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The Ellipsoid Method. Ellipsoid º squashed sphere Start with ball containing (polytope) K . y i = center of current ellipsoid. Given K , find x Î K. If y i Î K then DONE ; (return y i ) If y i K , use separating hyperplane to chop off infeasible half-ellipsoid. K. - PowerPoint PPT Presentation
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The Ellipsoid MethodEllipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xK.
If yiK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
K
The Ellipsoid Method
If yiK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
Ellipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xK.
K
The Ellipsoid MethodEllipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.If yiK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
K
Given K, find xK.
The Ellipsoid MethodEllipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
K
If yiK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
Given K, find xK.
The Ellipsoid MethodEllipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
K
If yiK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
Given K, find xK.
The Ellipsoid Method for Linear Optimization
Ellipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.
Max c.x subject to xK.
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
If yiK,
The Ellipsoid Method for Linear Optimization
Ellipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.
Max c.x subject to xK.
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
If yiK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid.
c.x ≥ c.yi
The Ellipsoid Method for Linear Optimization
Ellipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.
Max c.x subject to xK.
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
If yiK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid.
c.x ≥ c.yi
The Ellipsoid Method for Linear Optimization
Ellipsoid squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Max c.x subject to xK.
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
If yiK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid.
P
x1, x2, …, xk: points lying in P. c.xk is a close to optimal value.
x1
x2
xk
x*
