Team LinearBy

Marks Speight, Taylor Bishop

Linear Programming Problem :

Optimize a linear function subject to linear constraints.

Constraints• Linear equation• Linear inequalities

Linear Program Problem:

• Convex Hull • Analytic Geometry • Simplex method

Convex Hull

• A convex set is a set of points (finite or infinite) in the plane if for any two points P and Q in the set, the entire line segment with end point a P an Q belongs to the set.

• An extreme point of a convex set is a point of this set that is not a middle point of any line segment with endpoints in the set. List the extreme points in a clockwise order

• Applicable to arbitrary sets of points that happen to line in the same line

Methods of finding extreme points

• Brute Force

• Analytic Geometry

Brute Force

A line segment connecting two point pi and pj of a set of n points is a part of the convex hull’s boundary if all other points of the set lie on the same side of the straight line through these two points. Repeating this test for every pair of points yields a list of line segments that make up the convex hull’s boundary

Geometric method

A linear programming problem with a nonempty bounded feasible region always has a solution, which can be found at one of the extreme points of its feasible region

Solution: The points of intersection of the constraint lines is solved by a system of linear equations. The feasible region is obtained by the intersection of half-planes and the quadrants of the Cartesian plane defined by the sing of the constraints, and the points of intersection on the constraint lines are solve by a system of linear equations

Geometric (cont.)

O(n3)

For each of pairs of distinct points , we find the sign of ax + by – c for each of the other n – 2 points

Extreme point obstacles• a mechanism for generating extreme

points of the feasible region

• the number of extreme points in known to grow exponentially with the size of the problem

• Brute force is unrealistic for linear programming problems of non-trivial sizes

Geometric Obstacles

• Empty: constraints include contradictory requirements

• Unbounded feasible region

Real-life optimization problems

May have thousands of variables

And thousands of constraints

Which means astronomical numbers of corner points

Making the above technique impossible

Even on super-computers.

The Simplex Method inspects a small faction of the extreme points of the feasible region before reaching an optimal oneIt uses methods from linear algebra

the study of systems of linear equationsusing matrices

Simplex Method

Maximize: 3x+5ySubject to: x+3y<=6

x+4<=4 x,y>=0

Simplex Method

Pivot obstacles

If you have 5 inequalities and 3 variables You will have to introduce 5 slack variables

Creating a system of 5 equations with 8 variables.

You set 3 variables to zero at a time and solve. Reducing the system to a system of 5 equations with 5 variables to

obtain a basic solution.

You will have to do this (8)(7)(6)/(3)(2)(1)=8C5=56 times to get 56 basic solutions.

This might take awhile by hand but only a few seconds at most for a computer. Seems good then right?

Pivot obstacles (cont.)

BUT compared to some real situations for which the simplex method is used: 100 constraint equations and 300 variables Which will give 300C200 = 4 x 81 solutions

If a computer finds one basic solution every millionth of a second it would take 4/3 x 1068 years to complete.

Simplex Method average time complexity: O(n2) Worst case of: O(2n)

Why 380?

These problems were introduced in chapter 3.3(Brute Force Convex Hull Extreme Point Method),chapter 6.6 (Linear programming), 10.1(Simplex Method), 10.1(Geometric Method)

“Linear programming is used to allocate resources, plan production, schedule workers, plan investment portfolios and formulate marking(and military) strategies. The versatility and economic impact of linear programming in today’s industrial world is truly awesome”-Eugene Lawler Expert on combinatorial optimization and a founder of the field.

Questions1) Does every linear programming problem have an optimal solution that can be found at a vertex of its feasible region? No

2) A Simplex method problem has 5 variables and 7 constraints. How many Columns are in the tableau? 14

3)How many pivots for a Simplex Method problem with 5 constraints and 8 variables? 8C5=56

4) Convex Hull is used for what kind of programming problem? Linear

5) Why is the time complexity of Geometric Method O(n2)? Plane cutting