A Convex and Concave Functions
Convex Function. A functionf (X)is said to be convex if for any pair of points
X 1 =
x 1 (^1 )
x 2 (^1 )
..
.
xn(^1 )
and X 2 =
x 1 (^2 )
x 2 (^2 )
..
.
xn(^2 )
and allλ, 0≤λ≤1,
f[λX 2 + ( 1 −λ)X 1 ] ≤λf (X 2 ) +( 1 −λ)f (X 1 ) (A.1)
that is, if the segment joining the two points lies entirely above or on the graph of
f (X). Figures A.1aand A.2aillustrate a convex function in one and two dimensions,
respectively. It can be seen that a convex function is always bending upward and
hence it is apparent that the local minimum of a convex function is also a global
minimum.
Concave Function. A functionf (X)is called a concave function if for any two
pointsX 1 andX 2 , and for all 0≤λ≤ 1 ,
f[λX 2 + ( 1 −λ)X 1 ] ≥λf (X 2 ) +( 1 −λ)f (X 1 ) (A.2)
that is, if the line segment joining the two points lies entirely below or on the graph
off (X).
Figures A.1band A.2bgive a concave function in one and two dimensions, respec-
tively. It can be seen that a concave function bends downard and hence the local
maximum will also be its global maximum. It can be seen that the negative of a con-
vex function is a concave function, and vice versa. Also note that the sum of convex
functions is a convex function and the sum of the concave functions is a concave
function. A functionf (X)is strictly convex or concave if the strict inequality holds
in Eqs. (A.1) or (A.2) for anyX 1 =X 2. A linear function will be both convex and
concave since it satisfies both inequalities (A.1) and (A.2). A function may be con-
vex within a region and concave elsewhere. An example of such a function is shown
in Fig. A.3.
Testing for Convexity or Concavity. In addition to the definition given, the following
equivalent relations can be used to identify a convex function.
Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao 779
Copyright © 2009 by John Wiley & Sons, Inc.
