1 / 9

Linear Programming Interior-Point Methods

Linear Programming Interior-Point Methods. D. Eiland. Linear Programming Problem. LP is the optimization of a linear equation that is subject to a set of constraints and is normally expressed in the following form :. Minimize :. Subject to :. Barrier Function.

kieu
Télécharger la présentation

Linear Programming Interior-Point Methods

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. Linear ProgrammingInterior-Point Methods D. Eiland

  2. Linear Programming Problem LP is the optimization of a linear equation that is subject to a set of constraints and is normally expressed in the following form : Minimize : Subject to :

  3. Barrier Function To enforce the inequality on the previous problem, a penalty function can be added to Then if any xj 0, then trends toward As , then is equivalent to

  4. Lagrange Multiplier To enforce the constraints, a Lagrange Multiplier (-y) can be added to Giving a linear function that can be minimized.

  5. Optimal Conditions Previously, we found that the optimal solution of a function is located where its gradient (set of partial derivatives) is zero. That implies that the optimal solution for L(x,y) is found when : Where :

  6. Optimal Conditions (Con’t) By defining the vector , the previous set of optimal conditions can be re-written as

  7. Newton’s Method Newton’s method defines an iterative mechanism for finding a function’s roots and is represented by : When ,

  8. Optimal Solution Applying this to we can derive the following :

  9. Interior Point Algorithm This system can then be re-written as three separate equations : Which is used as the basis for the interior point algorithm : • Choose initial points for x0,y0,z0 and the select value for τ between 0 and 1 • While Ax - b != 0 • Solve first above equation for Δy [Generally done by matrix factorization] • Compute Δx and Δz • Determine the maximum values for xn+1, yn+1,zn+1 that do not violate the constraints x >= 0 and z >= 0 from : With : 0 < a <=1

More Related