STOCHASTIC

PSEUDO-CONVEX FUNCTIONS 283
PROPERTY 1. Let C be convex. If 8(x) is convex on C, then 6(x) is pseudoconvex
in C, but not conversely.
Proof. If 8(1) is convex on C, then by (1.4),
(x 2 - x'YvJix 1 ) & 0 implies B(x 2 ) S 9(x'),
which is precisely (1.1). That the converse is not necessarily true can be
seen from the example
$(x) = x + x\ x (= E\
which is pseudo-convex on A' 1 but not convex. 1
PROPERTY 2. Let C be convex. If B(x) is pseudo-convex on C, then B(x) is
strictly quasi-convex (and hence quasi-convex) on C, but not conversely.
Proof. Let 0(x) be pseudo-convex on C. We shall assume that 9(x) is not
strictly quasi-convex on C and show that this leads to a contradiction. If
B(x) is not strictly quasi-convex on C, then it follows from (1.7) that there
exist x l ?* x 2 in C such that
(2.1) 0{x 2 ) < 8(x y ),
and
(2.2) B(x) £ B{x'),
for some iCL, where
(2.3) L = \x | x = Xx 1 + (1 - A)x 2 , 0 < A < l\.
Hence there exists an x £ L such that
(2.4) 6(x) = max 8{x),
where
(2.5) L = LU\x l ,x 2 }.
Now define
(2.6) /(A) = «((1 - \)x l + Xx 2 ), 0SJS1.
Hence
(2.7) 9(x) = /(A),
where
(2.8) x = (1 - X)z' + Xx 2 , 0 < X < 1.
1 To see that x + a: 3 is pseudo-convex, note that Vxd(x) = 1 + 3x 2 > 0. Hence
(l - i^'V^x") jg 0 implies that x & x° and x 3 i (x')', and thus
$(x) - 9(3°) = Or + x') - (i° + (x°)') g 0.
CONVEXITY AND THE KUHN-TUCKER CONDITIONS