代写program、Matlab程序设计代做

日期：2024-04-19 08:23

Table 2: Upper and Lower Bounds on Pizza Items

Item Upper Bounds* Lower Bound*

sauce 1.986 1.140

dough 5.249 N/A

cheese 2.270 1.703

pepperoni 0.983 N/A

ham 1.135 N/A

bacon 0.993 N/A

g.pepper 1.561 N/A

onion 0.993 N/A

celery 1.561 N/A

mushroom 1.135 N/A

tomato 1.703 N/A

pineapple 1.703 N/A

meat 1.490 0.993

veg. N/A 0.993

fungi N/A 0.922

* Amount in hundreds of grams.

Table 3: Nutritional Decomposition of Pizza Items **

Item Calc Iron Prot Vit A Thia Niac Ribo Vit C

cheese 517.700 .222 20.000 3000.000 .022 6.000 .244 -

sauce 14.000 1.800 2.000 800.000 0.100 1.400 .060 6.000

dough 18.233 3.826 14.224 - .586 8.852 .628 -

pepperoni 10.000 2.500 15.000 - - 2.000 - -

ham 9.031 2.291 14.692 - .740 4.009 0.178 -

bacon 13.000 1.189 8.392 - .361 1.828 .114 -

g.pepper 9.459 .675 1.351 209.460 .081 .540 .081 127.030

onion 27.273 .545 1.818 18.182 .363 .545 .036 10.000

celery 40.000 .250 - 125.000 .025 .500 .025 10.000

mushroom 6.000 .800 3.000 - .100 4.300 .460 3.000

tomato 13.333 .533 1.333 450.000 .066 .800 .04 22.667

pineapple 12.016 .310 .387 25.194 .081 .193 .019 6.977

** Units as in Table 1.

Table 4: Costs of Pizza Items

Item Cost in cents/100 grams

cheese 95.53

sauce 72.24

dough 19.74

pepperoni 90.43

ham 90.30

bacon 90.75

green pepper 51.46

onion 10.93

celery 28.99

mushrooms 63.96

tomatoes 45.16

pineapple 53.26

2

2. Portfolio Selection Problem. An individual with $100,000 to invest has identified three mutual

funds as attractive opportunities. Over the last five years, dividend payments (in cents per dollar invested)

have been as shown in Table 5, and the individual assumes that these payments are indicative of what

can be expected in the future. This particular individual has three requirements:

(1) the combined expected yearly return from her/his investments must be no less than $2,000, i.e.,

the amount $100,000 would earn at 2 percent interest, and

(2) the variance in future, yearly, dividend payments should be as small as possible, and

(3) the amount invested in Investment 1 must be at least the amount invested in Investment 3.

How much should this individual fully invest her/his $100,000 in each fund to achieve these requirements?

Table 5: Dividend Payments

Years

1 2 3 4 5

Investment 1 5 8 8 3 1

Investment 2 4 3 6 2 0

Investment 3 5 6 4 3 2

[Hint: Let xi, i = 1, 2, 3, designate the amount of funds to be allocated to investment i, and let xik denote

the return per dollar invested from investment i during the kth time period in the past (k = 1, 2,..., 5).

If the past history of payments is indicative of future performance, the expected return per dollar from

investment i is

Ei = 1

5

X

5

k=1

xik.

The variance in future payments can be expressed as

f(x1, x2, x3) = X

3

i=1

X

3

j=1

2

ijxixj = x>Cx,

where the covariances 2

ij are given by

2

ij = 1

5

X

5

k=1

xikxjk  1

52

X

5

k=1

xik! X

5

k=1

xjk!

. ]

(a) Using the following table, calculate the covariance matrix C = [2

ij ].

Table 6: Intermediate Calculations

k x1k x2k x3k x2

1k x2

2k x2

3k x1kx2k x1kx3k x2kx3k

1 5 4 5 25 16 25 20 25 20

2 8 3 6 64 9 36 24 48 18

3 8 6 4 64 36 16 48 32 24

4 3 2 3 9 4 9 6 9 6

5 1 0 2 1 0 4 0 2 0

total 25 15 20 163 65 90 98 116 68

(b) Set up a standard form optimization problem (i.e. quadratic optimization problem) that will determine the best investment mix.

(c) Solve the problem using the MATLAB quadratic programming routine quadprog. Interpret your

results in plain English.

3

3. Consider the optimization problem

(P1) min x2Rm

Xm

i=1

xi

s.t. Ym

i=1

xi = b,

xi  0, i = 1, . . . , m,

where b > 0 is some constant. The product notation means that Qm

i=1 xi = x1x2 ...xm. Assume that the

problem (P1) has a global minimizer.

(a) Find a constrained stationary point xˉ of (P1).

(b) Using only first-order information, explain why the constrained stationary point xˉ of part (a) is a

global minimizer for (P1).

(c) Hence or otherwise, show that, if x1,...,xm  0, then

1

m

Xm

i=1

xi

?Ym

i=1

xi

?1/m

.

4. Consider the following inequality constrained optimization problem

(P2) min x2Rn f(x)

s.t. gi(x) ? 0, i = 1, . . . , m,

where f : Rn ! R and gi : Rn ! R are di?erentiable functions. Let x? 2 Rn be a feasible point of

(P2) at which Karush-Kuhn-Tucker conditions are satisfied with Lagrange multipliers ?

i , i = 1, 2,...,m.

Assume that the functions f and gi’s satisfy the following generalized convexity condition:

For each x 2 Rn,

f(x)  f(x?)  rf(x?)

>?(x, x?)

gi(x)  gi(x?)  rgi(x?)

>?(x, x?)

for some function ? : Rn ? Rn ! Rn. Show that x? is a global minimizer for (P2).

NOTES: Essential information for accessing files from the MATH3161/MATH5165 Course Web page

and for using Matlab.

? Matlab can be accessed from your own laptop using the myAccess service. (see the link on the

Course Web-page, UNSW Moodle, Computing facilities (labs, virtual apps, software).

? Matlab M-files can be obtained from Matlab Worksheets in Class Resources at the Course Webpage, UNSW Moodle. The Matlab files for Q1, Problem Sheet 1 (ss24.m) and for Q5, Problem

Sheet 6 (qp24.m) are available at this page in the assignment folder.

? Matlab is run by typing

matlab

at the UNIX prompt. Inside Matlab use ‘help command’ to get help,

e.g.

help optim

help linprog

help quadprog

4

? To run a Matlab .m file from within Matlab simply type the name of the file:

ss24

This assumes the file ss24.m is in the current directory (use the UNIX command ‘ls’ to see what

files you have; if it is not there get a copy of the file from Matlab worksheets page at the Course

Web page and save it as ss24.m).

? An entire Matlab session, or a part of one, can be recorded in a user-editable file, by means of the

diary command. The recording is terminated by the command diary off. A copy of the output

produced by Matlab can be stored in the file ‘ss24.out’ by typing diary ss24.out For example

diary ss24.txt

ss24

diary off

will save a copy of all output in the file ss24.txt

? The file ss24.out may be viewed using ‘more’ or any text editor (xedit, vi) or printed using the ‘lpr’

command.

5