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The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK.

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Presentation on theme: "The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK."— Presentation transcript:

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.


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