Because of the significance of convexity in nonlinear programming, it is useful to be able to identify convex sets and convex (or concave) functions easily. Apply the definitions given in this chapter to establish the following properties:
a) If C1 and C2 are two convex sets, then the intersection of C1 and C2 (that is, points lying in both) is a convex set. For example, since the set C1 of solutions to the inequality.
is convex and the set C2 of points satisfying
is convex, then the feasible solution to the system
which is the intersection of C1 and C2, is also convex. Is the intersection of more than two convex sets a convex set?
b) Let f1(x), f2(x), . . . , fm(x) be convex functions and let α1, α2, . . . , αm be nonnegative numbers; then the function
is convex. For example, since f1(x1, x2) = x 2 1 and f2(x1, x2) = |x2| are both convex functions of x1 and x2, the function.
c) Let f1(x) and f2(x) be convex functions; then the function f (x) = f1(x) f2(x) need not be convex. [Hint. See part (a(v)) of the previous exercise.]
d) Let g(y) be a convex and nondecreasing function [that is, y1 ≤ y2 implies that g(y1) ≤ g(y2)] and let f (x) be a convex function; then the composite function h(x) = g[ f (x)] is convex. For example, since the function e y is convex and nondecreasing, and the function f (x1, x2) = x 2 1 + x 2 2 is convex, the function.