Time complexity of simplex algorithm

Albi thomas

M.tech (TM)

Roll no.11

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

Overview Polynomial time-complexity(bound) Eg.gaussian elimination Exponential time complexity Eg. Buchberger's algorithm Feasibility Simplex Method Simplex Weaknesses

Exponential Iterations Convex Sets and Hulls

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

Feasibility in 2-Space 2x1 + x2 ≤ 4

In an LP environment, restrict to Quadrant I since x1, x2 ≥ 0

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

Five total constraints; therefore 5 faces to the polyhedron

Feasibility in 3-Space

0,,

5

42subject to

523maximize

321

3

21

321

xxx

x

xx

xxx

Simplex Illustrated: Initial Dictionary

321

35

214

523

5

24

xxxz

xx

xxx

Current solution:

x1 = 0x2 = 0x3 = 0

Simplex Illustrated: First Pivot

521

53

214

52325

5

24

xxxz

xx

xxx

Current solution:

x1 = 0x2 = 0x3 = 5

Simplex Illustrated: Second Pivot

5423

221

53

421

221

1

531

5

2

xxxz

xx

xxx

Current solution:

x1 = 2x2 = 0x3 = 5

Simplex Illustrated: Final Pivot

541

53

412

52733

5

24

xxxz

xx

xxx

Final solution (optimal):

x1 = 0x2 = 4x3 = 5

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

Simplex Weaknesses: Exponential Iterations: Klee-Minty Reviewed   

Cases with high complexity (2n-1 iterations) Normal complexity is O(m3) How was this problem solved?

1 2 3

1

1 2

1 2 3

1 2 3

100 10

1

20 100

200 20 10,000

, , 0

x x x z

x

x x

x x x

x x x

Geometric Interpretation & Klee-Minty

Saw non-optimal solution earlier

How can we represent the Klee-Minty problem class graphically?

0,,

5

42subject to

523maximize

321

3

21

321

xxx

x

xx

xxx

Step 1: Constructing a Shape Start with a cube.

What characteristics do we want the cube to have?

What is the worst case to maximize z?

1 1 2 2 3 3

1

2

3

1 2 3

1

1

1

, , 0

c x c x c x z

x

x

x

x x x

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

Step 2: Increasing Objective Function: Modifying the Cube      Squash the cube

New dictionary

1 2 3

1

1 2

1 2 3

1 2 3

100 1,000 10,000

1

0.2 1

0.02 0.2 1

, , 0

x x x z

x

x x

x x x

x x x

[0, 1, 0.8] [0, 1, 0.82]

[1, 0.8, 0][0, 1, 0]

[1, 0, 0.98][0, 0, 1]

[1, 0, 0][0, 0, 0]

4 1

5 1 2

6 1 2 3

1

1 0.2

1 0.02 0.2

x x

x x x

x x x x

Step 3: Achieving 2n-1 Iterations: Altering the Algebra    Let

Convert

to ( )j j j

j N

z v d s x

j j js x x

j jj N

z v d x

31 2

1 2 3

11

4 4

1 21 2

5 5 5

31 21 2 3

6 6 6 6

1 2 3

100 1,000 10,000

1

10.2

10.02 0.2

, , 0

xx xz

s s s

sx

s s

s sx x

s s s

ss sx x x

s s s s

x x x

The Final Solution Most desirable: Least desirable:

1 4 2 5 3 6

1 2 3

1

1 2

1 2 3

1 2 3

1, 0.01, 0.0001

100 10

1

20 100

200 20 10,000

, , 0

s s s s s s

x x x z

x

x x

x x x

x x x

1 4,x x

3x

Thank you