CS 312: Algorithm Analysis

Lecture #30: Linear Programming: Intro. to the Simplex Algorithm

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Slides by: Eric Ringger, with contributions from Mike Jones and Eric Mercer

Announcements Homework #21 (Project 6, Part 1)

Due online today by 5pm No late submissions accepted (just like any homework asst.) Key available online after 5pm

Important: LP Notes available on the schedule Works through the ideas up through the Simplex algorithm Includes the pseudo-code for Simplex (and Pivot)

Project #6: Linear Programming (Part 2) Early day: Friday Due: next Monday Verification suggestion: use another LP solver

Objectives

Review various forms for representing an LP problem

Understand the Simplex method algebraically

Introduce the concept of a Pivot Walk through an example to completion

Forms of an LP Problem1. English form2. Algebraic form3. Standard form up to step #1

also algebraic

Example: Standard Form (up to step #1)

Forms of an LP Problem1. English form2. Algebraic form3. Standard form up to step #1

also algebraic4. Geometric / graphical form

Only if 2-D, maybe 3-D5. Matrix-vector form

Read off directly from standard form up to step #1

Example: Matrix-Vector Form

Forms of an LP Problem1. English form2. Algebraic form3. Standard form up to step #1

also algebraic4. Geometric / graphical form

Only if 2-D, maybe 3-D5. Matrix-vector form

Read off directly from standard form up to step #16. Padded matrix-vector form

Adapted directly from matrix-vector form Reflects the participation of slack variables

Example: Padded Matrix-Vector form

Purposes:• for input to the Simplex algorithm• for computation in the Simplex algorithm

Forms of an LP Problem1. English form2. Algebraic form3. Standard form up to step #1

also algebraic4. Geometric / graphical form

Only if 2-D, maybe 3-D5. Matrix-vector form

Read off directly from standard form up to step #16. Padded matrix-vector form

Adapted directly from matrix-vector form Reflects the participation of slack variables

7. Standard form up to step #2 (aka “slack form”) Also algebraic

Standard Form (step #2)

Sketch of Simplex Algorithm

Start with Basic solution (the origin of the feasible region)

1. Pick a non-basic variable2. Raise its value as high as we can3. Pick a basic variable4. Algebraically pivot5. Repeat at #1 until we cannot raise the value of

any non-basic variables

Basic Solution

Basic Solution

Identify a Non-Basic Variable

Identify a Non-Basic Variable

Select a Basic Variable

Select a Basic Variable

Solve for the Non-Basic Variable: “Pivot”

Solve for the Non-Basic Variable: “Pivot”

Pivot: Rewrite the Other Equations

Pivot: Rewrite the Other Equations

After the Pivot: Objective Values

After the Pivot: Objective Values

Repeat

Repeat

Repeat Again (Third time)

Repeat Again (Third time)

What next?

Observations At the beginning of every round of Simplex,

The value of each non-basic variable in the current solution is 0.

The space for the transformed problem is spanned by unit vectors in the directions of the non-basic variables

The current solution is at the origin of that space The new feasible region is defined in that space

Pivot is designed to keep our attention focused on the origin of each successive space

Assignments

Project #6 Now re-read the Simplex and Pivot pseudo-

code to see how these algebraic steps are performed in the code

HW #22