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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. Min c . x subject to x Î 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. Min c.x subject to xÎK. If yiK, find a constraint a.x ≤ b of K violated by yi. Use hyperplanea.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 hyperplanea.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 hyperplanea.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 hyperplanea.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 hyperplanea.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 hyperplanea.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 hyperplanea.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. xk x1 New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. x* K x1, x2, …, xk: points lying in K. c.xk is a close tooptimal value.

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