Math 565
Fritz Keinert
due Thursday, Apr 30, 2015
Homework 5
For each problem that uses Matlab or some other tool, you should hand in a printout
of the relevant script or function file(s), or a transcript of your interactive session, plus
whatever outputs or plots are requested. Put the problems in the proper order, and
label all printouts clearly. The final output should have full accuracy (format long);
intermediate results can be shorter, if you want.
1. (Similar to Nocedal 13.2) I showed in class that the dual of the linear program
maximize cT x
subject to Ax ≤ b
x≥0
is
minimize bT λ
subject to AT λ ≥ c
λ ≥ 0.
Show that the dual of the dual is the original primal problem again.
(5)
2. (Similar to Nocedal 13.9) Solve the linear program
maximize 5x1 + x2
subject to x1 + x2 ≤ 5,
2x1 + (1/2)x2 ≤ 8,
x ≥ 0.
by the simplex method as explained in class. That is, write out the initial tableau for
the problem, and manipulate it until you have the answer. Start at x = (0, 0). Make
sure to state the answer, not just give the final tableau.
(5)
3. (Nocedal 14.1) Consider the following linear program
minimize x1
subject to x1 + x2 = 1,
x ≥ 0.
Show that the solution is
0
x =
,
1
∗
∗
λ = 0,
1
s =
.
0
∗
Verify that the system F (x, λ, s) = 0 also has the solution
1
0
x=
, λ = 1, s =
.
0
−1
(5)
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Math 565
—
Homework 5
—
due Thursday, Apr 30, 2015
4. (Nocedal 15.2) Consider the problem
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minimize sin(x1 + x2 ) + x23 + (x4 + x45 + x6 /2)
3
subject to 8x1 − 6x2 + x3 + 9x4 + 4x5 = 6,
3x1 + 2x2 − x4 + 6x5 + 4x6 = −4.
In example 15.3 in Nocedal, the variables x3 and x6 are eliminated.
Instead, eliminate x2 and x5 to obtain an unconstrained problem in the remaining
variables x1 , x3 , x4 , x6 .
(5)
5. Do optimization project 3: Design of a four-bar linkage (see pages appended at
the end). In detail:
(a) For general p, q, r, s, derive and program formulas that will calculate φ as a
function of θ.
(b) Derive and program formulas for computing φmax , φmin , θ+ , θ− , α, A, β0 , φmodel ,
and d2rms .
Compute the values of all these things for p = 0.5, q = r = s = 1. The values should
come out as shown in the picture. You don’t have to produce a picture.
before beta optimization
2
1.8
1.6
φ
max
1.4
1.2
1
α
0.8
0.6
0.4
φmin
0.2
0
β0
0
θ+
1
2
θ−
3
4
5
6
(c) For fixed p, q, r, s as above, find the value β which minimizes d2rms . Use starting
guess β0 . The optimal β will be quite close to β0 .
(d) In addition to the constraints listed in the project 3 handout, derive the constraints
that come from | cos γ| ≤ 1.
Hint: They involve h, which depends on θ, but you can compute and use the maximum
and minimum values of h to get constraints that do not depend on θ.
(e) Keep s = 1 fixed, and minimize d2rms as a function of p, q, r. You can use the
starting guesses from part (b). Other ones will probably also work.
Hint: Inside the routine that computes d2rms (p, q, r), you need to optimize with respect
to β.
(30)
Math 565
—
Homework 5
—
due Thursday, Apr 30, 2015
3
6. (Extra Credit) Solve the problem of the week #14 (see the end). 2 points extra
credit for solving it numerically. Another 2 points if you can prove it analytically.
Programming is quite easy. I found an analytic proof, but it was pretty hard.
Hint: what do you know about the relationships between eigenvalues, determinant
and trace, in particular for the case of 2 × 2 matrices?
(2+2)
Problem 14. Let
a11 a12
A=
a21 a22
and B =
b11 b12
,
b21 b22
with −1 ≤ aij , bij ≤ 1. Find the maximum possible
value of
det(A · B − B · A),
and provide examples of matrices A and B for which
the maximum is acheived.
Disclaimer! This problem was submitted by Prof.
Irvin Hentzel. I think we know the answer to this
problem but we do not have a proof. I will list
scores according to the following rule: “If M is the
maximum answer submitted, with evidence, then all
students who submit an answer, with evidence, in
the interval [0.95M, M ] will be listed as successful
solvers.”
Solutions due by 10:00am Monday, April 27, 2015.
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