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Generalizations of convex and related functions
369
Generalizations of convex and related functions R.N. K A U L and Surjeet K A U R Department of Mathematics, University of Delhi, Delhi.110007, India Received May 1980 Revised March 198 !
The concept oI semilocally convexity was recently introduced by Ewing [2]. This paper defines semilocally quasiconvex, semilocally pseuiloconvex and other related functions and investigates some of their properties.
away with convexity. For example, the result [3, Theorem 4.1.8] regarding the characterization of convex functions in terms of its epigraphs is slightly sharpened in the form of Theorem 1.2. The characterization [3, Theorem 9.1.3] of quasiconvexity by its level sets is slightly generalized in the form of Theorem 2.1. A few more generalizations presented in this paper pertain to some known results ([3, Theorems 9.1.4, 9.2.4, 9.3.3]; [I, Theorem 3.5.9]).
1. Notations, definitions and preliminaries Introduction The cc~ncept of convexity plays an important role in th~ optimization theory. The class of c~.,avex functions enjoys some special and imrortant properties. It is a well-known fact that ~, c,~nvex function can be characterized by the convexity of its epigraph. Various generalizations in the form of quasiconvex and pseudoconvex functions have been studied by several authors [3,4] and their properties investigated. In a recent paper Ewing [2] has defined semilocally convex and related functions and applied these to study problems in variational and control theory. The object of this paper is to define semilocally quasiconvex functions, semilocaUy pseudoconvex functions etc. and study their properties. We follow closely the approach of Mangasarian [3] in the presentation of our results. Mangasarian [3] has defined quasiconvexity at a point of a set which is not necessarily convex. It is however to be noted that many of the interesting results studied in [3] regarding quasiconvex functions do make use of the fact that the underlying set I" is convex. By defining semilocally quasiconvex functions at a point of a set which is not necessarily convex, it is possible to obtain results for sets which are locally star shaped and thus do The authors are grateful to the referee for some useful suggestions.
Let R n denote the n dimensional Euclidean space and let Ne(~ ) denote the neighbourhood of .~ ~ R ~, i.e.
Let 0 be a numerical ,function defined on a set F c R", then the right differential of 0 at ~ in the direction x - ~, denoted by (dO) + (~,x - ~) is defined as
(dO) + ( ~ , x - ~) = lim
0((l
X--.O+
(l.l) provided the limit exists. Similarly the left differential of 0 at .~ in the direction x - , ~ , denoted by ( d 0 ) - ( ~ , x - . ~ ) , is given by (dO)- (~,x - ~) :
lim x-.o-
?, (1.2)
If both right differential and left differential of 8 at X in the direction x - ,~ exist and are equal, then the common value, denoted by (dO)(',~,x - ~), is the differential of 8 at .~ in the direction x - ,~ and we write =
where
North-Holland Publishing Company European Journal of Operational Research 9 (I 982) 369-377
037%2217/82/0000-0000/$02.75 © 1982 North-Holland
~X!
'""
~X n
3 70
R.N. Kaui. S. Kaur / Generalizations of convex and related functions
is the gradient vector of O at :~. Definition 1.1 ([2,Definition 1. I ]). A subset I" c R" is said to be locally star shaped at ~ ~. F if correspending to .~ and each x ~/I', :~ a maximum po.,.itive number a ( g , x ) g I such that (1-h)~+2kx
~ F,
Proof. Let g be any point of I'. Since each f~,i = 1,2,..., m, is s~c at ,~ ~ I', therefore I" is locally star shaped at ,~ and corresponding to each x ~ I" and for each i,3 a positive number d i ( . ~ , x ) g ! such that ~((1 -X)~+
O<2k
Xx) - (1 - X ) / ~ ( ~ ) +
X/~(x),
O
!f a(.~,x) = 1 for each x ~ F, then F is said to be star shaped at ~ and if F is star shaped at each E: F, then F is said to be convex.
Taking ~(.V,x) = m i n i m u m d~(~,x) and multiplying the inequality (I.4) by a i and summing over i, we obtain
Definition 1.2 ([2, Definition 1.3]). A numerical function 0 defined on a set F c R" is said to be semilocally convex (sic) at ~ ~ I" if I" is locally star shaped at ~ (refer Definition 1.1) and if correspcnding to g and each x ~ F,:~ a positive number d ( ~ , x ) g a ( ~ , x ) such that
f ( ( l -- J~).~ + h x ) g (l -- ~ ) f ( X ) + h f ( x ) ,
~((~ -
x ) ~ + Xx) ~ (~ - x)o(.~) + x o ( x ) ,
O
(I.3)
It may, however, be noted here that there do exist sets which are locally star shaped at each of their points, but which are not convex. For exampie, the set {x I x ~ a--- {0},x 3 g 1 } is locally star shaped at each of its points but is not convex. If d(.~, x ) - a(.~, x) = 1 for each x ~ I', then 0 is said to be convex at ~. If 0 is sic at each g ~ I', then 0 is said to be sic on F. If we have strict inequality in (1.3), then 0 is said to be strictly semilocally convex (sslc) at ~ ~_ F. If d ( g , x ) - a(.~, x) - 1 for each x E F, then 0 is strictly convex at ~ and if 0 is sslc at each .~ ~ F, then 0 is said to be ssic on I'. Theorem 1.1 ([2, Theorem 1.2]). Let 0 be a ni:merical function defined on a set F C R". I f O is sic at X ~ I', then ( d 0 ) + ( . ~ , x - . ~ ) exists and O ( x ) - O ( ~ ) ; ~ ( d 0 ) + ( g , x - . ~ ) and if O is sslc at g ~ F, then O(x)-O(~)>(dO)+(g,x-g)
forx~.
Definition 1.3. An m dimensional vector function f = ( fi, f2,..-, f,,) defined on a set F c a " is said to be sic on F if for each i = 1,2 .... , m,f~ is sic on I'. Lemma 1.1. Let f = ( f n , f 2 , . . . , f,n) be an mdimensional vector function defined on a set F C RL l f f is sic on F and a~;~O,i= 1,2,...,m, then the function f = ~7=ia~fi is sic on F.
0
Theorem 1.2. For a numerical function 0 defined on a set F C R n to be sic on F, it is necessary and sufficient that its epigraph
Go = { ( x , ~ ) l x ~ r , ~ E a , 0 ( x )
~}
c a "+'
iS locally star shaped at each of its points.
l~roof. To prove sufficiency let us assume that Gois locally star shaped at each of its points. Let ~ be any point of F, then ( ~ , 0 ( ~ ) ) E Go. Also (x,0(x)) E Go for each x E F. Since Go is locally star shaped at (g,0(~)), 3 a maximum positive number d ( g , x ) g I such that (1
-
XX~,0(~)) + X(x,O(x)) ~ G0,0< X < d(~,x),
i.e. (1 - h ) ~ + ~,x E I" and
o ( ( l - x ) ~ + x ~ ) ,~ (1 - x ) o ( ~ ) + xo(.,),
0
Since 0 is sic at g,3 a maximum positive number a ( ~ , x ) g I and a positive number d ( g , x ) ~ a ( X , x ) such that
(1 - X ) ~ + X x ~ 1", 0 < X < a ( ~ , x )
R.N. Kaul, S. Kaur / Generalizations of convex and related functions
Example 1.1. Consider the function O:R-.-(0)-. R defined by
and 0((1
- -
)g).~+ )gx) g (i
- -
371
)g)O(.g) + )gO(x),
O ( x ) = x 3, x÷O.
0<)g
Here A o = { x l x ~ R . - - ( O ) , x 3 sga} is locally star shaped at each of its points but 0 is not sic on
0((! - 2k).~ 4- )gx) ~g (i -- )g)~ 4- )g~,
n-(0).
i.e.
Theorem 1.4. I f (0~)~ t is a family of numerical functions which are sic and bounded from above on a set F C Rn, then the numerical function O( x ) : sup~lOAx) is sic on r.
((1-)g).~ + )gx, (1--)g)~ + )g~) e G o, 0
The following lemma is required in the proof of the above theorem.
Thus ( 1 - h ) ( 2 , ~ ) 4 - ) g ( x , ~ ) ~ G,,
0
Hence Go is locally star shaped at (~,0(2)) and therefore at each of its points. Theorem 1.3. I f a numerical function 0 defined on a set F C R" is sic on F, then for each a ~ R, the set A., = {x/x ~ r,o(x) ~ a} is locally star shaped at each of its points but the converse is not necessarily true.
Lemma 1.2. I f {Si}~ z is a family of subsets in R n which are locally star shaped at each of their points. then their intersection S = f q , s t S ~is also locally star shaped at each of its points.
Proof. Let ~ be any arbitrary but fixed point of S and x be any other point of S. Since g,x belong to Si for each i, 3 a maximum positive number d~(~,x) g I such that (I--X)2+XxGS,,
0
:1.8) Proof. Let ,~,x E A,,, then ~,x ~ r and
O(2)~a,
O(x)~a.
The result in (1.8) also holds in the form (1.6)
Since 0 is sic at g, therefore :t a maximum positive number a(g, x) g I and a positive number d(g, x) a(g, x) such that
(l-)g)2+Xx~S,
0
where d ° ( g , x ) = inf di(.~,x). iEl
r, Hence and
( 1 - A ) X + J ~ x ~ '..
0((1 -- )g)~ + )gx) ~g (1 -- )g)0(2) + )gO(x), 0<)g < d ( ~ , x ) .
(1.7)
Using (I.6) the inequality in (1.7) yields
0 <)g < d ° ( g , x )
which completes the proof of the lemma Proof of Theorem 1.4. Since each 0~ is sic on F, therefore according to Theorem 1.2, their epigraphs
O((l --)g)g + h x ) g a ,
Therefore (i - h ) ~ + ) g x E A,, Hence A o is locally star shaped at ,~ and therefore at each of its points. However, the converse is not true (see Example 1. I).
co, =
are locally star shaped at each of their points. Therefore, by Lemma 1.2, their intersection
CI
=
iE!
=
372
R.N. Kaul, S. Kaur / Generalizations of convex and relatedfunctions
is locally star shaped at each of its points. This gives that epigraph of 0 is locally star shaped at each of its points. Hence, by Theorem 1.2, 0 is sic on F. We end this section by the following remarks. Remark 1.1. Every convex function is sic but not conversely. Example 1.2. Consider the function 0: R -- {0 } -. R defined by
If d(~, x) = a(.~, x) = ! for each x E F, then 0 is said to be quasiconvex at • E F. If 0 is slqc at each ~ E F, then 0 is said to be slqc on F. Theorem 2.1. Let 0 be a numerical function defined
on a set F C i~. Then 0 is slqc on F if and only if the set A , = {xJx E F,O(x)~a} is locally star shaped at each of its points, for every a E R. Proof. Let 0 be slqc on F and let .~, x • A,,. Then ~, x ~ F and 0(~) ~a,
O(x) = [ X2 t0
for X < 0 , for x > 0 .
Here 0 is sic on R ~- {0}" as its epigraph is locally star shaped at each of its points but is not convex. Remark 1.2. Every sslc function is sic but not conversely.
O(x) ~ a .
(2.1)
Without any loss of generality we may assume that O ( x ) ~ O ( ~ ) . Since 0 is slqc at .~, therefore 3 a maximum positive number a ( ~ , x ) ~ 1 and a posi-
tive number d ( . ~ , x ) ~ a ( ~ , x ) such that
(l-X)~+Xx~r,
0
and
0((l-x)~+x~),~0(~), 0
Using (2.1), it is immediately obvious that
O(x) = 2
( l - x ) ~ + x x ~A,., 0
forx~0.
Example 1.4. Consider the function 0: R .-- {0} ~- R defined by
Thus A,, is locally star shaped at g and therefore at each of its points. Conversely suppose that for each a E R, the set Aa = {xlx ~ F,0(x)~a} is locally star shaped at each of its points. Let ~,x E F such that O(x) g O(.~). Choosing a = O(.~) we find that
O(x)=x 2 forx~0.
xEA,
Here 0 is sslc on R---{0} but not strictly convex on R--- {0}.
Since the set A, is locally star shaped at .~, therefore :1 a maximum positive number d ( g , x ) g 1 such that
Here 0 is sic on R ,~ {0} but not sslc on R-- {0}. Remark 1.3. Every strictly convex function is sslc but not conversely.
and .~ E A,.
( l - x ) ~ + X x ~ A., 0
0(~),~0(~) } o
From the definition of A°, the last result implies that
(l-X)~+Xx~r and
0((l-x)~+x~),~a, o
0((l- x)~+Xx),~0(~), 0
373
R.N. Kaul, S. Kaur / Generalizations of concex and related functions
Theorem 2.2. Let 0 be a numerical function defined on a set F c W and let O be slqc on F. I f 0( ~) is a strict local minimum of 0 over F, then 0(~) is a strict global minimum of 0 over F. However, a local minimum of O over F is not necessarily a global minimum.
Proof. Let 0(.~) be a strict local minimum of 0 over F. Then 3 an e > 0 and a neighbourhood N,(~) of ~ such that x
jv(
)n r
Here 0 is slqc on R ~ {0,1 } and the point x = 2 is a strict local minimum which is also a strict global minimum. Remark 2.1. Every quasiconvex function is slqc but not conversely.
Example 2.3. The numerical function 0 defined by O ( x ) = {1 0
forx=0, for x ~= 0
is slqc on R but not quasiconvex on R. Let if possible 0(£) be not a s'rict global minimum of 0 over F, then :l~ E F such &at 0(~) ~ 0(.~). Since 0 is slqc at £,3 a maximum positive number a ( £ , ~ ) ~ 1 and a positive number d ( £ , 2 ) < a(£,.~) such that
Remark 2.2. Every sic function is slqc but not conversely. Example 2.4. The function 0(x) = x3, x ~ 0 defined on R--- {0} is slqc on R - - {0 } but is not sic on R'-- {0}.
and 0 ( ( 1 - },)£ + },.~)~ 0(.Y),
0 < X < d(£,.~). (2.2)
But for sufficiently small h, satisfying 0 < ~, < d(.~, .~), we have (I - ~).~+ h.f ~ N,(.~) ~ r .
(2.3)
Definition 2.2. A numerical function 0 defined on a set I" c R n is said to be semilocally explicitly quasiconuex (sleqc) at ~ ~ I" if F is locally star shaped at ~ and if for ~ and each x ~ F , 3 a positive number d ( ~ , x ) ~< a ( ~ , x ) < 1 (refer Definition 1.1) such that
The result in (2.3) implies that 0 ( ( 1 - h ) X + ~,.~)> 0 ( £ ) ,
0
O
} =*0((1 - X ) . ~ + h x )
<0(£).
which contradicts (2.2). Hence 0(.~) is a strict global minimum of 0 over I'. However, a local minimum of 0 over F may not be a global minimum of 0 over F where 0 is slqc on F (see Example 2.1).
Theorem 2.3. Let 0 be a numerical function defined
Example 2.1. Consider the function 0 : R ~ {0,1 } --, R defined by
on a set F c R". Let 0 be sleqc at ~ E I'. I f 0( ~) is a local minimum of 0 over I', then 0(~) is a global minimum of 0 over F.
O(x)=
x 0 x-
I
forx<0, for0 1.
Here 0 is slqc on R--- {0,1 } but x = ½ is a point of local minimum which is not a point of global minimum of 0.
Example 2.2. Consider the function 0:R--- {0,1 } -, R defined by for x < 0 ,
[Ix-21
for 0 < x < I, for x > 1.
If d ( £ , x ) = a ( £ , x ) = 1 for each x ~ F, then 0 is said to be explicitly quasiconvex at £. If 0 is sleqc at each £ ~ F, then 0 is said to be sleqc on F.
Proof. Let 0(,~) be a local minimum of 0 over F. Then 3 a neighbourhood N,(~) of ~ such that
x
nr
O(x).
(2.4)
Suppose that 0(.~) is not a global minimum of 0 over V. Then 3 2 E if" such that
o( t < Since 0 ;3 ~,leqc at ~, 3 a maximum positive number a(.,~,,~) ~ 1 and a positive number d(.~, 2) a(~', ~) st~ch that
374
R.N. Kaul, S. Kaur / Generalizationsof convex and relatedfunctions
and
o((I-x)~+x~)
But for sufficiently small X satisfying 0 < h < d(2, ~), we have
(1 - h)2+ X~ ~N,(2) n I'.
But it is possible to choose ~, sufficiently small such that
This gives that
(1- X)~+h~ ~N,(2)n
0((1 -- ~,)2 + X~) ~ 0 ( 2 )
r,
0
This yields the inequality
O((l-X)~+x~) ~o(~), o
O(x) ~o(~) ~÷ ~
} ~o((~ - x)~ + Xx) < o(~).
which contradicts the result in (2.6). Hence 0(2) is the unique global minimum of 0 over F. Remark 2.4. If 0 is slsqc, then 0 is slqc but not conversely. Example 2.5. The function 0 defined on the closed interval I = [0,1] ~ R by
O(x) =0 is slqc on I but not slsqc on I. Remark 2.5. Every sslc function is slsqc but not conversely.
Example 2.6. The function 0 defined on the closed interval I = [0,1] ~ R by
O
o(~) = -x~
If d(g, x) = a(2, x) = 1 for each x ~ I', then # is said to be strongly quasiconvex at 5~. If O is slsqc at each 2 ~ F, then 0 is said to be
is slsqc on I but not sslc on I.
slsqc on F.
Theorem 2.4. Let 0 be a numerical function defined on a set F C ! ~ and let 0 be slsqc at 2 ~ F. I f 0( 2) is a local minimum of O over F, then 0 ( 2 ) is the unique global minimum of 0 over F.
Proof. Since 0(2) is a local minimum of 0 over I', therefore 3 a neighbourhood N~(2) of 2 such that
x ~lv,(x) n r ~ o ( x ) ~o(~). Suppose 0(2) is not the unique global minimum, then :! a point ~ ~ F such that ~ ~ 2 and 0(.~)
o(2). Since 0 is slsqc at 2, therefore 3 a maximum positive number a(2, ~) g 1 and a positive number d(2, ~) g a(2, ~) such that
(l-h)2+X~ ~r,
0
and 0((1 - h ) ~ +
h~) < 0 ( 2 ) ,
0
Theorem 2.5. Let 0 be a numerical function defined on a set F C !~. I f the right differential (dO) + (2,x - 2 ) of 0 at the point 2 exists in the direction x - 2 for each x E F and if (i) 0 is slqc at 2 ~. F, then
o(~) ~o(~) ~(d0) + (~,~- ~) ~o, (if) 0 is sleqc at 2 E F, then
O(x)
o(~)¢o(~) } =(d0)+(~,~_~) ¢o. x~2
Proof. (i) Let x E F such that 0(x) ,~ 0(2). Since 0 is slqc at 2 ~ F, there¢~e 3 a maximum positive number a(2, x) ~ 1 and a positive number d(~, x) a(2, x) such that
(1 - X ) 2 + X x ~ I ' ,
0
R.N. Kaul, S. Kaur / Generalizations of convex and related functions
375
i.e.
and
o((~ - x ) ~ + x~) ~ o(~),
0 < k < d(.~,x). (2.7)
Also from (1.1) we have
(dO) + (~,x-~)-- lim
o((~-x)~+x.)-o(~) k
k-,O +
(2.8) Making use of (2.7) in (2.8) it immediately follows that
(dO)'~(~,x- ~) ~0.
O((l-h)~+hx)
0 <~, < e ( ~ , x ) .
Therefore 0 is slsqc at ~ E F and hence slqc at .~F. Theorem 2.7. Let 0 be a numerical function defined on a set F C R" which is locally star shaped at E F. I f the right differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and if
0(~) <0(~) =(o0) ÷ (~,x- ~) <0 for each x E F,
The proofs of the results (ii) and (iii) are on the similar lines.
then 0 is sleqc at ~ E F.
The proof is similar to the above theorem. Theorem 2.6. Let 0 be a numerical function defined on a set F C !1" which is locally star shaped at ~ F, I f the right differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and if
o(~) ~o(~)~÷~ } ~(dO)+(~,~-~) <0, then 0 is slsqc at x ~ F and therefore slqc at ~ ~ F.
3. Semilocally pseudoconvex functions (slpc)
Definition 3.1. Let 0 be a numerical function defined on a set F c R". 0 is said to be semilocally pseudoconvex (slpc) at ~ ~ F if for each x ~ F, the right differential (dO) + ( , ~ , x - ~ ) of 0 at ~ in the direction x - .~ exists and (dO) + ( ~ , x - ~ ) ~ , 0 = 0 ( x ) > ~ 0 ( ~ ) .
Proof. Let ~,x ~ F such that 0(x) ~ 0(.~),x ~ .~. Since F is locally star shaped at .~ ~ F, therefore 3 a maximum positive number a(~, x ) < 1 such that
(i-x)~+Xx~r, 0
~(~,x,X) = o ( ~ ) - 0 ( ( ~ - x ) ~ + x~). We observe that this function vanishes at h = O. The fight de:vative of ~ ( ~ , x , k ) with respect to h at k = O is =
lira
,(~,~,x)-,(~,~,o) k
k-.,O +
= lira X-,O+
o(~)-o((l-x)~+Xx)
= - (d0)÷(~,~-~)>0. Thus ¢b(~,x,k)> 0 if k is in some open interval (0,e). We take the supremum of all such e not exceeding a ( 2 , x ) in the definition of locally star shaped as e(2,x) and we get
o(~)-o((l-x)~+x~)
>o,
o
(3.1) ,i
If 0 is slpc at each ~ E F, then 0 is said to be slpc on r. If 0 is differentiable and slpc on F, then 0 is pseudoconvex on F. Remark 3.1. If 0 is pseudoconvex on F, then 0 is slpc on F but not conversely. Example 3.1. The function 0(x) = ix! defined on R is slpc on R but not pseudoconvex on R. Theorem 3.1. Let 0 be a numerical function defined on a set F C R". I f 0 is sic on F, then 0 is slpc on F. The converse is not necessarily true.
Proof. Let ~ be any arbitrary point of F. Since 0 is slc at ~, therefore by Theorem l.l, the right differentiable ( d 0 ) + ( , ~ , x - ~) of 0 at .~ in the direction x - ~ exists for each x E F and
O ( x ) - o(~) ~ (dO) ~ ( ~ , x - ~). Therefore
(d0) ÷ (.~,x- r.-) ~ 0 = 0(x) ~ 0(~).
R.N. Kaui, S. Kaur / Generalizations of convex and relatedfunctions
376
i.e. 0 is slpc at .~ and hence on F. However, the converse is not true (see Example 3.2).
Applying again the fact that 0 is slpc on F to the inequalities in (3.6) in turn, we obtain
Exampl~ 3,2, The function 0 defined on the closed interval I = [0,1] -, R by
(d0)+ ( ~ , ~ - $ )
e(x)=
-x 2 -x
<0
(3.7)
and
(dO)+ (.~,.~- ~) < O.
is slpc on I but not sic on L
But ~ - . ~ = ( - h / ( 1 - h ) ) ( ~ - . ~ ) ,
(3.8) therefore the
Theorem 3.2. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at each of its points. I f 0 is slpc on F, then 0 is sleqc on F but the converse is not true.
inequalities in (3.7) and (3.8) give a contradiction. Hence the result. However, the converse is not true (see Example 3.3).
Proof. Let 0 be slpc on F and let us suppose that 0 is not sleqc at some point ~ ~ F. Since F is locally star shaped at ~, :1 a maximum positive number a(~, x) ~ 1 such that
Example 3.3. The numerical function 0 defined on R ~ { 1 } by 0(x) = x 3 is sleqc on It-- { 1} but not slpc on R • {1}.
(1 --X)~+2kx E F,
Theorem 3.3. Let 0 be a numerical function defined
O
Since 0 is not sleqc at ~, 3 x in F such that 0(x) < 0(~) but 0(~) ~ 0 ( ~ )
(3.2)
O
From (3.2) we observe that O(Yc)>O(x) which yields (d0)+(~,x-~)
<0
F C R n which is locally star shaped at 0 is slpc at ~ and if 0(~) is a local of 0 over F, then 0( ~) is a global miniover F.
Proof. It follows from the above theorem that 0 is sleqc at .~. Applying Theorem 2.3, we find that 0(~) is a global minimum of 0 over I'.
where a?=(1--),°).f+~,°x,
on a set E F. I f minimum mum of 0
(3.3)
because 0 is slpc on I'. But
0)
Definition 3.2. Let 0 be a numerical function defined on a set I' c R n. 0 is said to be strictly semilocaily pseudoconvex (sslpc) at ~ E F if for each x • F, the fight differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and (dO)+(~,x-~)~O~O(x)>O(~)
for x ~ .
xo
(3.9)
so the inequality in (3.3) can also be expressed in the form (dO)+ ( ~ , . ~ - ~) > 0.
(3.4)
The function 0 is slpc on F and therefore the inequality in (3.4) implies that 0(~)~0(~).
(3.5)
Combining (3.2) and (3.5) we obtain 0 ( ~ ) = 0(~). Also it follows from (3.4) that there exists a positive number o sufficiently small such that 0(~) > 0(~t) = 0 ( ~ ) where
(3.6)
If 0 is sslpc at each ~ (E F, then 0 is said to be sslpc on F. If 0 is differentiable and sslpc on F, then 0 is strictly pseudoconoex on F.
Remark 3.2. If 0 is strictly pseudoconvex on F, then 0 is sslpc on F but the converse is not necessarily true. Example 3.4. The numerical function 0 defined on R as follows
O(x) = { x - I 1-x
forx~l, forxg I
is sslpc but not strictly pseudoconvex on R.
R.N. Kaui, S. Kaur / Generalizations of convex and related functions
377
Theorem 3.4. Let 0 be a numerical function defined on a set F C R n I f 0 is ssic on ~', then 0 is sslpc on F. The converse is not necessarily true.
which contradicts (3.11). Hence the result. However, the converse is not tru~ (see Example 3.6).
The proof is on the similar lines as the proof of Theorem 3.1 except that we use the second part of Theorem 1.1 instead of first part of Theorem 1.1. The following example illustrates that the converse is not true.
Example 3.6. The numerical function 0 defined on the closed interval I : [0,1] by
Example 3.5. The numerical function 0 defined on the open interval (0,1) by
O(x)=
-x
-x
is sslpc on (0,1) but not sslc on (0,1). Theorem 3.5. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at each of its points. I f 0 is ssipc on F, then 0 is slsqc on F but the converse is not true.
Proof. Let 0 be sslpc on F and let us suppose that 0 is not slsqc at some point g E F. Since F is locally star shaped at g, 3 a maximum positive number a(.~, x) g 1 such that (l-x)~+xx~r,
O(x) =
is slsqc on I but not sslpc on I. Theorem 3.6. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at ~ E F . I f 0 is sslpc at ~ and if 0(~) i~ a local minimum of O over F, then 0( ~) is the unique global minimum of 0 over F. Proof. It follows from the above theorem that 0 is slsqc at .~. Applying Theorem 2.4, we find that 0(.~) is the unique global minimum of 0 over F.
Remark 3.3. Every sslpc function is slpc but not conversely. Example 3.7. The numerical function 0 defined on R as follows
0 < ~,
Since 0 is not slsqc at .~, 3 x E F such that 0(x) ~0(~),
l--x
x÷~
f o r x ~ 1, for - l g x g l , for x g - l
is slpc but not sslpc on R.
but 0 ( ~ ) ~0(.~)
(3.10)
where .~=(1-h°).g+2k°x,
0
From (3.10) we observe that O(~¢)~O(x) which yields (dO) + ( ~ , x - . ~ ) < 0 because O is sslpc on F. But x - .~ = (1 - h°)(x - .~) so the last inequality can also be expressed in the form (dO) + ( ~ , x - g ) < 0.
But
so we can write (3.12) in the form
A numerical function 0 defined on a set F c R" is said to be semilocally concave if - 0 is semilocally convex. The corresponding definitions in case of sslc, slqc, sleqc, slsqc, slpc, sslpc follow easily. Some of the results in the form of lemmas and theorems have their counterparts in terms of these functions.
(3.11)
The function 0 is sslpc on I', therefore the inequality (3.10) gives (dO) + (,'~,X-- .~) < 0.
4. Conclusion
(3.12)
References [i] M.S. Bazaraa and C.M. Shetty, Nonlinear Programming Theory and Algorithms (Wiley, New York, 1979). [2] G.M. Ewing, Sufficient conditions for global minima of suitable convex functionals from variational and control theory, SIAM Rev. 19 (2) (1977). [3] O.L. Mangasarian, Nonlinear Programming (McGraw-Hill, New York, 1969). [4] ol. Ponstein, Seven kinds of convexity, SIAM Rev. 9 (1967) !!5-119.
Generalizations of convex and related functions R.N. K A U L and Surjeet K A U R Department of Mathematics, University of Delhi, Delhi.110007, India Received May 1980 Revised March 198 !
The concept oI semilocally convexity was recently introduced by Ewing [2]. This paper defines semilocally quasiconvex, semilocally pseuiloconvex and other related functions and investigates some of their properties.
away with convexity. For example, the result [3, Theorem 4.1.8] regarding the characterization of convex functions in terms of its epigraphs is slightly sharpened in the form of Theorem 1.2. The characterization [3, Theorem 9.1.3] of quasiconvexity by its level sets is slightly generalized in the form of Theorem 2.1. A few more generalizations presented in this paper pertain to some known results ([3, Theorems 9.1.4, 9.2.4, 9.3.3]; [I, Theorem 3.5.9]).
1. Notations, definitions and preliminaries Introduction The cc~ncept of convexity plays an important role in th~ optimization theory. The class of c~.,avex functions enjoys some special and imrortant properties. It is a well-known fact that ~, c,~nvex function can be characterized by the convexity of its epigraph. Various generalizations in the form of quasiconvex and pseudoconvex functions have been studied by several authors [3,4] and their properties investigated. In a recent paper Ewing [2] has defined semilocally convex and related functions and applied these to study problems in variational and control theory. The object of this paper is to define semilocally quasiconvex functions, semilocaUy pseudoconvex functions etc. and study their properties. We follow closely the approach of Mangasarian [3] in the presentation of our results. Mangasarian [3] has defined quasiconvexity at a point of a set which is not necessarily convex. It is however to be noted that many of the interesting results studied in [3] regarding quasiconvex functions do make use of the fact that the underlying set I" is convex. By defining semilocally quasiconvex functions at a point of a set which is not necessarily convex, it is possible to obtain results for sets which are locally star shaped and thus do The authors are grateful to the referee for some useful suggestions.
Let R n denote the n dimensional Euclidean space and let Ne(~ ) denote the neighbourhood of .~ ~ R ~, i.e.
Let 0 be a numerical ,function defined on a set F c R", then the right differential of 0 at ~ in the direction x - ~, denoted by (dO) + (~,x - ~) is defined as
(dO) + ( ~ , x - ~) = lim
0((l
X--.O+
(l.l) provided the limit exists. Similarly the left differential of 0 at .~ in the direction x - , ~ , denoted by ( d 0 ) - ( ~ , x - . ~ ) , is given by (dO)- (~,x - ~) :
lim x-.o-
?, (1.2)
If both right differential and left differential of 8 at X in the direction x - ,~ exist and are equal, then the common value, denoted by (dO)(',~,x - ~), is the differential of 8 at .~ in the direction x - ,~ and we write =
where
North-Holland Publishing Company European Journal of Operational Research 9 (I 982) 369-377
037%2217/82/0000-0000/$02.75 © 1982 North-Holland
~X!
'""
~X n
3 70
R.N. Kaui. S. Kaur / Generalizations of convex and related functions
is the gradient vector of O at :~. Definition 1.1 ([2,Definition 1. I ]). A subset I" c R" is said to be locally star shaped at ~ ~. F if correspending to .~ and each x ~/I', :~ a maximum po.,.itive number a ( g , x ) g I such that (1-h)~+2kx
~ F,
Proof. Let g be any point of I'. Since each f~,i = 1,2,..., m, is s~c at ,~ ~ I', therefore I" is locally star shaped at ,~ and corresponding to each x ~ I" and for each i,3 a positive number d i ( . ~ , x ) g ! such that ~((1 -X)~+
O<2k
Xx) - (1 - X ) / ~ ( ~ ) +
X/~(x),
O
!f a(.~,x) = 1 for each x ~ F, then F is said to be star shaped at ~ and if F is star shaped at each E: F, then F is said to be convex.
Taking ~(.V,x) = m i n i m u m d~(~,x) and multiplying the inequality (I.4) by a i and summing over i, we obtain
Definition 1.2 ([2, Definition 1.3]). A numerical function 0 defined on a set F c R" is said to be semilocally convex (sic) at ~ ~ I" if I" is locally star shaped at ~ (refer Definition 1.1) and if correspcnding to g and each x ~ F,:~ a positive number d ( ~ , x ) g a ( ~ , x ) such that
f ( ( l -- J~).~ + h x ) g (l -- ~ ) f ( X ) + h f ( x ) ,
~((~ -
x ) ~ + Xx) ~ (~ - x)o(.~) + x o ( x ) ,
O
(I.3)
It may, however, be noted here that there do exist sets which are locally star shaped at each of their points, but which are not convex. For exampie, the set {x I x ~ a--- {0},x 3 g 1 } is locally star shaped at each of its points but is not convex. If d(.~, x ) - a(.~, x) = 1 for each x ~ I', then 0 is said to be convex at ~. If 0 is sic at each g ~ I', then 0 is said to be sic on F. If we have strict inequality in (1.3), then 0 is said to be strictly semilocally convex (sslc) at ~ ~_ F. If d ( g , x ) - a(.~, x) - 1 for each x E F, then 0 is strictly convex at ~ and if 0 is sslc at each .~ ~ F, then 0 is said to be ssic on I'. Theorem 1.1 ([2, Theorem 1.2]). Let 0 be a ni:merical function defined on a set F C R". I f O is sic at X ~ I', then ( d 0 ) + ( . ~ , x - . ~ ) exists and O ( x ) - O ( ~ ) ; ~ ( d 0 ) + ( g , x - . ~ ) and if O is sslc at g ~ F, then O(x)-O(~)>(dO)+(g,x-g)
forx~.
Definition 1.3. An m dimensional vector function f = ( fi, f2,..-, f,,) defined on a set F c a " is said to be sic on F if for each i = 1,2 .... , m,f~ is sic on I'. Lemma 1.1. Let f = ( f n , f 2 , . . . , f,n) be an mdimensional vector function defined on a set F C RL l f f is sic on F and a~;~O,i= 1,2,...,m, then the function f = ~7=ia~fi is sic on F.
0
Theorem 1.2. For a numerical function 0 defined on a set F C R n to be sic on F, it is necessary and sufficient that its epigraph
Go = { ( x , ~ ) l x ~ r , ~ E a , 0 ( x )
~}
c a "+'
iS locally star shaped at each of its points.
l~roof. To prove sufficiency let us assume that Gois locally star shaped at each of its points. Let ~ be any point of F, then ( ~ , 0 ( ~ ) ) E Go. Also (x,0(x)) E Go for each x E F. Since Go is locally star shaped at (g,0(~)), 3 a maximum positive number d ( g , x ) g I such that (1
-
XX~,0(~)) + X(x,O(x)) ~ G0,0< X < d(~,x),
i.e. (1 - h ) ~ + ~,x E I" and
o ( ( l - x ) ~ + x ~ ) ,~ (1 - x ) o ( ~ ) + xo(.,),
0
Since 0 is sic at g,3 a maximum positive number a ( ~ , x ) g I and a positive number d ( g , x ) ~ a ( X , x ) such that
(1 - X ) ~ + X x ~ 1", 0 < X < a ( ~ , x )
R.N. Kaul, S. Kaur / Generalizations of convex and related functions
Example 1.1. Consider the function O:R-.-(0)-. R defined by
and 0((1
- -
)g).~+ )gx) g (i
- -
371
)g)O(.g) + )gO(x),
O ( x ) = x 3, x÷O.
0<)g
Here A o = { x l x ~ R . - - ( O ) , x 3 sga} is locally star shaped at each of its points but 0 is not sic on
0((! - 2k).~ 4- )gx) ~g (i -- )g)~ 4- )g~,
n-(0).
i.e.
Theorem 1.4. I f (0~)~ t is a family of numerical functions which are sic and bounded from above on a set F C Rn, then the numerical function O( x ) : sup~lOAx) is sic on r.
((1-)g).~ + )gx, (1--)g)~ + )g~) e G o, 0
The following lemma is required in the proof of the above theorem.
Thus ( 1 - h ) ( 2 , ~ ) 4 - ) g ( x , ~ ) ~ G,,
0
Hence Go is locally star shaped at (~,0(2)) and therefore at each of its points. Theorem 1.3. I f a numerical function 0 defined on a set F C R" is sic on F, then for each a ~ R, the set A., = {x/x ~ r,o(x) ~ a} is locally star shaped at each of its points but the converse is not necessarily true.
Lemma 1.2. I f {Si}~ z is a family of subsets in R n which are locally star shaped at each of their points. then their intersection S = f q , s t S ~is also locally star shaped at each of its points.
Proof. Let ~ be any arbitrary but fixed point of S and x be any other point of S. Since g,x belong to Si for each i, 3 a maximum positive number d~(~,x) g I such that (I--X)2+XxGS,,
0
:1.8) Proof. Let ,~,x E A,,, then ~,x ~ r and
O(2)~a,
O(x)~a.
The result in (1.8) also holds in the form (1.6)
Since 0 is sic at g, therefore :t a maximum positive number a(g, x) g I and a positive number d(g, x) a(g, x) such that
(l-)g)2+Xx~S,
0
where d ° ( g , x ) = inf di(.~,x). iEl
r, Hence and
( 1 - A ) X + J ~ x ~ '..
0((1 -- )g)~ + )gx) ~g (1 -- )g)0(2) + )gO(x), 0<)g < d ( ~ , x ) .
(1.7)
Using (I.6) the inequality in (1.7) yields
0 <)g < d ° ( g , x )
which completes the proof of the lemma Proof of Theorem 1.4. Since each 0~ is sic on F, therefore according to Theorem 1.2, their epigraphs
O((l --)g)g + h x ) g a ,
Therefore (i - h ) ~ + ) g x E A,, Hence A o is locally star shaped at ,~ and therefore at each of its points. However, the converse is not true (see Example 1. I).
co, =
are locally star shaped at each of their points. Therefore, by Lemma 1.2, their intersection
CI
=
iE!
=
372
R.N. Kaul, S. Kaur / Generalizations of convex and relatedfunctions
is locally star shaped at each of its points. This gives that epigraph of 0 is locally star shaped at each of its points. Hence, by Theorem 1.2, 0 is sic on F. We end this section by the following remarks. Remark 1.1. Every convex function is sic but not conversely. Example 1.2. Consider the function 0: R -- {0 } -. R defined by
If d(~, x) = a(.~, x) = ! for each x E F, then 0 is said to be quasiconvex at • E F. If 0 is slqc at each ~ E F, then 0 is said to be slqc on F. Theorem 2.1. Let 0 be a numerical function defined
on a set F C i~. Then 0 is slqc on F if and only if the set A , = {xJx E F,O(x)~a} is locally star shaped at each of its points, for every a E R. Proof. Let 0 be slqc on F and let .~, x • A,,. Then ~, x ~ F and 0(~) ~a,
O(x) = [ X2 t0
for X < 0 , for x > 0 .
Here 0 is sic on R ~- {0}" as its epigraph is locally star shaped at each of its points but is not convex. Remark 1.2. Every sslc function is sic but not conversely.
O(x) ~ a .
(2.1)
Without any loss of generality we may assume that O ( x ) ~ O ( ~ ) . Since 0 is slqc at .~, therefore 3 a maximum positive number a ( ~ , x ) ~ 1 and a posi-
tive number d ( . ~ , x ) ~ a ( ~ , x ) such that
(l-X)~+Xx~r,
0
and
0((l-x)~+x~),~0(~), 0
Using (2.1), it is immediately obvious that
O(x) = 2
( l - x ) ~ + x x ~A,., 0
forx~0.
Example 1.4. Consider the function 0: R .-- {0} ~- R defined by
Thus A,, is locally star shaped at g and therefore at each of its points. Conversely suppose that for each a E R, the set Aa = {xlx ~ F,0(x)~a} is locally star shaped at each of its points. Let ~,x E F such that O(x) g O(.~). Choosing a = O(.~) we find that
O(x)=x 2 forx~0.
xEA,
Here 0 is sslc on R---{0} but not strictly convex on R--- {0}.
Since the set A, is locally star shaped at .~, therefore :1 a maximum positive number d ( g , x ) g 1 such that
Here 0 is sic on R ,~ {0} but not sslc on R-- {0}. Remark 1.3. Every strictly convex function is sslc but not conversely.
and .~ E A,.
( l - x ) ~ + X x ~ A., 0
0(~),~0(~) } o
From the definition of A°, the last result implies that
(l-X)~+Xx~r and
0((l-x)~+x~),~a, o
0((l- x)~+Xx),~0(~), 0
373
R.N. Kaul, S. Kaur / Generalizations of concex and related functions
Theorem 2.2. Let 0 be a numerical function defined on a set F c W and let O be slqc on F. I f 0( ~) is a strict local minimum of 0 over F, then 0(~) is a strict global minimum of 0 over F. However, a local minimum of O over F is not necessarily a global minimum.
Proof. Let 0(.~) be a strict local minimum of 0 over F. Then 3 an e > 0 and a neighbourhood N,(~) of ~ such that x
jv(
)n r
Here 0 is slqc on R ~ {0,1 } and the point x = 2 is a strict local minimum which is also a strict global minimum. Remark 2.1. Every quasiconvex function is slqc but not conversely.
Example 2.3. The numerical function 0 defined by O ( x ) = {1 0
forx=0, for x ~= 0
is slqc on R but not quasiconvex on R. Let if possible 0(£) be not a s'rict global minimum of 0 over F, then :l~ E F such &at 0(~) ~ 0(.~). Since 0 is slqc at £,3 a maximum positive number a ( £ , ~ ) ~ 1 and a positive number d ( £ , 2 ) < a(£,.~) such that
Remark 2.2. Every sic function is slqc but not conversely. Example 2.4. The function 0(x) = x3, x ~ 0 defined on R--- {0} is slqc on R - - {0 } but is not sic on R'-- {0}.
and 0 ( ( 1 - },)£ + },.~)~ 0(.Y),
0 < X < d(£,.~). (2.2)
But for sufficiently small h, satisfying 0 < ~, < d(.~, .~), we have (I - ~).~+ h.f ~ N,(.~) ~ r .
(2.3)
Definition 2.2. A numerical function 0 defined on a set I" c R n is said to be semilocally explicitly quasiconuex (sleqc) at ~ ~ I" if F is locally star shaped at ~ and if for ~ and each x ~ F , 3 a positive number d ( ~ , x ) ~< a ( ~ , x ) < 1 (refer Definition 1.1) such that
The result in (2.3) implies that 0 ( ( 1 - h ) X + ~,.~)> 0 ( £ ) ,
0
O
} =*0((1 - X ) . ~ + h x )
<0(£).
which contradicts (2.2). Hence 0(.~) is a strict global minimum of 0 over I'. However, a local minimum of 0 over F may not be a global minimum of 0 over F where 0 is slqc on F (see Example 2.1).
Theorem 2.3. Let 0 be a numerical function defined
Example 2.1. Consider the function 0 : R ~ {0,1 } --, R defined by
on a set F c R". Let 0 be sleqc at ~ E I'. I f 0( ~) is a local minimum of 0 over I', then 0(~) is a global minimum of 0 over F.
O(x)=
x 0 x-
I
forx<0, for0
Here 0 is slqc on R--- {0,1 } but x = ½ is a point of local minimum which is not a point of global minimum of 0.
Example 2.2. Consider the function 0:R--- {0,1 } -, R defined by for x < 0 ,
[Ix-21
for 0 < x < I, for x > 1.
If d ( £ , x ) = a ( £ , x ) = 1 for each x ~ F, then 0 is said to be explicitly quasiconvex at £. If 0 is sleqc at each £ ~ F, then 0 is said to be sleqc on F.
Proof. Let 0(,~) be a local minimum of 0 over F. Then 3 a neighbourhood N,(~) of ~ such that
x
nr
O(x).
(2.4)
Suppose that 0(.~) is not a global minimum of 0 over V. Then 3 2 E if" such that
o( t < Since 0 ;3 ~,leqc at ~, 3 a maximum positive number a(.,~,,~) ~ 1 and a positive number d(.~, 2) a(~', ~) st~ch that
374
R.N. Kaul, S. Kaur / Generalizationsof convex and relatedfunctions
and
o((I-x)~+x~)
But for sufficiently small X satisfying 0 < h < d(2, ~), we have
(1 - h)2+ X~ ~N,(2) n I'.
But it is possible to choose ~, sufficiently small such that
This gives that
(1- X)~+h~ ~N,(2)n
0((1 -- ~,)2 + X~) ~ 0 ( 2 )
r,
0
This yields the inequality
O((l-X)~+x~) ~o(~), o
O(x) ~o(~) ~÷ ~
} ~o((~ - x)~ + Xx) < o(~).
which contradicts the result in (2.6). Hence 0(2) is the unique global minimum of 0 over F. Remark 2.4. If 0 is slsqc, then 0 is slqc but not conversely. Example 2.5. The function 0 defined on the closed interval I = [0,1] ~ R by
O(x) =0 is slqc on I but not slsqc on I. Remark 2.5. Every sslc function is slsqc but not conversely.
Example 2.6. The function 0 defined on the closed interval I = [0,1] ~ R by
O
o(~) = -x~
If d(g, x) = a(2, x) = 1 for each x ~ I', then # is said to be strongly quasiconvex at 5~. If O is slsqc at each 2 ~ F, then 0 is said to be
is slsqc on I but not sslc on I.
slsqc on F.
Theorem 2.4. Let 0 be a numerical function defined on a set F C ! ~ and let 0 be slsqc at 2 ~ F. I f 0( 2) is a local minimum of O over F, then 0 ( 2 ) is the unique global minimum of 0 over F.
Proof. Since 0(2) is a local minimum of 0 over I', therefore 3 a neighbourhood N~(2) of 2 such that
x ~lv,(x) n r ~ o ( x ) ~o(~). Suppose 0(2) is not the unique global minimum, then :! a point ~ ~ F such that ~ ~ 2 and 0(.~)
o(2). Since 0 is slsqc at 2, therefore 3 a maximum positive number a(2, ~) g 1 and a positive number d(2, ~) g a(2, ~) such that
(l-h)2+X~ ~r,
0
and 0((1 - h ) ~ +
h~) < 0 ( 2 ) ,
0
Theorem 2.5. Let 0 be a numerical function defined on a set F C !~. I f the right differential (dO) + (2,x - 2 ) of 0 at the point 2 exists in the direction x - 2 for each x E F and if (i) 0 is slqc at 2 ~. F, then
o(~) ~o(~) ~(d0) + (~,~- ~) ~o, (if) 0 is sleqc at 2 E F, then
O(x)
o(~)¢o(~) } =(d0)+(~,~_~) ¢o. x~2
Proof. (i) Let x E F such that 0(x) ,~ 0(2). Since 0 is slqc at 2 ~ F, there¢~e 3 a maximum positive number a(2, x) ~ 1 and a positive number d(~, x) a(2, x) such that
(1 - X ) 2 + X x ~ I ' ,
0
R.N. Kaul, S. Kaur / Generalizations of convex and related functions
375
i.e.
and
o((~ - x ) ~ + x~) ~ o(~),
0 < k < d(.~,x). (2.7)
Also from (1.1) we have
(dO) + (~,x-~)-- lim
o((~-x)~+x.)-o(~) k
k-,O +
(2.8) Making use of (2.7) in (2.8) it immediately follows that
(dO)'~(~,x- ~) ~0.
O((l-h)~+hx)
0 <~, < e ( ~ , x ) .
Therefore 0 is slsqc at ~ E F and hence slqc at .~F. Theorem 2.7. Let 0 be a numerical function defined on a set F C R" which is locally star shaped at E F. I f the right differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and if
0(~) <0(~) =(o0) ÷ (~,x- ~) <0 for each x E F,
The proofs of the results (ii) and (iii) are on the similar lines.
then 0 is sleqc at ~ E F.
The proof is similar to the above theorem. Theorem 2.6. Let 0 be a numerical function defined on a set F C !1" which is locally star shaped at ~ F, I f the right differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and if
o(~) ~o(~)~÷~ } ~(dO)+(~,~-~) <0, then 0 is slsqc at x ~ F and therefore slqc at ~ ~ F.
3. Semilocally pseudoconvex functions (slpc)
Definition 3.1. Let 0 be a numerical function defined on a set F c R". 0 is said to be semilocally pseudoconvex (slpc) at ~ ~ F if for each x ~ F, the right differential (dO) + ( , ~ , x - ~ ) of 0 at ~ in the direction x - .~ exists and (dO) + ( ~ , x - ~ ) ~ , 0 = 0 ( x ) > ~ 0 ( ~ ) .
Proof. Let ~,x ~ F such that 0(x) ~ 0(.~),x ~ .~. Since F is locally star shaped at .~ ~ F, therefore 3 a maximum positive number a(~, x ) < 1 such that
(i-x)~+Xx~r, 0
~(~,x,X) = o ( ~ ) - 0 ( ( ~ - x ) ~ + x~). We observe that this function vanishes at h = O. The fight de:vative of ~ ( ~ , x , k ) with respect to h at k = O is =
lira
,(~,~,x)-,(~,~,o) k
k-.,O +
= lira X-,O+
o(~)-o((l-x)~+Xx)
= - (d0)÷(~,~-~)>0. Thus ¢b(~,x,k)> 0 if k is in some open interval (0,e). We take the supremum of all such e not exceeding a ( 2 , x ) in the definition of locally star shaped as e(2,x) and we get
o(~)-o((l-x)~+x~)
>o,
o
(3.1) ,i
If 0 is slpc at each ~ E F, then 0 is said to be slpc on r. If 0 is differentiable and slpc on F, then 0 is pseudoconvex on F. Remark 3.1. If 0 is pseudoconvex on F, then 0 is slpc on F but not conversely. Example 3.1. The function 0(x) = ix! defined on R is slpc on R but not pseudoconvex on R. Theorem 3.1. Let 0 be a numerical function defined on a set F C R". I f 0 is sic on F, then 0 is slpc on F. The converse is not necessarily true.
Proof. Let ~ be any arbitrary point of F. Since 0 is slc at ~, therefore by Theorem l.l, the right differentiable ( d 0 ) + ( , ~ , x - ~) of 0 at .~ in the direction x - ~ exists for each x E F and
O ( x ) - o(~) ~ (dO) ~ ( ~ , x - ~). Therefore
(d0) ÷ (.~,x- r.-) ~ 0 = 0(x) ~ 0(~).
R.N. Kaui, S. Kaur / Generalizations of convex and relatedfunctions
376
i.e. 0 is slpc at .~ and hence on F. However, the converse is not true (see Example 3.2).
Applying again the fact that 0 is slpc on F to the inequalities in (3.6) in turn, we obtain
Exampl~ 3,2, The function 0 defined on the closed interval I = [0,1] -, R by
(d0)+ ( ~ , ~ - $ )
e(x)=
-x 2 -x
<0
(3.7)
and
(dO)+ (.~,.~- ~) < O.
is slpc on I but not sic on L
But ~ - . ~ = ( - h / ( 1 - h ) ) ( ~ - . ~ ) ,
(3.8) therefore the
Theorem 3.2. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at each of its points. I f 0 is slpc on F, then 0 is sleqc on F but the converse is not true.
inequalities in (3.7) and (3.8) give a contradiction. Hence the result. However, the converse is not true (see Example 3.3).
Proof. Let 0 be slpc on F and let us suppose that 0 is not sleqc at some point ~ ~ F. Since F is locally star shaped at ~, :1 a maximum positive number a(~, x) ~ 1 such that
Example 3.3. The numerical function 0 defined on R ~ { 1 } by 0(x) = x 3 is sleqc on It-- { 1} but not slpc on R • {1}.
(1 --X)~+2kx E F,
Theorem 3.3. Let 0 be a numerical function defined
O
Since 0 is not sleqc at ~, 3 x in F such that 0(x) < 0(~) but 0(~) ~ 0 ( ~ )
(3.2)
O
From (3.2) we observe that O(Yc)>O(x) which yields (d0)+(~,x-~)
<0
F C R n which is locally star shaped at 0 is slpc at ~ and if 0(~) is a local of 0 over F, then 0( ~) is a global miniover F.
Proof. It follows from the above theorem that 0 is sleqc at .~. Applying Theorem 2.3, we find that 0(~) is a global minimum of 0 over I'.
where a?=(1--),°).f+~,°x,
on a set E F. I f minimum mum of 0
(3.3)
because 0 is slpc on I'. But
0)
Definition 3.2. Let 0 be a numerical function defined on a set I' c R n. 0 is said to be strictly semilocaily pseudoconvex (sslpc) at ~ E F if for each x • F, the fight differential (dO) + ( ~ , x - ~) of 0 at ~ in the direction x - ~ exists and (dO)+(~,x-~)~O~O(x)>O(~)
for x ~ .
xo
(3.9)
so the inequality in (3.3) can also be expressed in the form (dO)+ ( ~ , . ~ - ~) > 0.
(3.4)
The function 0 is slpc on F and therefore the inequality in (3.4) implies that 0(~)~0(~).
(3.5)
Combining (3.2) and (3.5) we obtain 0 ( ~ ) = 0(~). Also it follows from (3.4) that there exists a positive number o sufficiently small such that 0(~) > 0(~t) = 0 ( ~ ) where
(3.6)
If 0 is sslpc at each ~ (E F, then 0 is said to be sslpc on F. If 0 is differentiable and sslpc on F, then 0 is strictly pseudoconoex on F.
Remark 3.2. If 0 is strictly pseudoconvex on F, then 0 is sslpc on F but the converse is not necessarily true. Example 3.4. The numerical function 0 defined on R as follows
O(x) = { x - I 1-x
forx~l, forxg I
is sslpc but not strictly pseudoconvex on R.
R.N. Kaui, S. Kaur / Generalizations of convex and related functions
377
Theorem 3.4. Let 0 be a numerical function defined on a set F C R n I f 0 is ssic on ~', then 0 is sslpc on F. The converse is not necessarily true.
which contradicts (3.11). Hence the result. However, the converse is not tru~ (see Example 3.6).
The proof is on the similar lines as the proof of Theorem 3.1 except that we use the second part of Theorem 1.1 instead of first part of Theorem 1.1. The following example illustrates that the converse is not true.
Example 3.6. The numerical function 0 defined on the closed interval I : [0,1] by
Example 3.5. The numerical function 0 defined on the open interval (0,1) by
O(x)=
-x
-x
is sslpc on (0,1) but not sslc on (0,1). Theorem 3.5. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at each of its points. I f 0 is ssipc on F, then 0 is slsqc on F but the converse is not true.
Proof. Let 0 be sslpc on F and let us suppose that 0 is not slsqc at some point g E F. Since F is locally star shaped at g, 3 a maximum positive number a(.~, x) g 1 such that (l-x)~+xx~r,
O(x) =
is slsqc on I but not sslpc on I. Theorem 3.6. Let 0 be a numerical function defined on a set F C R n which is locally star shaped at ~ E F . I f 0 is sslpc at ~ and if 0(~) i~ a local minimum of O over F, then 0( ~) is the unique global minimum of 0 over F. Proof. It follows from the above theorem that 0 is slsqc at .~. Applying Theorem 2.4, we find that 0(.~) is the unique global minimum of 0 over F.
Remark 3.3. Every sslpc function is slpc but not conversely. Example 3.7. The numerical function 0 defined on R as follows
0 < ~,
Since 0 is not slsqc at .~, 3 x E F such that 0(x) ~0(~),
l--x
x÷~
f o r x ~ 1, for - l g x g l , for x g - l
is slpc but not sslpc on R.
but 0 ( ~ ) ~0(.~)
(3.10)
where .~=(1-h°).g+2k°x,
0
From (3.10) we observe that O(~¢)~O(x) which yields (dO) + ( ~ , x - . ~ ) < 0 because O is sslpc on F. But x - .~ = (1 - h°)(x - .~) so the last inequality can also be expressed in the form (dO) + ( ~ , x - g ) < 0.
But
so we can write (3.12) in the form
A numerical function 0 defined on a set F c R" is said to be semilocally concave if - 0 is semilocally convex. The corresponding definitions in case of sslc, slqc, sleqc, slsqc, slpc, sslpc follow easily. Some of the results in the form of lemmas and theorems have their counterparts in terms of these functions.
(3.11)
The function 0 is sslpc on I', therefore the inequality (3.10) gives (dO) + (,'~,X-- .~) < 0.
4. Conclusion
(3.12)
References [i] M.S. Bazaraa and C.M. Shetty, Nonlinear Programming Theory and Algorithms (Wiley, New York, 1979). [2] G.M. Ewing, Sufficient conditions for global minima of suitable convex functionals from variational and control theory, SIAM Rev. 19 (2) (1977). [3] O.L. Mangasarian, Nonlinear Programming (McGraw-Hill, New York, 1969). [4] ol. Ponstein, Seven kinds of convexity, SIAM Rev. 9 (1967) !!5-119.