Download - Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar [email protected] Slides available online
![Page 1: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/1.jpg)
Polyhedral OptimizationLecture 3 – Part 3
M. Pawan Kumar
Slides available online http://cvn.ecp.fr/personnel/pawan/
![Page 2: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/2.jpg)
Solving Linear Programs
s.t. A x ≤ b
maxx cTxOptimization
Feasibility asks if there exists an x such that
cTx ≥ K
A x ≤ b
Optimization via binary search on K
Feasible solution
For a given K
![Page 3: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/3.jpg)
Feasibility via Ellipsoid Method
Feasible region of LP
![Page 4: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/4.jpg)
Feasibility via Ellipsoid Method
Ellipsoid containing feasible region of LP
![Page 5: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/5.jpg)
Feasibility via Ellipsoid Method
Centroid of ellipsoid
![Page 6: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/6.jpg)
Feasibility via Ellipsoid Method
Separating hyperplane for centroid
![Page 7: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/7.jpg)
Feasibility via Ellipsoid Method
Smallest ellipsoid containing “truncated” ellipsoid
![Page 8: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/8.jpg)
Feasibility via Ellipsoid Method
Centroid of ellipsoid
![Page 9: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/9.jpg)
Feasibility via Ellipsoid Method
Separating hyperplane for centroid
![Page 10: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/10.jpg)
Feasibility via Ellipsoid Method
Smallest ellipsoid containing “truncated” ellipsoid
![Page 11: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/11.jpg)
Feasibility via Ellipsoid Method
Centroid of ellipsoid
![Page 12: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/12.jpg)
Feasibility via Ellipsoid Method
Terminate when feasible solution is found
![Page 13: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/13.jpg)
• Separating hyperplane in polynomial time– Check each of the ‘m’ LP constraints in O(n) time
• New ellipsoid in polynomial time– Shor (1971), Nemirovsky and Yudin (1972)
• Polynomial iterations (Khachiyan 1979, 1980)– Volume of ellipsoid reduces exponentially
• Only requires a separation oracle– Constraint matrix A can be very large
Ellipsoid Method
![Page 14: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/14.jpg)
• Separation implies easy optimization
• What about the reverse?
• Matroid polytopes admit greedy optimization
• Do they allow easy separation?
• Why are we even interested in this?
Optimization vs. Separation
![Page 15: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/15.jpg)
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
![Page 16: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/16.jpg)
Polar Polyhedron
Polyhedron P = {x: Ax ≤ b}
Polar Polyhedron P* = {y: for all x P, ∈ xTy ≤ 1}
Assume 0 is in the interior of P
(P*)* = P Proof?
b > 0
No “loss of generality” as P can be translated
![Page 17: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/17.jpg)
P is a subset of (P*)*
If x P, then for all ∈ y P* we have ∈ xTy ≤ 1
(P*)* = {z: for all y P*, ∈ zTy ≤ 1}
Therefore, x (P*)*∈
![Page 18: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/18.jpg)
(P*)* is a subset of P
Let there be an x P∉
There must exist a separating hyperplane
cTx > d cTz ≤ d, for all z P ∈
Since 0 interior of P, d > 0 ∈
Without loss of generality, d = 1
![Page 19: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/19.jpg)
(P*)* is a subset of P
Let there be an x P∉
There must exist a separating hyperplane
cTx > 1 cTz ≤ 1, for all z P ∈
c P* ∈
x (P*)*∉
Why?
Why? Hence proved
![Page 20: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/20.jpg)
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
![Page 21: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/21.jpg)
Optimization Problem over P
Polyhedron P = {x: Ax ≤ b}
max cTx
x P ∈
![Page 22: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/22.jpg)
Separation Problem over P*
Polar Polyhedron P* = {y: for all x P, ∈ xTy ≤ 1}
Given y, return ‘YES’ if y P* ∈
Otherwise, return separating hyperplane
![Page 23: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/23.jpg)
Using Optimization for Separation
Set c = y
max cTx
x P ∈
C* =
If C* ≤ 1, then return ‘YES’
If C* > 1, then return x*
Optimal solution x*
![Page 24: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/24.jpg)
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
![Page 25: Polyhedral Optimization Lecture 3 – Part 3 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader035.vdocuments.us/reader035/viewer/2022062715/56649d825503460f94a67548/html5/thumbnails/25.jpg)
Poly-Time Equivalence
OptimizationonP
SeparationonP*
Polarity
OptimizationonP*
Ellipsoidmethod
Separationon
(P*)* = P
Polarity
Ellipsoidmethod