Lecture19 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 19 Dr. Ted Ralphs IE417 Lecture 19 1 Reading for this lecture • Sections 9.1-9.2 IE417 Lecture 19 2 Constrained Optimization • In Chapter 9, we look at methods based on applying unconstrained methods to constrained problems. • Idea : Penalize violations of the constraints in the objective function. • Consider min f ( x ) s.t. h ( x ) = 0 • Try min { f ( x ) + [ h ( x )] 2 } . Will this work? IE417 Lecture 19 3 Penalty Functions • A suitable penalty function α is α ( x ) = Σ φ ( g i ( x )) + Σ ψ ( h i ( x )) where – φ ( y ) = 0 if y ≤ – φ ( y ) > if y > – ψ ( y ) = 0 if y = 0 – ψ ( y ) > if y 6 = 0 IE417 Lecture 19 4 Performance of Penalty Methods • Suppose we simply solve min { f ( x ) + μα ( x ) : x ∈ X } for some suitable penalty function and some μ > . • Consider solving the following problem: max Θ( μ ) s.t. μ ≥ where Θ( μ ) = inf { f ( x ) + μα ( x ) : x ∈ X } ....
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This note was uploaded on 08/06/2008 for the course IE 417 taught by Professor Linderoth during the Fall '08 term at Lehigh University .

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Lecture19 - Advanced Mathematical Programming IE417 Lecture...

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