My mentor assigned me with the task of studing the content of appendix A and part of appendix B in Bertsekas' Nonlinear Programing, which cover the basics of convex analysis and its prerequisities. The appendix, in order, goes through closed and open sets, eigenvales and square matrices, symmetric and positive definite matrices, going through the mean value theorem, second order expansion, descent lemma, implicit function theorem, contraction mapping theorem, until convex sets and funtions, caratheodory's theorem, some properties of closure and continuity, finishing with projection theorem. But i'm not familiar with most of it, and have yet to cover a significant part, and because of that, I want material to assist me with teory and pratice.
The resources i'm after preferably would have lots of examples and exercises, it could cover content that wouldn't help me with the appendix, but in that case you could tell which sections are worth reading, similarly it dosen't have to cover all of the topics I mentioned, but knowing what isn't covered is helpful. Idealy I like texts that provide reflections that sharpen intuition, change the way you think about something, help with visualization, or simply add examples and strengthen pratice. To help understand what I idealy want here is an example that have all the of the aspects above:
In Spivaks book Calculus on Manifolds when he introduces the definitions of interior, exterior and boundary, he points out that these have counter intuitive meaning, which you will understand in the exercises tha he mentions.
If $A \subset Rn$ and $x \in R^n$, then one of three possibilities must hold (Figure 1-2) :
- There is an open rectangle $B$ such that $x \in B \subset A$.
- There is an open rectangle $B$ such that $x \in B \subset R^n - A$.
- If $B$ is any open rectangle with $x \in B$, then B contains points of both $A$ and $R^n - A$.
Those points satisfying (1) constitute the interior of $A$, those satisfying (2) the exterior of $A$, and those satisfying (3) the boundary of $A$. Problems 1-16 to 1-18 show that these terms may sometimes have unexpected meanings.
Referred exercises:
1- 16. Find the interior, exterior, and boundary of the sets:
$\{x \in R^n: |x| <= 1\}$
$\{x \in R^n: |x| = 1\}$
$\{x \in R^n: \textrm{each } x^i \textrm{ is rational}\}$.
1-17. Construct a set $A \subset [0,1]\times[0,1]$ such that $A$ contains at most one point on each horizontal and each vertical line but boundary $A = [0,1]\times[0,1]$. Hint: It suffices to ensure that $A$ contains points in each quarter of the square $[0,1]\times[0,1]$ and also in each sixteenth, etc.
1-18. If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that each rational number in $(0,1)$ is contained in some $(a_i,b_i)$, show that boundary $A = [0,1] - A$.
After that I got a muth clearer picture of what those three concepts mean. I had a similar experience with Cauculus of the same author, and with Linear Algebra from Hoffman and Kunze which would go back and forth through the concepts showing how linear systems, matrices and linear transformations have a intricate correspondence.
An extra caracteristic missing from the appendix that is important is the connection betwen the different topics.
I am sorry for the really long and possibly unclear question. I hope I didn't give the impression that I'm strict about what is a good suggestion. If you think your suggestion doesn't fit some of the categories, just post it. It could be anything really, a collection of exercises, a book, a short text, anything.
Convex Optimization by Boyd and Vandenberghe is an excellent text. A free PDF is available from Boyd’s website.
They also have a recent text on linear algebra that may be helpful if you want more background there.