Linear Programming Interior-Point Methods

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