# Interior-point method for LP

Authors: John Plaxco, Alex Valdes, Wojciech Stojko. (ChE 345 Spring 2014)

Steward: Dajun Yue, Fengqi You

Date Presented: May 25, 2014

## Contents |

# Introduction

Interior point methods are a type of algorithm that are used in solving both linear and nonlinear convex optimization problems that contain inequalities as constraints. The LP Interior-Point method relies on having a linear programming model with the objective function and all constraints being continuous and twice continuously differentiable. In general, a problem is assumed to be strictly feasible, and will have a dual optimal that will satisfy Karush-Kuhn-Tucker (KKT). The problem is solved (assuming there IS a solution) either by iteratively solving for Karush-Kuhn-Tucker (KKT) conditions or to the original problem with equality instead of inequality constraints, and then applying Newton's method to these conditions.

Interior point methods came about from a desire for algorithms with better theoretical bases than the simplex method. While the two strategies are similar in a few ways, the interior point methods involve relatively expensive (in terms of computing) iterations that quickly close in on a solution, while the simplex method involves usually requires many more inexpensive iterations. From a geometric standpoint, interior point methods approach a solution from the interior or exterior of the feasible region, but are never on the boundary.

# Uses

# Algorithm

# Example

# Conclusion

## Sources

1. R.J. Vanderbei, Linear Programming: Foundations and Extensions (Chp 17-22). Springer, 2008.

2. J. Nocedal, S. J. Wright, Numerical optimization (Chp 14). Springer, 1999.

3. S. Boyd, L. Vandenberghe, Convex Optimization (Chp 11). Cambridge University Press, 2009