Math 484 - Spring 2017 - Class log
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(1. Wednesday 01/18/2017) -
Course intro, Theorem 1.1.1 and 1.1.4, Definition 1.1.3, Started Theorem 1.1.5
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(2. Friday 01/20/2017) -
Theorem 1.1.5, review of dot product and orthogonality, properties of a norm,
distance between points in R^n, open ball, interior set, interior, open and closed sets,
bounded sets and compact sets, the Bolzano-Weierstrass property,
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(3. Monday 01/23/2017) -
gradient, Definition 1.2.2, Theorem 1.2.3, line segment joining x^* and x
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(4. Wednesday 01/25/2017) -
Taylor's formula (Theorem 1.2.4), Def 1.2.8 (pos. def., neg. def., pos. semidef, neg. semidef and indefinite),
Theorem 1.2.9, kth principal minor
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(5. Friday 01/27/2017) -
Theorem 1.3.3 (proof not given), Theorem 1.3.6, brief intro to saddle points (will continue on Wednesday)
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(6. Wednesday 02/01/2017) -
Saddle point definition, Theorem 1.3.7,
coercive functions definition,
examples, Theorem 1.4.4
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(7. Friday 02/03/2017) -
Section 1.5 eigenvalue and positive definiteness,
review of facts about symmetric matrices,
Theorem 1.5.1, Example,
Section 2.1 Convex sets,
Definition of convex sets and examples
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(8. Monday 02/06/2017) -
More examples of convex sets: Closed/Open Halfspaces, Balls, The intersection
of convex sets is convex, solutions of systems of equations.
Definitions weighted average/Convex combinations.
Theorem 2.1.3 - the convex combination of points
in a convex set C is in C. Definition: convex hull,
For any set D, the convex hull of D is
the set of all convex combinations of the points in D.
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(9. Wednesday 02/08/2017) -
Review of convex hull and convex combinations,
Examples of convex and concave functions in 1-d. Definition of convex in 1-d.
Theorem 2.3.1 (convex functions are continuous) - proof not given.
Characterization of convex function that are differentiable,
and characterization of convex functions that are twice differentiable.
Definition of a convex function in R^n,.
Theorem 2.3.3 - proof not given (similar to proof of 2.1.3).
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(10. Friday 02/10/2017)
Theorem 2.3.4, Statement of proof of Theorem 2.3.5 (proof not given in class),
Corollary (2.3.6), Theorem (2.3.7), Statement of Theorem 2.3.10
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(11. Monday 02/13/2017)
Theorem 2.3.10 (Proof of c) only, some examples and discussion of problem on homework,
Theorem 2.4.1 (AM-GM inequality)
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Midterm 1 (02/14/2017)
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(12. Wednesday 02/15/2017)
AM-GM inequality for optimization - examples
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(13. Friday 02/17/2017)
Finished section 2.4 and started section 2.5,
AM-GM inequality for optimization - examples - and general case,
Definitions primal geometric program and dual geometric program
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(14. Friday 02/20/2017)
Passed back Test 1 and discussed solutions.
Definitions and primal dual inequality section 2.5,
started Theorem 2.5.2
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(15. Wednesday 02/22/2017)
Finished Theorem 2.5.2, Steps to solve a geometric program
and geometric programming example where (GP) has no solutions.
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(16. Friday 02/24/2017)
Geometric programming example where (DGP) has many examples,
Introduction to convex programming - some material from section 5.2 of book
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(17. Monday 02/27/2017)
KKT Theorem 5.2.13(Saddle point form) and 5.2.14 (gradient form)
(Assuming every superconsistent convex program has a sensitivity vector - which we will prove next)
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(18. Wednesday 03/01/2017)
Two examples of the use of 5.2.14 (This is the last material that will appear on midterm 2).
Introduction to Theorem 5.1.1.
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(19. Friday 03/03/2017)
Theorem 5.1.1, Corollary 5.1.2 and Theorem 5.1.3
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(20. Monday 03/06/2017)
Corollary 5.1.4, Theorem 5.1.5, Definition: closure, Corollary 5.1.6 (proof) and Theorem (5.1.7) (proof not given),
statement of Theorem 5.1.8
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(21. Wednesday 03/08/2017)
Theorem 5.1.8, Theorem 5.1.9, statement of Theorem 5.1.10
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(22. Friday 03/10/2017)
Theorem 5.1.10, Definitions supremum, infimum, P(z), MP(z), the domain of MP(z), examples
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(23. Monday 03/13/2017)
Examples of the domain of MP(z), The definition of a sensitivity vector and
Theorem 5.2.11. The first parts of theorem 5.2.6
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(24. Wednesday 03/15/2017)
The rest of 5.2.6, Remark 5.2.7 and Theorem 5.2.8 and Theorem 5.2.16. The extended AM-GM
inequality (5.3.1)
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(25. Monday 03/27/2017)
Section 5.3 - Example 5.3.3, Definition GP and DGP, First part of Theorem 5.3.5
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(26. Wednesday 03/29/2017)
Theorem 5.3.5, Brief into to 5.4 and Duality Method
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(27. Friday 03/31/2017)
Definitions (DP), h(λ), MD, feasible,consistent and a solution with respect to (DP),
Theorem 5.4.6, Corollary 5.4.7, Example 5.4.2 Linear programming, Duality Theorem for linear programming,
Intro to example 5.4.5
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(28. Monday 4/3/2017)
Example 5.4.5, Definition 5.4.9 - Duality Gap, Example 5.4.8, Penalty Method - g+(x),
penalty parameter, penalty term, Absolute value penalty function, Lemma 6.1.3 Courant-Beltrami Penalty Function
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Midterm 3 (4/4/2017)
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(29. Wednesday 4/5/2017)
Redo Duality Gap example 5.4.8, Penalty method example 6.2.2, Theorem 6.2.3, Definition Generalized Penalty
Function
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(30. Monday 4/10/2017)
Corollary 6.2.4, Intro to Section 6.3, Theorem 6.3.1
Redo Duality Gap example 5.4.8, Penalty method example 6.2.2, Theorem 6.2.3, Definition Generalized Penalty
Function
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(31. Wednesday 4/12/2017)
Lemma 6.3.2, Theorem 6.3.4, Statement and beginning of Theorem 6.3.5
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(32. Friday 4/14/2017)
Finished Theorem 6.3.5, Started chapter 7, Lagrange multipliers example,
Tangent plane, Tangent space, Normal space
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(33. Monday 4/17/2017)
A path in a surface, Theorem 7.1.5
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(34. Wednesday 4/19/2017)
Examples, Theorem 7.2.1 and discussion of relationship with 5.2.14 and first part of proof.
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(35. Friday 4/21/2017)
Finished Proof of Theorem 7.2.1, Intro to chapter 3 and Newton's Method, Netwon's method
in 1d and the Jacobian Matrix
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(37. Monday 4/24/2017)
Description of a Newton's Method sequence, Example, Newton's method for function minimization,
Theorems 3.1.5 and 3.1.6
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Midterm 3 (4/25/2017)
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(37. Wednesday 4/26/2017)
Section 3.2 The method of Steepest descent, description of sequence, Example 3.2.2,
Theorem 3.2.3 (The method of steepest descent moves in perpendicular steps),
Theorem 3.2.5 (Steepest descent is a descent method),
Statement of the convergence theorem 3.2.6 and its corollary 3.2.7
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(38. Friday 4/28/2017)
Proof of Corollary 3.2.7, Section 3.3 - the four criterion for a good update rule,
Theorem 3.3.1 (Wolfe's Theorem).
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(39. Wednesday 5/3/2017)
Passed back exam 4 and a brief course overview