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Lecture20 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 20 Dr. Ted Ralphs
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IE417 Lecture 20 1 Reading for this lecture Sections 9.3
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IE417 Lecture 20 2 Exact Penalty Functions The penalty functions φ ( g i ( x )) = max { 0 , g i ( x ) } , and ψ ( h i ( x )) = | h i ( x ) | are called exact penalty functions . Exact penalty functions achieve optimality for a finite value of μ . For a convex program, μ must exceed the maximum of the absolute values of the optimal Lagrange multipliers.
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IE417 Lecture 20 3 Augmented Lagrangian Methods The problem with the foregoing penalty functions is that they are not differentiable everywhere. Idea : Shift the penalty term a little in order to reduce its dominance of the objective function. Consider the penalty function ψ ( h i ( x )) = [ h i ( x ) - Θ i ] 2 . Assuming only equality constraints, the penalized objective function can then be written as F ( x, v ) = f ( x ) + Σ v i h i ( x ) + μ Σ[ h i ( x )] 2
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IE417 Lecture 20 4 Augmented Lagrangian Methods (continued) Note that if ( x * , v * )
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