The document discusses the time complexity of the simplex algorithm for solving linear programming problems. It begins by defining time complexity as the number of arithmetic operations required to solve a problem. It then provides an overview of different time complexities such as polynomial time and exponential time. The rest of the document focuses on using geometric interpretations to understand the simplex algorithm and analyze cases where it exhibits exponential running time. It illustrates concepts like the region of feasibility and simplex pivoting through examples. It also reviews the Klee-Minty example, which shows that the simplex algorithm can require an exponential number of iterations in the worst case.
2. Introduction
Time complexity of an algorithm counts the
number of arithmetic operations sufficient for
the algorithm to solve the problem
Understand properties of LP in terms of
geometry
Use geometry as aid to solve LP
Some concepts new
4. Region of Feasibility
Graphical region describing all feasible
solutions to a linear programming problem
In 2-space: polygon, each edge a constraint
In 3-space: polyhedron, each face a constraint
5. Feasibility in 2-Space
2x1 + x2 ≤ 4
In an LP environment,
restrict to Quadrant I
since x1, x2 ≥ 0
6. Simplex Method
Every time a new dictionary is generated:
Simplex moves from one vertex to another vertex
along an edge of polyhedron
Analogous to increasing value of a non-basic
variable until bounded by basic constraint
Each such point is a feasible solution
Average time taken is linear in 2 space
7. Feasibility in 3-Space
maximize 3 x1 + 2 x2 + 5 x3
subject to 2 x1 + x2 ≤4
x3 ≤ 5
x1 , x2 , x3 ≥ 0
Five total constraints;
therefore 5 faces to the
polyhedron
12. Simplex Review and Analysis
Simplex pivoting represents traveling along
polyhedron edges
Each vertex reached tightens one constraint
(and if needed, loosens another)
May take a longer path to reach final vertex
than needed
13.
14.
15. Simplex Weaknesses: Exponential
Iterations: Klee-Minty Reviewed
100 x1 + 10 x2 + x3 = z
x1 ≤ 1
20 x1 + x2 ≤ 100
200 x1 + 20 x2 + x3 ≤ 10, 000
x1 , x2 , x3 ≥ 0
Cases with high complexity (2n-1 iterations)
Normal complexity is O(m3)
How was this problem solved?
16. Geometric Interpretation & Klee-Minty
Saw non-optimal
solution earlier
How can we represent
the Klee-Minty
problem class
graphically? maximize 3 x1 + 2 x2 + 5 x3
subject to 2 x1 + x2 ≤ 4
x3 ≤ 5
x1 , x2 , x3 ≥ 0
17. Step 1: Constructing a Shape
c1 x1 + c2 x2 + c3 x3 = z
Start with a cube.
x1 ≤ 1
x2 ≤ 1
What characteristics do x3 ≤ 1
we want the cube to x1 , x2 , x3 ≥ 0
have?
What is the worst case
to maximize z?
18. Step 1: Constructing a Shape
Goal 1: Create a shape
with a long series of
increasing facets
Goal 2: Create an LP
problem that forces this
route to be taken