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SCHOOL OF MATHEMATICS AND STATISTICS

MATH5165

OPTIMIZATION

Term 1, 2024

ASSIGNMENT

Your answers to this assignment must be submitted via Moodle before 5:00pm, April 19, 2024. Late assignments will not be accepted except on documented medical or compassionate grounds.

Assignments must include a signed cover sheet available from the School of Mathematics and Statistics web site at:

http://www.maths.unsw.edu.au/sites/default/files/assignment.pdf

School date stamp is not needed on the cover sheet.

Marking of this Assignment

Each question on this assignment is worth 25 marks. A full mark of 100 on this assignment is worth 5% of the total marks for MATH5165. Present your work (especially for questions 1 and 2) as a self contained, well-written report including the problem statement, solution summary (≤ 250 words), clear interpretation of solutions, model formulation, definition of problem variables and your computer output. State clearly the assumptions that you make. Any Matlab files that you modify/write should be included as an appendix.

1. Minimum Cost Pizza Problem. Using only the items given in the tables below, formulate an optimization problem in standard form to create a minimum cost pizza which satisfies both the nutritional requirements of Table 1 and bounds on item quantities given in Table 2. Use the nutritional data of Table 3 and the cost data of Table 4 in your model.

Use the MATLAB linear optimization routines linprog to solve the problem. Interpret your results.

** Units as in Table 1.

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?

[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

The variance in future payments can be expressed as

where the covariances  are given by

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

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

(c) Solve the problem using the MATLAB quadratic programming routine quadprog. Interpret your results in plain English.

3. Consider the optimization problem

where b > 0 is some constant. The product notation means that  xi =1 xi = x1x2 . . . xm. Assume that the problem (P1) has a global minimizer.

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

(b) Using only first-order information, explain why the constrained stationary point  of part (a) is a global minimizer for (P1).

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

4. Consider the following inequality constrained optimization problem

where f : R n → R and gi : R n → R are differentiable functions. Let x* ∈ R n be a feasible point of (P2) at which Karush-Kuhn-Tucker conditions are satisfied with Lagrange multipliers , i = 1, 2, . . . , m. Assume that the functions f and gi ’s satisfy the following generalized convexity condition:

For each x ∈ R n ,

for some function η : 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 Web page, 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

• 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.