Linear Programming

Piyush Kumar

Welcome to COT 5405

Optimization

For example

Min

subject to

0

Tc x

Ax b

x

This is what is known as a standard linear program.

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

minimize z = 7x1 + x2 + 5x3

subject to x1 - x2 + 3x3 >= 10

5x1 + 2x2 - x3 >= 6

x1, x2, x3 0

What happens at (2,1,3)?What does it tell us about z* = optimal value of z?

LP Upper bounds

Any feasible solution to LP gives an upper bound on z*

So now we know z* <= 30. How do we construct a lower bound? z* >= 16? [Y/N]?

Lower bounding an LP

7x1+x2+5x3 >= (x1-x2+3x3) + (5x1+2x2-x3)

>= 16

Find suitable multipliers ( >0 ?) to construct lower bounds. How do we choose the multipliers?

The Dual

maximize z’ = 10y1 + 6y2

subject to y1 + 5y2 <= 7

-y1 + 2y2 <= 1

3y1 – y2 <= 5

y1, y2 0

What is the dual of a dual? Every feasible solution of the dual gives a lower bound on z*

The Primal

minimize z = 7x1 + x2 + 5x3

subject to x1 - x2 + 3x3 >= 10

5x1 + 2x2 - x3 >= 6

x1, x2, x3 0

Every feasible solution of the primal is an upper bound on the solution to the dual.

Primal – Dual picture

0 Z*

DualSolutions

Primal Solutions

Strong OptimalityPrimal = Dual at opt

Duality

A variable in the dual is paired with a constraint in the primal Objective function of the dual is determined by the right hand side of the primal constraints The constraint matrix of the dual is the transpose of the constraint matrix in the primal.

Duality PropertiesSome relationships between the primal and dual problems:

1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable

2. If the optimal value of the primal is unbounded then the dual is infeasible.

3. If the optimal value of the dual is unbounded then the primal is infeasible.

In Matrix terms

Min

subject to

0

Tc x

Ax b

x

1 1, ,A c xmxn nx nx

LP Geometry

Forms a n 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

The normals to the halfspaces defining the polyhedron are formed by the coefficents of the constraints. Rows of A form the normals to the hyperplanes defining the primal LP pointing inside the polyhedron.

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.

Ellipsoid Method

Courtesy S. Boyd

Barrier Algorithms

Simplex solution path

Barrier central path

o Predictor

o Corrector

Optimum

Interior Point Methods

Back to LP Basics

Standard form of LP

Min

subject to

0

Tc x

Ax b

x

1 1, ,A c xmxn nx nx

Standard form of the Dual

Max b

subject to

0

T

T

y

A y c

y

1 1, ,A c xmxn nx nx

Weak Duality

( )T T T T T Tb y Ax y x A y x c c x

We will not prove strong duality in this classbut assume it.

Complementary solutions

For any primal feasible (but suboptimal) x, its complementary solution y is dual infeasible, with cx=ybFor any primal optimal x*, its complementary solution y* is dual optimal, with cx*=y*b=z*

Duality Gap = cx-yb

Complementary slackness

x*, y* are feasible, then they are optimal for (P) and (D) iff

For I = 1..m if yi* > 0

oThen aix* = bi

For J = 1..n if xj* > 0

oThen y*Aj = ci

ai are rows of A and Aj are the columns of A

Complementary slackness

x*, y* are simultaneously optimal for (P) and (D) iff y*(Ax* - b) = 0 (y*A – c)x* = 0

Summary: If a variable is positive, its dual constraint is tightOr if a constraint is loose its dual variable is zero.

Complementary Slackness

Proof? y*(Ax* - b) - (y*A – c)x*

= y*Ax* - y*b - y*Ax* + cx*

= cx* - y*b = 0( But all terms are non-negative )Hence all must be zero!

Primal-Dual Algorithms

Find a feasible solution for both P and D. Try to satisfy the complementary slackness conditions.

Algorithm Design Techniques

LP Relaxation Rounding

oRound the fractional solution obtained by solving LP-relaxation.

oRuns fast Primal Dual Schema

o(iteratively constructs primal n dual solutions)

objective

Linear Program

LP optimum

feasible solutions

y

x

objective

IP optimum

Integer Program

rounding down optimum of LP relaxation

feasible solutions =

y

x

optimum ofLP relaxation

Linear Relaxations

What happens if the optimal of a LP-Relaxation is Integral? There are a class of IPs for which this is guaranteed to happen Transportation problems MaxFlow problems In general (Unimodularity) … Exact

Relaxation

Lower Bounds

Assume minimization problemAny relaxation of the original IP has a _____________ optimal objective function value than the optimal objective function value of the original IP

z*relaxation z*

z*relaxation is called a __________________ on z*

Difference between these two values is called the relaxation gap

Upper Bounds

Any feasible solution to the original IP has a _____________ objective function value than the optimal objective function value of the original IP

zfeasible z*

zfeasible is called an __________________ on z*

Heuristic techniques can be used to find “good” feasible solutions Efficient, may be beneficial if optimality can be

sacrificed Usually application- or problem-specific

Vertex Cover

Introduction to LP Rounding A simple 2-approximation using LP Better than 2-factor approx?