Polyhedral OptimizationLecture 1 – Part 1

M. Pawan Kumar

pawan.kumar@ecp.fr

Slides available online http://cvn.ecp.fr/personnel/pawan/

• What is it about?

• What are the prerequisites?

• What type of material will be provided?

• How will the students be evaluated?

• Tips and tricks

Course Description

Problem 1

Given a set of ‘n’ real numbers S

Is there a non-empty subset X S such that ⊆

∑x X∈ x ≤ C

Problem 1

Given a set of ‘n’ real numbers S

Is there a non-empty subset X S such that ⊆

∑x X∈ x ≤ C

{-28, 53, -58, -99, 13, 27, -55, -31, -91, 12, -87, -68}

Problem 1

Given a set of ‘n’ real numbers S

Is there a non-empty subset X S such that ⊆

∑x X∈ x ≤ -250

Solution?

{-28, 53, -58, -99, 13, 27, -55, -31, -91, 12, -87, -68}

Solution

Add up all the negative numbers

Check if the number is less than or equal to C

If there are negative numbers in S

Else

Pick smallest number

n

Easy problem (run-time is polynomial in ‘n’)

Run

ning

tim

e

Problem 2

Given a sequence of ‘n’ real numbers S

Is there a non-empty subsequence X such that

∑x X∈ x ≤ C

Problem 2

Given a sequence of ‘n’ real numbers S

Is there a non-empty subsequence X such that

∑x X∈ x ≤ C

-28 53 -58 -99 13 27 -55 -31 -91 12 -87 -68

Problem 2

Given a sequence of ‘n’ real numbers S

Is there a non-empty subsequence X such that

∑x X∈ x ≤ -175

-28 53 -58 -99 13 27 -55 -31 -91 12 -87 -68

Solution?

Solution

For every i, j such that 1 ≤ i ≤ j ≤ n

Compute the sum of subsequence from i to j

Check if the sum is less than or equal to C

n

Easy problem (run-time is polynomial in ‘n’)

Run

ning

tim

e

Problem 3

Given a set of ‘n’ real numbers S

Is there a non-empty subset X S such that ⊆

∑x X∈ x = 0

Problem 3

Given a set of ‘n’ real numbers S

Is there a non-empty subset X S such that ⊆

∑x X∈ x = 0

{-28, 53, -58, -99, 13, 27, -55, -31, -91, 12, -87, -68}

Solution?

Solution

For every non-empty subset X S ⊆

Compute ∑x X∈ x

Check if the sum is equal to 0

Run

ning

tim

e

n

Hard problem (run-time is exponential in ‘n’)

Problems

Some problems are easy (formal definition soon)

Some problems are hard (formal definition soon)

Many natural questions arise

Which ones will be answered in this course?

Is The Given Problem Easy?

✗

Why Is The Given Problem Easy?

Part I

How Easy is an Easy Problem?

✗

Part II of the “Optimization” course

How Hard is a Hard Problem?

Part II

• What is it about?

• What are the prerequisites?

• What type of material will be provided?

• How will the students be evaluated?

• Tips and tricks

Course Description

Discrete Mathematics

What is a directed graph?

What is an undirected graph?

Walks, paths, and other basic definitions

Some definitions will be provided as needed

Linear Algebra

What is a vector?

What is a matrix?

Rank, base, span and other basic definitions

Some definitions will be provided as needed

Optimization

What is a convex set?

What is a convex function?

Linear function? Polyhedron?

Linear programming?

Basics will be covered in lecture 1

Programming

Theory course, no programming assignments

But some basic programming skills required

Enough to understand pseudo-code

And estimate its run-time

• What is it about?

• What are the prerequisites?

• What type of material will be provided?

• How will the students be evaluated?

• Tips and tricks

Course Description

Course Website

Detailed syllabus

Slides for all the lectures

Links to scribes of related courses

No lecture “notes”

Example exam questions

• What is it about?

• What are the prerequisites?

• What type of material will be provided?

• How will the students be evaluated?

• Tips and tricks

Course Description

Evaluation

Final exam worth 20 marks

“Easy” questions worth 10 marks

“Hard” questions worth 10 marks

Grading on a curve

• What is it about?

• What are the prerequisites?

• What type of material will be provided?

• How will the students be evaluated?

• Tips and tricks

Course Description

Elie Wiesel

Attendance

No minimum attendance requirement

There will be no “roll call”

All the information is available online

But not in an easy-to-follow form

Tip 1: Attend all lectures from start to end

During the Class …

Tip 2: Clarify your doubts

Reasons

No such thing as a “silly question”

Flynn effect

It’s part of my job

It helps me improve the course material

Evaluation

Final exam worth 20-x marks

x is a student-dependent variable

Increases with an interesting answer

Increases with an interesting question

Tip 3: Maximize ‘x’ to minimize stress

Questions?