Lecture2 - Advanced Mathematical Programming IE417 Lecture...

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Unformatted text preview: Advanced Mathematical Programming IE417 Lecture 2 Dr. Ted Ralphs IE417 Lecture 2 1 Reading for This Lecture • Primary Reading – Chapter 2, Sections 1-3 • Secondary Reading – Chapter 1 – Appendix A IE417 Lecture 2 2 Preliminaries IE417 Lecture 2 3 Real Vector Spaces • A real vector space is a set V , along with – an addition operation that is closed, commutative, and associative . – an element ∈ V such that a + 0 = a, ∀ a ∈ V . – an additive inverse operation such that ∀ a ∈ V, ∃ a ∈ V such that a + a = 0 . – a closed, scalar multiplication operation such that ∀ λ,μ ∈ R ,a,b ∈ V * λ ( a + b ) = λa + λb * ( λ + μ ) a = λa + μa * λ ( μa ) = ( λμ ) a * 1 a = a IE417 Lecture 2 4 Norms on Vector Spaces • A norm on a vector space is a function k · k : V → R satisfying – k v k ≥ ∀ v ∈ V – k v k = 0 if and only if v = 0 – k v + w k ≤ k v k + k w k , ∀ v,w ∈ V – k λv k = | λ | · k v k • Norms are used for measuring the “ size ” of an object or the “ distance ” between two objects in a vector space. • These are the normal properties you would expect such a measure to have. IE417 Lecture 2 5 Examples of Vector Spaces • R n • Z n • R n × n • { y ∈ R m : Ax = y, ∃ x ∈ R n } • Unless otherwise noted, we will be dealing with R n IE417 Lecture 2...
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Lecture2 - Advanced Mathematical Programming IE417 Lecture...

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