\(\def\Real{\mathbb{R}}\)
Finishing Noninear Programming: Guler, Chapter 9.
Class notes (pretty incomplete, to be used just as a study guide), here and here.…
\(\def\Real{\mathbb{R}}\)
\(\def\Real{\mathbb{R}}\)
Simplex method (see Matousek/Gaertner, Understanding and using linear programming.
Starting Noninear Programming: Guler, Chapter 9.
Class notes (pretty incomplete, to be used just as a study guide), here and here.
Homework (due by midnight of Sunday, Mar. 10).
\(\def\Real{\mathbb{R}}\)
Linear Programming: Guler, Chapter 6 and Matousek/Gaertner, Understanding and using linear programming.
Class notes (pretty incomplete, to be used just as a study guide), here and here.
Exercises
\(\def\Real{\mathbb{R}}\)
Finishing Guler, Chapter 4 and covering Chapter 6.
Class notes (pretty incomplete, to be used just as a study guide), here and here.
Exercises:
Finishing Guler, Chapter 3 and starting Chapter 4.
Class notes (pretty incomplete, to be used just as a study guide), here and here.
Using linear constraints compatibility in packing problems, see e.g. here.
Exercises:
Following Chapter 2 of Guler.
Class notes (pretty incomplete, to be used just as a study guide), here and here.
Exercises:
\(\def\Real{\mathbb{R}}\)
Spaces of differentiable functions on an interval \(I=[a,b]\subset\Real\):
\[
C(I,\Real), C^1(I,\Real),\ldots, C^n(I,\Real),\ldots
\]
For a function in \(C^1(I,\Real)\),
\[
f(y)=f(x)+\int_x^yf'(s)ds.
\]
Iterating (for functions in \(C^n(I,\Real)\), we obtain
\[
f(y)=f(x)+f'(x)(y-x)+\frac{f”(x)}{2!}(y-x)^2+\ldots+\frac{f^{(n-1)}(x)}{(n-1)!}(y-x)^{n-1}+
\int_{x<s_1<s_2<\ldots<s_n<y}f^{(n)}(s_1)\frac{(y-s_1)^{n-1}}{(n-1)!}ds.
\]
Implications:
Domain: \(U\subset\Real^n\), an open set.
Reminder: open, closed, bounded, compact sets.
Exercise: is the set \(\{|x|\leq 1, |y| \lt 1\}\) open?…
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