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390 Chapter Ten: Linear Programming

Example 10.2.3

maximize: z = 1 — 12x — 3y
( x + y > > — 1
1
— y
x
x,y>0
In this problem the set of feasible solutions is the same set S
as in the previous example. However the maximum value of
z in S occurs at x = 0, y = 1: this is an optimal solution of
the problem.

Exercises 10.2

In Exercises 10.2.1-10.2.3 sketch the convex subset of all
feasible solutions of a linear programming problem with the
given constraints.
( x- y < - 2
1. < 2x+ y < 3
(x,y>0
{ x x+ - 2y < < 6
3
y
x,y>0
{ x + y + z < 5
<
z
0
x - y -
x,y,z > 0
4. Find all the extreme points in the programs of Exercises
10.2.1 and 10.2.2 .
5. Suppose the objective function in Exercise 10.2.2 is
z = 2x + 3y. Find the optimal solution when z is to be
maximized.
n
6. Let S be any subspace of R . Prove that S is convex.
Then give an example of a convex subset of R 2 containing
(0,0) which is not a subspace.