Guide for Week 6
Math 408
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Reading Assignment:
Homework Assignment:
Vocabulary Words
- Elements of Multivariable Calculus
- norms for vectors and matrices and the closed unit ball
- continuity
- open, closed, bounded, and compact sets
- cluster points
- directional derivatives and partial derivatives
- derivatives and their representations as matrices of partial derivatives
- the little-o notation and the little-o class
- the gradient and the Hessian matrix
- first-order expansion and the first-order Taylor approximation
- the delta method for commputing derivatived
- differential calculus and the chain rule
- the mean value theorem in its various forms
- Lipschitz continuity
- the quadratic bound lemma
- second-order expansion and the second-order Taylor approximation
- Optimality Conditions for Unconstrained Problems
- 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
- convex functions and sets (and strict convex functions)
- the epi-graph and essential domain of a function
- existence of directional derivatives for convex functions
- the convexity and optimality theorem (proof not required)
- first- and second-order conditions for convexity checking
- gradients and hessians of linear least squares functions and general
quadratic functions
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Key Concepts:
- Elements of Multivariable Calculus
- first- and second-order expansions and approximations
- the Mean Value Theorem in its various forms
- the delta method for computing derivatives
- Optimality Conditions for Unconstrained Problems
- Coercivity and existence
- first- and second-order optimality conditions
- convexity and optimality
- Coercivity and existence
- first- and second-order optimality conditions
- convexity and optimality
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Skills to Master:
- implementing the conjugate gradient algorithm
- computing derivatives and Taylor approximations
- checking compactness
- locating and classifying critical points
- checking coercivity
- checking convexity
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Quiz:
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This quiz covers Sections 5 and 6 of Chapter 3 of
the online notes
as well as the basic continuity and differentiablility material
from Appendix B (Elements of Multivariable Calculus) and the notion
of a directional derivative.
There are two questions on the quiz. The first questions your knowledge
of the definitions and theoretical content and the second is similar to
the kinds of questions asked in
Homework set 5.