1. The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xÎK.
If yiÎK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. K

2. The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xÎK.
If yiÎK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.

3. The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xÎK.
If yiÎK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.

4. The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xÎK.
If yiÎK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.

5. The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Given K, find xÎK.
If yiÎK then DONE; (return yi)
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.

6. The Ellipsoid Method for Linear Optimization
Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Max c.x subject to xÎK.
If yiÎK, K
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.

7. The Ellipsoid Method for Linear Optimization
Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Max c.x subject to xÎK.
If yiÎK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid. K
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.
c.x ≥ c.yi

8. The Ellipsoid Method for Linear Optimization
Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Max c.x subject to xÎK.
If yiÎK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid. K
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid.
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,T.
c.x ≥ c.yi

9. The Ellipsoid Method for Linear Optimization
Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Max c.x subject to xÎK. x2
If yiÎK, use objective function cut
c.x ≥ c.yi to chop off K, half-ellipsoid.
If yiK, use separating hyperplane to chop off infeasible half-ellipsoid. x1 xk
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. P x*
x1, x2, …, xk: points lying in P. c.xk is a close to optimal value.