1. ABC Investments. ABC Inc. is considering several investment options. Each option has a minimum and maximum investment allowed (only if the option is chosen). These restrictions, along with the expected return are summarized in the following table ( gures are in millions of dollars):
Do not solve Q1!!!
Option
Minimum
Maximum
Expected
investment
investment
return (%)
1
3
27
13
2
2
12
9
3
9
35
17
4
5
15
10
5
12
46
22
6
4
18
12
Because of the high-risk nature of Option 5, company policy requires that the total amount invested in Option 5 be no more that the combined amount invested in Options 2, 4 and 6. In addition, if an investment is made in Option 3, it is required that at least a minimum investment be made in Option 6. ABC has $80 million to invest and obviously wants to maximize its total expected return on investment. Which options should ABC invest in, and how much should be invested?
2. Heat pipe design. A heated uid at temperature T (degrees above ambient temperature) ows in a pipe with xed length and circular cross section with radius r. A layer of insulation, with thickness w, surrounds the pipe to reduce heat loss through the pipe walls (w is much smaller than r). The design variables in this problem are T , r, and w.
The energy cost due to heat loss is roughly equal to 1T r=w. The cost of the pipe, which has a xed wall thickness, is approximately proportional to the total material, i.e., it is given by 2r. The cost of the insulation is also approximately proportional to the total insulation material, i.e., roughly 3rw. The total cost is the sum of these three costs.
The heat ow down the pipe is entirely due to the ow of the uid, which has a xed velocity, i.e., it is given by 4T r2. The constants i are all positive, as are the variables T , r, and w.
Now the problem: maximize the total heat ow down the pipe, subject to an upper limit Cmax on total cost, and the constraints
Tmin T Tmax; rmin r rmax wmin w wmax; w 0:1r
.
a) Express this problem as a geometric program, and convert it into a convex optimization problem. Recall that a generic geometric program has the following form
x
Xj
cj0x1
x2
xn
min
j01 j02
j0n
s:t:
Xj
cjix1ji1 x2ji2 xnji3
1; i = 1; ; m
Xj
djkx1jk1 x2jk2 xnjk3 = 1; k = 1; ; p
xq > 0;
q = 1; :::; n
cji > 0; djk > 0; 8ji; jk
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CS/ECE/ISyE 524 Introduction to Optimization L. Roald, Spring 2022
b) Consider a simple instance of this problem, where Cmax = 500 and 1 = 2 = 3 = 4 = 1. Also assume for simplicity that each variable has a lower bound of zero and no upper bound. Solve this problem using JuMP. Use the Ipopt solver and the command @NLconstraint(...) to specify nonlinear constraints such as log-sum-exp functions. What is the optimal T , r, and w?
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