STOCHASTIC

15. Besides functions that are convex (Q, strictly convex (SC), quasi-convex (QC), and
pseudo-convex (PC), it is of interest to consider the related functions that are strictly quasiconvex,
strictly pseudo-convex, and XC convex. Let fix) be a numerical function defined
on the convex set C •= R". Then/is strictly quasi-convex (SQC) if
fix 2 ) < fix 1 ) => /[Xx 1 +(l~X)x 2 1 < fix 1 ), VO < A < 1, x l ,x z e C;
/is strictly pseudo-convex (SPC) if/is differentiable and
fix 2 ) ^ fix 1 ) => (x 1 -x 1 yVf(x l ) f^x 1 + (1 -A)x 2 ] < fix 1 ),
Vx 1 ,x 2 €C, x 1 ? x 2 , and 0 < X < 1.
(a) Show that a local minimum of a quasi-convex function is not necessarily a global
minimum by considering the function
la if 0 § x < i
fix) — { where a ¥= b.
\b if i £ x £ 1
(b) Suppose a local minimum x 1 of a quasi-convex function / is not a global minimum.
Then / is constant in the intersection of Xi and the line segment between x l and any
global minimum x*.
(c) Show that a local minimum of a strictly quasi-convex or a strictly pseudo-convex
function is a global minimum.
(d) Show that D = {x|#j(x)£0, i=\,...,m) is convex if the gs are quasi-concave.
(e) Show that the seven kinds of convexity may be related as follows: SC =*• C, C => PC,
PC=>SQC, and SQC=*QC; and SC=>XC, XC=*SQC, and PC+XC=>SPC. Assume
differentiability as needed, and that the SQC functions are lower semicontinuous, otherwise
SQC i> QC.
(f) Show that the seven kinds of convexity are indeed different by verifying the results
in the following table, where a plus indicates that the function has the stated property.
Function Region R SC C SPC PC SQC QC XC
/,(*)= 10 0S*£1 - - - - - +
l-Ot-1) 2 1S*S2
Mx,y) = -x 2 Ogx.^gl - - - - + +
hix,y) = -x 2 -x OStjgl - - - + + +
Uix) = 0 OSxgl - + - + + +
fsix) = -x Ogxgl - + + + + + +
Uix) = x 2 0