1 / 7

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.

caroun
Download Presentation

The Ellipsoid Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related