measure
is (-2/n)-concave.
8. In conclusion, consider the Wishart distribution, which is the multivariate analog of the xZ-distribution. The density
function Ps(Y) of the Wishart distribution is defined by the formula
n --p --2
(det Y) -- 2
Ps(Y) =
2 2
n
a
exp {-- 71
Sp (C-~Y)}
detJCt 2 '
P
i=1
if Y is positive definite and Ps(Y) = 0 otherwise. Here C and Y are p × p matrices, both symmetric and positive definite.
Assume that n _> p + 2. It is shown in [30] that for any X E (0, 1) and any positive definite matrices Yo, Y1 we have the
inequality det((1 - X)Yo + XY1) > (det Yl)l-X(det Y2) k.
The function e x p { - ( 1 / 2 ) S p ( C - I Y ) } is obviously log-concave. Therefore Ps(Y) is log-concave and the measure
generated by this function is also log-concave on the (convex) set of positive definite matrices.
867
APPLICATIONS TO PROBABILITY OPTIMIZATION AND FUZZY OPTIMIZATION
Consider the stochastic programming problem
f o (x) = Po {0o E Oo [ ~o (x 0o) ~ Coi --~ max,
3¢
F~ (x) = P~ {Oi E O, I [i (x, 0~} ~> Cd ~> a~ :> O,
i = 1. . . . . m,
where Pi is the probability measure on the set O i Q R ni, 0 i E Oi, the function f is such that part of its arguments (the vector
x) are variables and the remaining arguments (the vector 0i) are statistically specified. The function Fi(x ) is defined on the set
X i of points x such that there exists 0i(x) for which f(x, 0i(x)) _> C i. The conditions x E X i, i = 0,...,m, are therefore
implicitly embedded in the formulation of the problem.
This is one of the basic formulations of stochastic programming problems and thus has many applications [31]. It is
difficult to solve because the function Fi(x ) in general is not concave and even discontinuous, despite concavity and continuity
of fi. However, the following proposition provides an opportunity for effective solution of the problem. According to this
proposition, the degree of concavity of the probability distribution function is determined by the degree of concavity of the
measure.
T H E O R E M 8. Assume that the measure P has a convex support 0 C R m and it is a-concave on O, the function f:
X x 0 --, R 1 is quasiconcave on X x @, and X is a convex set in R n. Then the probability distribution function F(x) = P{0 E
O lf(x, 0) ___ C} defined on the convex set
DF = {X E X I ~0 (x} E 0 : f (x, 0 0:)) ~ C},
is a-concave on D F.
Proof Introduce the set-valued mapping H(x) = {0 (E @ If(x, 0) > C}, x E X, dom H = {x (E X l30(x): f(x, 0(x)) _>
C} = D F. It is convex, because its graph is a convex set in Rn+m: g f H = {(x, 0) E X X O[f(x, 0) ~ C}. Take Xo, x 1 E
dom H and X E (0, 1), denote x x = (1 - X)xo + Xx 1. By convexity of H, (1 - X)H(xo) + XH(Xl) C H(xx). Hence, by c~concavity of the measure P, we obtain for a ¢ 0, _+~ that F(xh) = P{E(xx) } > P{(1 - X)H(xo) + XH(xl) } >
[(1 - X)PC'{H(xo)} + XP~{H(xl)}] l& > [(1 - X)F"(xo) + XF"(Xl)]l/% For c~ = 0, +_ oo the proof is similar.
Using Lemma 9 and Theorem 7, we can also treat this theorem as a corollary of Theorem 5. The formulation given
above emphasizes the fact that the degree of concavity of the probability distribution function is determined by the degree of
concavity of the corresponding probability measure.
Numerical algorithms for probability optimization are given in [7, 12, 13, 19, 32-34]. Theorem 8 may be used to prove
that these algorithms converge to the global extremum.
Consider the problem of fuzzy vector optimization with nonfuzzy alternatives x E X C R n, which are evaluated by
a nonfuzzy vector criterion f(x) = {fi(x)} E R m with fuzzy goals for each criterion defined by the monotone membership
functions ~,i(t). The optimal alternative is obtained by solving the following problem [35]:
% (x) = rain % (fi (x))--+ max.
i
xEX
Our results lead to conditions when this problem has a unique extremum. Specifically, assume that the functions
are (min i ai)-concave in x and therefore have a unique extremal region. The functions Sol(y) may be chosen in the form
~oo
where Ki(t) is the density function of some probability measure Pi, ~i is the corresponding random variable distributed according
to Pi. If the density function Ki(t) is such that the measure Pi is ai-concave (see Theorem 7), then by Theorem 8 the function
~oi(y) is ai-concave.
868
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869