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- 首页 >> Java编程 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 x¯ of (P1).
(b) Using only first-order information, explain why the constrained stationary point x¯ 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
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 x¯ of (P1).
(b) Using only first-order information, explain why the constrained stationary point x¯ 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