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Problem Problems: LP THEORY
Chapter 4: Geometry of LP
LP THEORY
1. Show that if a linear programming problem has exactly one optimal solution, then this solution
must be an extreme point of the feasible region. You can assume that the problem is in the
standard form max x cx, subject to Ax  b, x  0 .
Solution:
This is Theorem 4.4.2 in the Lecture Notes. A complete proof is given on page 4.13.
2. Prove that if a linear programming problem has more than one optimal solution then it has
infinitely many optimal solutions. You can assume that the problem is in the standard form
max cx subject to Ax£b, x•0 .
Solution:
Let x' and x'' be two distinct optimal solutions. Then, clearly
cx' = cx"
Ax' £b , Ax"£b
, x"•0
x' •0
If we now consider any 0 < a <1 then from the above it follows that x(a)"= ax' +(1 - a)x"
satisfies the following conditions:
cx' = cx(a) = cx"
Ax(a)£b
x(a)•0
Hence x(a) is an optimal solution for the problem, observing that there are infinitely many such
solutions.
3. Consider the linear programming problem max cx subject to Ax  b, x  0 . Show that if this
problem has exactly one optimal solution, then this solution is an extreme point of the feasible
region. Suppose that two extreme points of the feasible region are optimal. How many optimal
solutions are there in this case? Solution in the lecture notes. If there are two optimal solutions
then there are infinitely many optimal solutions (all convex combinations of the two given
optimal solutions, see question 2).
4. State the Fundamental Theorem of Linear Programming and briefly (no more than 5 lines)
explain its role in linear programming. Solution in the lecture notes.
Solved Problems: 4.1