\(\def\Real{\mathbb{R}}\)
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).

Sketch the contour plot for the function
\[
f(a,b)=\min_{(x,y)\in P} ax+by.
\]
for \(P\) being convex hull of the points
\[
(0,1),(0,1),(1,0),(2,1);
\]  ellipse \(\{x^2+3y^2\leq1\}\);
 negative \(x\)ray, \(\{(x,0), x\leq 0\}\).
 convex hull of the points

Dualize the following LP:
\[
\begin{array}{rrc}
x_1&2x_2&\to\max\\
\mathrm{subject\ to}&&\\
x_1&x_2&\leq 1\\
x_1&+x_2&\leq 1\\
x_1&+x_2&\geq 1\\
x_1&&\leq 1\\
&x_2&\leq 1\\
\end{array}.
\]
Solve the dual LP.  Find the cone \(K^*\subset\Real^3\) if \(K\) is the ball of radius \(1\) around the point \((2,2,2)\)
 Minimize
\[
x_1+x_2+x_3+x_4
\]
over the set
\[
x_1^2+x_2^2+x_3^2+x_4^2=4,\quad x_1x_2x_3x_4\leq1.
\]
Solutions to homework.
Is our midterm next week?
Can we get some information about the midterm? When and where exactly will it be and what chapters does it cover? Is it open book and notes? If it is, can we use any electronic device? Also, what is the best way to study for this exam?
Thanks a lot!
Are we going to have another homework before the midterm?
Yes.
In problem1, should we discuss whether a and b are >0 or <0? thanks!
All \(a,b\)’s should be covered, whether \(gt, \lt\) or \(=0\).
This page mentions the due date is Tuesday, March 10. However, March 10 is a Sunday. Is this assignment due Sunday or Tuesday?
Sunday.
For the exam, it mentions it is open book. Are we allowed to a laptop to access the Ebook?
*to use a laptop to access the Ebook
You can use laptops or tablets with wireless off.
For problem 1, how much work do we need to show? If I can get the answer by analyzing the graph, do I have to show my work for the contour graph by solving for all the different cases of a and b?
As much as to make sure we understand _how_ you arrived at the answer.
Also, can we do Problem 3 geographically?
I mean geometrically.
As long as the solution is correct, and your line of reasoning clear.
For Problem 2, do we have x_1, x_2 \geq 0?
Can we have the solution for this homework before the midterm?
The UIUC matlab requires internet connection to the matlab server. Can we use wifi but only for the matlab during the exam?
OK