Advanced Optimization

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Advanced Optimization

• Week 4
• Regina Burachik

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Outline

• One-sided Directional Derivatives
• Epigraphs and level sets
• Supremum of convex functions

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One-sided directional derivative
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• One sided derivative always exists for a convex
  function!
• See Lemma 2.2.5 in Lecture notes

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Sets induced by functions

• Graph
• Epigraph
• Level Sets

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Graph
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Epigraph
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Epigraph
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Level Sets
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Level Set
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Some Facts

• A function is convex if and only if its epigraph
  is a convex set
• If a function is convex, then its level sets are
  convex (converse not true)

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A Convex Optimization Problem
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Solution Set, Optimal Value
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• The set of solutions of a convex optimization
  problem is convex