Soham Uday Mehta. Linear Programming in 3 variables.

CS 270, SPRING 2013 PROJECT

A VISUAL TOOL FOR THE SIMPLEX METHOD IN LP

Soham Uday Mehta

Linear Programming in 3 variables

Subject to

Goals

Visualize the convex feasible region specified by constraints (in 3D)

Goals

Visualize the convex feasible region specified by constraints (in 3D)

Visualize Simplex Algorithm to solve the LP (simple version)

Visualization

Each constraint becomes a polygon bounding the convex feasible region

Visualization

Each constraint becomes a polygon bounding the convex feasible region

Start with 3 Quads for

Assume we have all polygons for current convex region and we want to add a new constraint

Visualization

Each constraint becomes a polygon bounding the convex feasible region

Start with 3 Quads for

Assume we have all polygons for current convex region and we want to add a new constraint

Need to clip all polygons by new constraint

Visualization

Find the ‘side’ of plane each vertex of polygon is on

If an edge of poly cuts plane, add new vertex and remove the ‘wrong’ side vertex

Visualization

Each constraint may also create a new polygon

Visualization

Each constraint may also create a new polygon

Visualization

Each constraint may also create a new polygon

Store ‘new’ vertices created by clipping existing polygons

Visualization

Each constraint may also create a new polygon

Store ‘new’ vertices created by clipping existing polygons

Remove duplicates

Visualization

Each constraint may also create a new polygon

Store ‘new’ vertices created by clipping existing polygons

Remove duplicates, re-order vertices, and create new poly

Simplex Algorithm

1. Start with a random vertex

Simplex Algorithm

1. Start with a random vertex

2. Find extreme directions at current vertex

3. Pick maximum improvement direction

4. If no improvement in any direction, stop

Simplex Algorithm

1. Start with a random vertex

2. Find extreme directions at current vertex

3. Pick maximum improvement direction

4. If no improvement in any direction, stop

5. Find max. feasible step and move to next vertex

6. Go back to 2

Choosing the start vertex

First add 1 auxiliary variable to each constraint to convert it to an equation, create matrix

Randomly choose from the and set to , solve the equations for rest

Choosing the start vertex

First add 1 auxiliary variable to each constraint to convert it to an equation, create matrix

Randomly choose from the and set to , solve the equations for rest

If matrix is not invertible or some , solve with different choice

The ones set to zero are called non-basic, rest are basic – store this in a boolean vector “B”

Finding directions at any vertex

Each non-basic variable gives a direction

Finding directions at any vertex

Each non-basic variable gives a direction

For , set 1 non-basic value to 1, other two to 0 and solve for rest (free upto a constant)

Pick direction which gives maximum

If all directions give , reached maximum, STOP

Finding the next vertex

Choose step size

Move to next vertex

Finding the next vertex

Choose step size

Move to next vertex

Update “B” vector to correspond to new vertex: non-basic generating the max direction becomes basic, the basic fixing becomes non-basic

Conclusion

Will be available for download

Conclusion

Will be available for download

Thanks for your attention

Comments / Questions?