Lecture3

# Lecture3 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 3 Dr. Ted Ralphs

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IE417 Lecture 3 1 Reading for This Lecture Primary Reading Chapter 2, Sections 4-7 Secondary Reading Chapter 1 Appendix A
IE417 Lecture 3 2 Hyperplanes and Half-spaces A hyperplane is a set of the form H = { x : p T x = α } where p is a nonzero vector in R n and α is a scalar. A hyperplane deﬁnes two closed half-spaces H - = { x : p T x α } and H + = { x : p T x α } . There are also corresponding open half-spaces . A hyperplane H = { x : p T x = α } is said to separate two nonempty sets S 1 and S 2 if S 1 ⊆ { x : p T x α } , and S 2 ⊆ { x : p T x α }

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3 Separation Theorem I Theorem 1. Let S be a nonempty, closed convex set in R n and y / S . Then there exists a unique point x * S with minimum distance from y . Furthermore, x is the minimizing point if and only if ( y - x * ) T ( x - x * ) 0 x S . What is this really saying?
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Lecture3 - Advanced Mathematical Programming IE417 Lecture...

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