Interior-point method for NLP

Author names: Cindy Chen
Steward: Dajun Yue and Fengqi You

Introduction

The interior point (IP) method for nonlinear programming was pioneered by Anthony V. Fiacco and Garth P. McCormick in the early 1960s. The basis of IP method restricts the constraints into the objective function (duality) by creating a barrier function. This limits potential solutions to iterate in only the feasible region, resulting in a much more efficient algorithm with regards to time complexity.

To ensure the program remains within the feasible region, a factor, $\mu$, is added to "penalize" close approaches to the boundaries. This approach is analogous to the use of an invisible fence to keep dogs in an unfenced yard. As the dog moves closer to the boundaries, the more shock he will feel. In the case of the IP method, the amount of shock is determined by $\mu$. A large value of $\mu$ gives the analytic center of the feasible region. As $\mu$ decreases and approaches 0, the optimal value is calculated by tracing out a central path. With small incremental decreases in $\mu$ during each iteration, a smooth curve is generated for the central path. This method is accurate, but time consuming and computationally intense. Instead, Newton's method is often used to approximate the central path for non-linear programming. Using one Newton step to estimate each decrease in $\mu$ for each iteration, a polynomial ordered time complexity is achieved, resulting in a small zig-zag central path and convergence to the optimal solution.

Algorithm

minimize $c^Tx - \mu\sum_{i=1} ln(x_i)$
subject to $Ax=b$

References

1. Shanno, David. "Who Invented the Interior Point Method?" Documenta Mathematica Extra Volume ISMP (2012): 55-64. link