Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar Slides available online

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’) Running time

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

Problem 2 Given a sequence of ‘n’ real numbers S Is there a non-empty subsequence X such that ∑ x ∈ X x ≤ 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’) Running time

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

Running time 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?