Difference between revisions of "Interior-point method for NLP"

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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 ([http://en.wikipedia.org/wiki/Duality_%28optimization%29 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.  
 
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 ([http://en.wikipedia.org/wiki/Duality_%28optimization%29 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.  
  
 
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To ensure the program remains within the feasible region, a factor, <math>\mu</math>, 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 <math>\mu</math>. A large value of <math>\mu</math> gives the analytic center of the feasible region. As <math>\mu</math> decreases and approaches 0, the optimal value is calculated by tracing out a central path. With small incremental decreases in <math>\mu</math> 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 <math>\mu</math> for each iteration, a polynomial ordered time complexity is achieved, resulting in a small zig-zag central path and convergence to the optimal solution.
Use value of mu to control how much value is given to the barrier, large mu means we stay far from boundaries. First solve with large value of mu gives us analytic center of the feasible region. As we decrease mu, we can find the optimal value by tracing out the central path. Traveling along the central path is time consuming and computational intense - can approximate by using Newton’s method for solving NLPs. To get polynomial ordered time complexity, we decrease mu slowly, using one Newton step each time we decrease mu. This results in small zig-zag convergence to optimal point. In practice, we can often decrease more rapidly and converge faster. Newton steps approximate central path through interior of feasible region.  
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=Algorithm=
 
=Algorithm=

Revision as of 22:06, 23 May 2015

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