Guide for Week 2
Math 408 Section A, January 14, 2013
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Reading Assignment:
Homework Assignment:
Vocabulary Words
- Multivariate calculus review
- vector norm
- 1, 2, and infinity norms
- norm equivalence in finite dimensions
- continuity
- open, closed, bounded, and compact sets
- cluster points
- Weierstrass Compactness Theorem
- Weierstrass Extreme Value Theorem
- little-o notation
- differentiablity: gradient, hessians, Jacobians
- Language and notation of nonlinear optimization
- linear function
- convex polyhedron
- quadratic function
- objective function
- feasible or constraint region
- Convex set and function
- linear and quadratic programming
- linear least squares problem
- global solution and strict global solution
- local solution and strict local solution
- first-order optimality condition for unconstrained problems
- critical points and stationary points
- 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
- the unitary diagonalization of symmetric matrices
- positive semi-definite and positive definite matrices
- Choleky factorization of a symmetric positive definite matrix
- matrix square roots
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Key Concepts:
- Multivariable calculus review
- 1, 2, and infinity norms
- continuity and differentiablity
- compactness
- cluster points
- Weierstrass Extreme Value Theorem
- gradients, hessians, and Jacobians
- Language and notation
- Convexity
- local and global (strict) extrema
- first-order optimality conditions
- critical points and stationary points
- Optimality Conditions for Unconstrained Problems
- Coercivity and existence
- first- and second-order optimality conditions
- convexity and optimality
- properties of symmetric matrices
- the linear least squares problem
- quadratic functions
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Skills to Master:
- locating and classifying critical points
- checking coercivity
- checking convexity
- computing gradients, hessians, and Jacobians
- working with linear least square objectives
- working with quadratic ojectives
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Quiz:
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The quiz will consist of 2 questions.
The first question will be related to the vocabulary
words The language and
notation of optimization and Optimality Conditions for Unconstrained Problems
(excuding Cholesky factorization and matrix square roots).
The second will be computational
focusing on the linear least squares objectives, properties of symmetric matrices,
computing hessians and gradients, and
computing first-order critical points.