Computational Geometry

Piyush Kumar(Lecture 5: Linear Programming)

Welcome to CIS5930

Linear Programming

Significance A lot of problems can be converted to LP

formulationo Perceptrons (learning), Shortest path, max flow, MST,

matching, …

Accounts for major proportion of all scientific computations Helps in finding quick and dirty solutions to NP-hard optimization problems Both optimal (B&B) and approximate (rounding)

Graphing 2-Dimensional LPs

Example 1:

x30 1 2

y

0

1

2

4

3

Feasible Region

x 0 y 0

x + 2 y 2

y 4

x 3

Subject to:

Maximize x + y

Optimal Solution

These LP animations were created by Keely Crowston.

Graphing 2-Dimensional LPs

Example 2:

Feasible Region

x 0 y 0

-2 x + 2 y 4

x 3

Subject to:

Minimize ** x - y

Multiple Optimal

Solutions!4

1

x31 2

y

0

2

0

3

1/3 x + y 4

Graphing 2-Dimensional LPs

Example 3:

Feasible Region

x 0y 0

x + y 20

x 5

-2 x + 5 y 150

Subject to:

Minimize x + 1/3 y

Optimal Solution

x

3010 20

y

0

10

20

40

0

30

40

y

x0

1

2

3

4

0 1 2

3

x3010 20

y

0

10

20

40

0

30

40

Do We Notice Anything From These 3 Examples?

x

y

0

1

2

3

4

0 1 2

3

Extreme point

A Fundamental Point

If an optimal solution exists, there is always a corner point optimal solution!

y

x0

1

2

3

4

0 1 2

3

x3010 20

y

0

10

20

40

0

30

40x

y

0

1

2

3

4

0 1 2

3

Graphing 2-Dimensional LPs

Example 1:

x30 1 2

y

0

1

2

4

3

Feasible Region

x 0y 0

x + 2 y 2

y 4

x 3

Subject to:

Maximize x + y

Optimal Solution

Initial Corner pt.

Second Corner pt.

And We Can Extend this to Higher Dimensions

Then How Might We Solve an LP?

The constraints of an LP give rise to a geometrical shape - we call it a polyhedron.

If we can determine all the corner points of the polyhedron, then we can calculate the objective value at these points and take the best one as our optimal solution.

The Simplex Method intelligently moves from corner to corner until it can prove that it has found the optimal solution.

But an Integer Program is Different

x

y

0

1

2

3

4

0 1 2

3

Feasible region is a set of discrete points.

Can’t be assured a corner point solution.

There are no “efficient” ways to solve an IP.

Solving it as an LP provides a relaxation and a bound on the solution.

Linear Programs in higher dimensions

maximize z = -4x1 + x2 - x3

subject to -7x1 + 5x2 + x3 <= 8

-2x1 + 4x2 + 2x3 <= 10

x1, x2, x3 0

In Matrix terms

Max

subject to

Tc x

Ax b

1 1, ,n dA c xx dx dx

LP Geometry

Forms a d dimensional polyhedron

Is convex : If z1 and z2 are two feasible solutions then λz1+ (1- λ)z2 is also feasible.Extreme points can not be written as a convex combination of two feasible points.

LP Geometry

Extreme point theorem: If there exists an optimal solution to an LP Problem, then there exists one extreme point where the optimum is achieved. Local optimum = Global Optimum

LP: AlgorithmsSimplex. (Dantzig 1947) Developed shortly after WWII in response to logistical

problems:used for 1948 Berlin airlift.

Practical solution method that moves from one extreme point to a neighboring extreme point.

Finite (exponential) complexity, but no polynomial implementation known.

Courtesy Kevin Wayne

LP: Polynomial Algorithms

Ellipsoid. (Khachian 1979, 1980) Solvable in polynomial time: O(n4 L) bit operations.

o n = # variables o L = # bits in input

Theoretical tour de force. Not remotely practical.

Karmarkar's algorithm. (Karmarkar 1984) O(n3.5 L). Polynomial and reasonably efficient

implementations possible.

Interior point algorithms. O(n3 L). Competitive with simplex!

o Dominates on simplex for large problems. Extends to even more general problems.

LP: The 2D case

1 2 n

, ,

Let's suppose we are given n linear inequalities h ,h ,..., h

:i i x i y ih a x a y b

Wlog, we can assume that c=(0,-1). So now we want to find theExtreme point with the smallest y coordinate.

Lets also assume, no degeneracies, the solution is given by twoHalfplanes intersecting at a point.

Incremental Algorithm?

How would it work to solve a 2D LP Problem? How much time would it take in the worst case? Can we do better?