1. The Ellipsoid Method Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi to chop off infeasible half-ellipsoid. K

2. The Ellipsoid Method Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi 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
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi 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
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi 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
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi to chop off infeasible half-ellipsoid.
If yiÎK, use objective function cut
c.x ≤ c.yi to chop off K, half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
c.x ≤ c.yi

6. The Ellipsoid Method Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK.
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi to chop off infeasible half-ellipsoid.
If yiÎK, use objective function cut
c.x ≤ c.yi to chop off K, half-ellipsoid. K
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Repeat for i=0,1,…,t.
c.x ≤ c.yi

7. The Ellipsoid Method Ellipsoid º squashed sphere
Start with ball containing (polytope) K.
yi = center of current ellipsoid.
Min c.x subject to xÎK. x2
If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplane a.x ≤ a.yi to chop off infeasible half-ellipsoid.
If yiÎK, use objective function cut
c.x ≤ c.yi to chop off K, half-ellipsoid. x1 xk
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. K x*
x1, x2, …, xk: points lying in K. c.xk is a close to optimal value.