Reading Assignment:

• Chapter 4: Optimality Conditions Due Friday, February 14.
• Convexity M Due Ferbruary 28
• Line Search Methods Due March 4

Homework Assignment:

• Do all problems in problem set 5 Due Friday, February 14.
• Do all problems in problem set 6 Due Friday, February 21.
• Do all problems in problem set 7 Due Friday, February 28

Vocabulary Words

• Optimality Conditions
  • Weierstrass extreme value theorem
  • coercive functions
  • The coercivity and compactness theorem (proof not required)
  • The coercivity and existence theorem (proof required)
  • The Basic first-order optimality result (proof not required)
  • The first-order necessary conditions for optimality
  • the second-order necessary and sufficient conditions for optimality
  • Tangent cone
  • Basic Constrained 1st-Order Optimality Conditions
  • Nonlinear Programming
  • Regularity and constraint qualifications LICQ and MFCQ
  • the Lagrangian and Lagrange multipliers
  • Constrained 1st-Order Optimality Conditions
  • Karush-Kuhn-Tucker (KKT) conditions
  • Constrained 2nd-Order Sufficient Conditions
• Convexity
  • convex set
  • epigraph ond domain of a function
  • convex functio
  • closed or lsc function
  • convex hull
  • support function
  • local = global min
  • sublinearity of the directional derivative
  • subdifferential inequality
  • first-order necessary and sufficient condition for optimality
  • tangent and normal cones
  • test for convexity
  • convex NLP
  • Slater CQ
  • necessary and sufficient first-order optimality in convex NLP
  • saddle points
  • the Lagrangian
  • the primal and dual problems
  • Lagrangian duality

Key Concepts:

• Optimality Conditions
  • First and second order conditons for optimality both unconstrained and constrained
  • The tangent cone
  • NLP regularity and constraint qualifications.
• Convexity
  • the epigraphical perspective
  • local = global
  • sublinearity of the directional derivative
  • test for convexity
  • Slater CQ
  • Lagrangian Duality

Skills to Master:

• computing derivatives and Taylor approximations
• locating critical points
• testing the optimality conditions
• checking regularity

• testing for convexity
• testing for optimality
• testing the Slater CQ
• constructing Lagrangian duals

Quiz:

• There will not be any questions on constraint qualifications on this quiz. There are two questions on the quiz. The first question concerns the theoretical content of Sections 4, 5, and 7 of Chapter 4 of the notes. The second question will be similar to the content of questions 1, 2, and 5 of homework set 7. In particular, be prepared to compute KKT points and their associated Lagrange multipliers.