polyhedral optimization lecture 1 – part 1 m. pawan kumar [email protected] slides available...
TRANSCRIPT
![Page 1: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/1.jpg)
Polyhedral OptimizationLecture 1 – Part 1
M. Pawan Kumar
Slides available online http://cvn.ecp.fr/personnel/pawan/
![Page 2: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/2.jpg)
• 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
![Page 3: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/3.jpg)
Problem 1
Given a set of ‘n’ real numbers S
Is there a non-empty subset X S such that ⊆
∑x X∈ x ≤ C
![Page 4: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/4.jpg)
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}
![Page 5: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/5.jpg)
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}
![Page 6: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/6.jpg)
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
![Page 7: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/7.jpg)
n
Easy problem (run-time is polynomial in ‘n’)
Run
ning
tim
e
![Page 8: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/8.jpg)
Problem 2
Given a sequence of ‘n’ real numbers S
Is there a non-empty subsequence X such that
∑x X∈ x ≤ C
![Page 9: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/9.jpg)
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
![Page 10: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/10.jpg)
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?
![Page 11: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/11.jpg)
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
![Page 12: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/12.jpg)
n
Easy problem (run-time is polynomial in ‘n’)
Run
ning
tim
e
![Page 13: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/13.jpg)
Problem 3
Given a set of ‘n’ real numbers S
Is there a non-empty subset X S such that ⊆
∑x X∈ x = 0
![Page 14: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/14.jpg)
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?
![Page 15: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/15.jpg)
Solution
For every non-empty subset X S ⊆
Compute ∑x X∈ x
Check if the sum is equal to 0
![Page 16: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/16.jpg)
Run
ning
tim
e
n
Hard problem (run-time is exponential in ‘n’)
![Page 17: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/17.jpg)
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?
![Page 18: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/18.jpg)
Is The Given Problem Easy?
✗
![Page 19: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/19.jpg)
Why Is The Given Problem Easy?
Part I
![Page 20: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/20.jpg)
How Easy is an Easy Problem?
✗
Part II of the “Optimization” course
![Page 21: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/21.jpg)
How Hard is a Hard Problem?
Part II
![Page 22: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/22.jpg)
• 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
![Page 23: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/23.jpg)
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
![Page 24: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/24.jpg)
Linear Algebra
What is a vector?
What is a matrix?
Rank, base, span and other basic definitions
Some definitions will be provided as needed
![Page 25: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/25.jpg)
Optimization
What is a convex set?
What is a convex function?
Linear function? Polyhedron?
Linear programming?
Basics will be covered in lecture 1
![Page 26: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/26.jpg)
Programming
Theory course, no programming assignments
But some basic programming skills required
Enough to understand pseudo-code
And estimate its run-time
![Page 27: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/27.jpg)
• 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
![Page 28: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/28.jpg)
Course Website
Detailed syllabus
Slides for all the lectures
Links to scribes of related courses
No lecture “notes”
Example exam questions
![Page 29: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/29.jpg)
• 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
![Page 30: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/30.jpg)
Evaluation
Final exam worth 20 marks
“Easy” questions worth 10 marks
“Hard” questions worth 10 marks
Grading on a curve
![Page 31: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/31.jpg)
• 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
![Page 32: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/32.jpg)
Elie Wiesel
![Page 33: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/33.jpg)
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
![Page 34: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/34.jpg)
During the Class …
Tip 2: Clarify your doubts
![Page 35: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/35.jpg)
Reasons
No such thing as a “silly question”
Flynn effect
It’s part of my job
It helps me improve the course material
![Page 36: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/36.jpg)
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
![Page 37: Polyhedral Optimization Lecture 1 – Part 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649e2a5503460f94b18ad6/html5/thumbnails/37.jpg)
Questions?