Reading Assignment:

• Optimality Conditions for Unconstrained Problems: Due Monday, February 2.
• Optimality Conditions for Constrained Problems: Due Friday, February 6.

Homework Assignment:

• Do all of the exercises in the notes on unconstrained optimization.
• Do all of the exercises in the notes on constrained optimization.

Vocabulary List:

• Optimality Conditions for Unconstrained Problems
  • The directional derivative.
  • The gradient and Hessian.
  • Eigen-value decomposition for symmetric matrices.
  • Positive definite and positive semi-definite matrices.
  • First-order necessary conditions for optimality.
  • Second-order necessary conditions for optimality.
  • Second-order sufficient conditions for optimality.
  • Convexity, sets and functions.
  • Subdifferential inequality.
  • conditions which imply that a function is convex
  • Show that a local minimizer of a covex function is a global minimizer.
  • Necessary and sufficient conditions for optimality in the convex case.
• Constrained Optimization
  • feasible directions
  • tangent cone
  • regularity
  • The Lagrangian
  • first-order conditions for optimality
  • KKT conditions (KKT pairs)
  • second-order conditions for optimality
  • first-order necessary and sufficient conditions under convexity
  • convexity implies the local solutions are global solutions

Key Concepts:

• Eigen-values and eigen-vectors of symmetric matrices.
• Optimality Conditions in the constrained and unconstrained cases
• Second-order Taylor expansion.
• Convexity

Skills to Master:

• Checking optimality conditions
• Checking if a matrix is positive semi-definite or positive semi-definite.
• checking regularity
• checking convexity

Quiz: