Operational research
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Time complexity of simplex algorithm
Albi thomas
M.tech (TM)
Roll no.11
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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
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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
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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
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Feasibility in 2-Space 2x1 + x2 ≤ 4
In an LP environment, restrict to Quadrant I since x1, x2 ≥ 0
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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
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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
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Simplex Illustrated: Initial Dictionary
321
35
214
523
5
24
xxxz
xx
xxx
Current solution:
x1 = 0x2 = 0x3 = 0
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Simplex Illustrated: First Pivot
521
53
214
52325
5
24
xxxz
xx
xxx
Current solution:
x1 = 0x2 = 0x3 = 5
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Simplex Illustrated: Second Pivot
5423
221
53
421
221
1
531
5
2
xxxz
xx
xxx
Current solution:
x1 = 2x2 = 0x3 = 5
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Simplex Illustrated: Final Pivot
541
53
412
52733
5
24
xxxz
xx
xxx
Final solution (optimal):
x1 = 0x2 = 4x3 = 5
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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
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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
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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
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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
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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
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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
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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
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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
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Thank you