ECE 490, Week of Feb 25

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

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).

  1. 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, \(\{(x,0), x\leq 0\}\).
  2. 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.
  3. Find the cone \(K^*\subset\Real^3\) if \(K\) is the ball of radius \(1\) around the point \((2,2,2)\)
  4. Minimize
    \[
    x_1+x_2+x_3+x_4
    \]
    over the set
    \[
    x_1^2+x_2^2+x_3^2+x_4^2=4,\quad x_1x_2x_3x_4\leq-1.
    \]

Solutions to homework.

20 Responses to ECE 490, Week of Feb 25

  1. Chuyuan Tao February 26, 2019 at 4:19 pm #

    Is our midterm next week?

  2. Hakan Tekgul February 26, 2019 at 9:07 pm #

    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!

  3. Xuan Wang February 28, 2019 at 4:36 pm #

    Are we going to have another homework before the midterm?

    • yuliy March 2, 2019 at 3:59 pm #

      Yes.

  4. Yiqing Xie March 2, 2019 at 8:45 pm #

    In problem1, should we discuss whether a and b are >0 or <0? thanks!

    • yuliy March 5, 2019 at 12:21 am #

      All \(a,b\)’s should be covered, whether \(gt, \lt\) or \(=0\).

  5. Sean Edward White March 4, 2019 at 6:34 pm #

    This page mentions the due date is Tuesday, March 10. However, March 10 is a Sunday. Is this assignment due Sunday or Tuesday?

    • yuliy March 5, 2019 at 12:21 am #

      Sunday.

  6. Sean Edward White March 6, 2019 at 7:50 pm #

    For the exam, it mentions it is open book. Are we allowed to a laptop to access the Ebook?

    • Sean Edward White March 6, 2019 at 7:54 pm #

      *to use a laptop to access the Ebook

    • yuliy March 7, 2019 at 12:30 am #

      You can use laptops or tablets with wireless off.

  7. Ruby Zhuang March 6, 2019 at 9:03 pm #

    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?

    • yuliy March 7, 2019 at 12:31 am #

      As much as to make sure we understand _how_ you arrived at the answer.

  8. Ruby Zhuang March 6, 2019 at 10:42 pm #

    Also, can we do Problem 3 geographically?

    • Ruby Zhuang March 6, 2019 at 10:42 pm #

      I mean geometrically.

    • yuliy March 7, 2019 at 12:32 am #

      As long as the solution is correct, and your line of reasoning clear.

  9. Xuan Wang March 8, 2019 at 1:41 pm #

    For Problem 2, do we have x_1, x_2 \geq 0?

  10. Xuan Wang March 11, 2019 at 12:06 am #

    Can we have the solution for this homework before the midterm?

  11. Zhijian Yang March 11, 2019 at 12:15 am #

    The UIUC matlab requires internet connection to the matlab server. Can we use wifi but only for the matlab during the exam?

    • yuliy March 11, 2019 at 9:08 am #

      OK

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