636 C H A P T E R 1 1 Nonlinear Programming

P R O B L E M SGroup A

On the given set S, determine whether each function isconvex, concave, or neither.

1 f(x) 5 x3; S 5 [0, ∞)

2 f(x) 5 x3; S 5 R1

3 f(x) 5 }1

x}; S 5 (0, ∞)

4 f(x) 5 xa (0 # a # 1); S 5 (0, ∞)

5 f(x) 5 ln x; S 5 (0, ∞)

6 f(x1, x2) 5 x31 1 3x1x2 1 x2

2; S 5 R2

7 f(x1, x2) 5 x21 1 x2

2; S 5 R2

8 f(x1, x2) 5 2x21 2 x1x2 2 2x2

2; S 5 R2

9 f(x1, x2, x3) 5 2x21 2 x2

2 2 2x23 1 .5x1x2; S 5 R3

10 For what values of a, b, and c will ax21 1 bx1x2 1 cx2

2

be a convex function on R2? A concave function on R2?

Group B

11 Prove Theorem 19.

12 Show that if f(x1, x2, . . . , xn) and g(x1, x2, . . . , xn) areconvex functions on a convex set S, then h(x1, x2, . . . , xn) 5f (x1, x2, . . . , xn) 5 g(x1, x2, . . . , xn) is a convex function on S.

13 If f(x1, x2, . . . , xn) is a convex function on a convex setS, show that for c $ 0, g(x, x2, . . . , xn) 5 cf(x1, x2, . . . , xn)is a convex function on S, and for c # 0, g(x1, x2, . . . , xn) 5cf(x1, x2, . . . , xn) is a concave function on S.

14 Show that if y 5 f(x) is a concave function on R1, thenz 5 }

f (

1

x)} is a convex function [assume that f(x) . 0].

15 A function f (x1, x2, . . . , xn) is quasi-concave on aconvex set S , Rn if x9 [ S, x0 [ S, and 0 # c # 1 implies

f [cx9 1 (1 2 c)x0] $ min[ f (x9), f (x0)]

Show that if f is concave on R1, then f is quasi-concave.Which of the functions in Figure 19 is quasi-concave? Is aquasi-concave function necessarily a concave function?

16 From Problem 12, it follows that the sum of concave

functions is concave. Is the sum of quasi-concave functionsnecessarily quasi-concave?

17 Suppose a function’s Hessian has both positive andnegative entries on its diagonal. Show that the function isneither concave nor convex.

18 Show that if f(x) is a non-negative, increasing concavefunction, then ln [ f(x)] is also a concave function.

19 Show that if a function f (x1, x2, . . . , xn) is quasi-concave on a convex set S, then for any number a the set Sa 5 all points satisfying f(x1, x2, . . . , xn) $ a is a convex set.

20 Show that Theorem 1 is untrue if f is a quasi-concavefunction.

21 Suppose the constraints of an NLP are of the formgi(x1, x2, . . . , xn) # bi(i 5 1, 2, . . . m). Show that if each ofthe gi is a convex function, then the NLP’s feasible regionis convex.

Group C

22 If f(x1, x2) is a concave function on R2, show that forany number a, the set of (x1, x2) satisfying f(x1, x2) $ a isa convex set.

23 Let Z be a N(0, 1) random variable, and let F(x) be thecumulative distribution function for Z. Show that on S 5

(2∞, 0], F(x) is an increasing convex function, and on S 5

[0, ∞), F(x) is an increasing concave function.

24 Recall the Dakota LP discussed in Chapter 6. Let v(L,FH, CH) be the maximum revenue that can be earned whenL sq board ft of lumber, FH finishing hours, and CHcarpentry hours are available.

a Show that v(L, FH, CH) is a concave function.

b Explain why this result shows that the value of eachadditional available unit of a resource must be a nonin-creasing function of the amount of the resource that isavailable.

a b c

F I G U R E 19