Math 287 * Nonlinear Optimization * Spring 2008

Course outline

The course covers theory and techniques for both unconstrained and constrained optimization in a general (nonlinear) setting. The emphasis is on methods that are useful for solving real problems, although important theoretical ideas such as convexity and the Karush-Kuhn-Tucker conditions are also covered.

The following topics will be covered.

• 1. Introduction
  • Optimization models
  • Conditions for optimality
  • Local versus global optimality
  • Convexity
  • Convergence analysis
• 2. Unconstrained one-dimensional methods
  • A. Derivative-zeroing methods
    • Newton's method
    • Bisection
    • Regula falsi
    • Secant method
  • B. Non-derivative methods
    • Bracketing framework
    • Golden section method
    • Quadratic interpolation
  • C. Cubic interpolation
• 3. Unconstrained multi-dimensional methods
  • A. Smooth functions
    • Line search framework
    • Taylor polynomials
    • Newton's method
    • Steepest descent
    • Quasi-Newton methods
    • Conjugate gradient methods
    • Convergence for line search framework
    • Trust region framework
  • B. Noisy functions
    • Nelder-Mead simplex method
    • Hooke-Jeeves pattern search
• 4. Constrained multi-dimensional optimization
  • A. Theory
    • Karush-Kuhn-Tucker conditions
  • B. Methods
    • Barrier functions
    • Penalty functions
    • Sequential quadratic programming