polyhedral optimization lecture 3 – part 3 m. pawan kumar [email protected] slides available...
TRANSCRIPT
Polyhedral OptimizationLecture 3 – Part 3
M. Pawan Kumar
Slides available online http://cvn.ecp.fr/personnel/pawan/
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
Feasibility via Ellipsoid Method
Feasible region of LP
Feasibility via Ellipsoid Method
Ellipsoid containing feasible region of LP
Feasibility via Ellipsoid Method
Centroid of ellipsoid
Feasibility via Ellipsoid Method
Separating hyperplane for centroid
Feasibility via Ellipsoid Method
Smallest ellipsoid containing “truncated” ellipsoid
Feasibility via Ellipsoid Method
Centroid of ellipsoid
Feasibility via Ellipsoid Method
Separating hyperplane for centroid
Feasibility via Ellipsoid Method
Smallest ellipsoid containing “truncated” ellipsoid
Feasibility via Ellipsoid Method
Centroid of ellipsoid
Feasibility via Ellipsoid Method
Terminate when feasible solution is found
• 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
• 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
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
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
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*)*∈
(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
(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
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
Optimization Problem over P
Polyhedron P = {x: Ax ≤ b}
max cTx
x P ∈
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
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*
• Polar Polyhedron
• Using Optimization for Separation
• Poly-Time Equivalence
Outline
Poly-Time Equivalence
OptimizationonP
SeparationonP*
Polarity
OptimizationonP*
Ellipsoidmethod
Separationon
(P*)* = P
Polarity
Ellipsoidmethod