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The Ellipsoid Method

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.

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The Ellipsoid Method

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  1. The Ellipsoid Method Ellipsoid º squashed sphere 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 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 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 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 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. xk x1 New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. x* P x1, x2, …, xk: points lying in P. c.xk is a close tooptimal value.

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