ISM 206 Lecture 6 Nonlinear Unconstrained Optimization.

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Presentation transcript:

ISM 206 Lecture 6 Nonlinear Unconstrained Optimization

Announcements This week: Nonlinear Optimization Unconstrained (today) Constrained (Thursday and Thursday!) Homework and solutions available Rescheduled midterm

Outline Today: Unconstrained –Helps us with constrained problems (recall primal/dual) One-dimensional rules Taylor’s series Searching for ‘zeros’ Multivariate problems –Necessary and Sufficient conditions –Line search methods

Reminder: Optimization Overview Variables: Objective: Subject to Constraints: Sometimes additional constraints: –Binary –Integer Today we move from linear to nonlinear constraints and objective functions

Single Variable unconstrained problems Remember basic calculus: Find the maximum / minimum of a polynomial function What are optimality conditions? Local vs. global optimum Smooth functions

Taylor’s Series Recall the Taylor expansion for a function This leads to necessary and sufficient conditions for optimality

Finding the turning points Bisection method Newton’s method

Questions and Break

Multivariate optimization Taylor’s series in higher dimensions Leads to more very similar optimality conditions!

Line search methods Steepest Descent Newton’s method Quasi-newton methods