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Lecture21

# Lecture21 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 21 Dr. Ted Ralphs

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IE417 Lecture 21 1 Reading for this lecture Sections 9.4-9.5
IE417 Lecture 21 2 Barrier Methods So far, we have talked about exterior penalty methods . Now, we move on to interior penalty methods or interior point methods . The idea is similar, except now we start with a feasible point and impose a steep penalty for approaching the boundary. Previously, we let the penalty multiplier go to infinity. Now, we will let the penalty itself go to infinity. For reasons which will be obvious, these methods only work with inequality constraints.

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IE417 Lecture 21 3 Barrier Functions A barrier function is B ( x ) = m i =1 φ ( g i ( x )) where φ is a continuous function of one variable, φ ( y ) 0 if y < 0 , lim y 0 + φ ( y ) = , Example : φ ( y ) = - 1 /y, φ ( y ) = log(min { 1 , - y } ) Consider θ ( μ ) = inf { f ( x ) + μB ( x ) : x X } What happens if we solve min θ ( μ ) s.t. μ 0
IE417 Lecture 21 4 Performance of Barrier Methods If f, g i and B are continuous, X

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Lecture21 - Advanced Mathematical Programming IE417 Lecture...

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