Guide for Week 7
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
- 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
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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
- First and second order conditons for optimality both unconstrained and constrained
- The tangent cone
- NLP regularity and constraint qualifications.
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Skills to Master:
- computing derivatives and Taylor approximations
- locating critical points
- testing the optimality conditions
- checking regularity
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Quiz:
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This quiz is based on material from the Multivariable Calculus Review and Chapter 4
of the notes section 1-3. The first question is based on the definitions and theorems
and the second is computational. The second question will be similar to questions
1-7 on Homework Set 6.