The Ellipsoid Method Ellipsoid squashed sphere Given K
- Slides: 9
The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK. Start with ball containing (polytope) K. yi = center of current ellipsoid. If yiÎK then DONE; (return yi) K If yi K, use separating hyperplane to chop off infeasible half-ellipsoid.
The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK. Start with ball containing (polytope) K. yi = center of current ellipsoid. If yiÎK then DONE; (return yi) 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.
The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK. Start with ball containing (polytope) K. yi = center of current ellipsoid. If yiÎK then DONE; (return yi) 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.
The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK. Start with ball containing (polytope) K. yi = center of current ellipsoid. If yiÎK then DONE; (return yi) 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.
The Ellipsoid Method Ellipsoid º squashed sphere Given K, find xÎK. Start with ball containing (polytope) K. yi = center of current ellipsoid. If yiÎK then DONE; (return yi) 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.
The Ellipsoid Method for Linear Optimization Max c. x subject to xÎK. Ellipsoid º squashed sphere Start with ball containing (polytope) K. yi = center of current ellipsoid. 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.
The Ellipsoid Method for Linear Optimization Max c. x subject to xÎK. Ellipsoid º squashed sphere Start with ball containing (polytope) K. yi = center of current ellipsoid. K c. x ≥ c. yi 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. New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. Repeat for i=0, 1, …, T.
The Ellipsoid Method for Linear Optimization Max c. x subject to xÎK. Ellipsoid º squashed sphere Start with ball containing (polytope) K. yi = center of current ellipsoid. K c. x ≥ c. yi 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. New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. Repeat for i=0, 1, …, T.
The Ellipsoid Method for Linear Optimization Max c. x subject to xÎK. Ellipsoid º squashed sphere Start with ball containing (polytope) K. yi = center of current ellipsoid. x 2 x 1 If yiÎK, use objective function cut c. x ≥ c. yi to chop off K, half-ellipsoid. xk If yi K, use separating hyperplane to chop off infeasible half-ellipsoid. New ellipsoid = min. volume ellipsoid containing “unchopped” x* P half-ellipsoid. x 1, x 2, …, xk: points lying in P. c. xk is a close to optimal value.
