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STAT804 Optimization and Operations Research

Semester 2, 2021

Assignment Task 2 ( 30%)

Instructions:

● The assignment should be submitted as a single PDF document through Blackboard by the due date. You should also submit all of your MATLAB and SAS program files.

● Do not submit your files as a zip folder.

● Your assignment file should include the Individual Assignment Coversheet: Blackboard/Assessment Policies, Regulations,Guides and Forms/Forms andCoversheets/Individual Assignment Coversheet

● Where programming in MATLAB or SAS is required, a screenshot/image of the relevant output should be provided in the PDF and a MATLAB/SAS program file should be submitted. MATLAB/SAS code will be assessed based on its accuracy and elegance.

● Handwritten submissions are appropriate for some questions. These should be scanned for inclusion in the PDF file.

● Failure to follow these instructions will result in the deduction of marks.

Late Submission:

Failure to submit the assignment on time will result in a mark of 0 for the assignment. If extenuating circumstances (e.g. illness) prevent the timely submission of your assignment, you can apply for special consideration. You may also apply for special consideration, if such circumstances result in your submission being incomplete. You can apply for special consideration via Blackboard.

Originality:

This assignment is an individual piece of work. You are encouraged to discuss the assignment with your lecturers and classmates, however, the work you submit must be your own. Assignments that show similarities to work submitted by other students will be investigated for plagiarism and treated very seriously. Plagiarism software, such as TurnItIn, may be used to electronically compare submissions to those of other students and to documents on the internet. Talk to the lecturer if you have any questions about this requirement. Before you submit this assignment, you should complete the Academic Integrity module on Blackboard.

1. Consider the following minimization problem

Take x0 = [1, 0, 0]^T as the initial point.

(a) Consider the vector p = [−1, 1, 1]^T.

i. Show that p is a descent direction of f at x0.

ii. Find the minimizer α1 of the following problem

and thus find your first iteration x1 = x0 +α1p.

(b) With the help of MATLAB, solve the problem using the Steepest Descent Method.

(c) Solve the problem using the MATLAB unconstrained NLP solver fminunc.

(d) Solve the problem using proc optmodel in SAS.

2. Consider the following optimization problem:

(a) Derive the KKT conditions for this problem.

(b) Use the KKT conditions to check whether x = (1, 1, 1)^T is an optimal solution.

(c) Use the KKT conditions to derive an optimal solution.

(d) Solve the problem in (a) using an appropriate solver in MATLAB.

(e) Solve the problem in (a) using an appropriate solver in SAS.

3. Consider the following constrained optimization problem:

subject to

and the point x∗ = [0, 1]^T.

(a) Does the LICQ hold at the point x∗? Justify your answer.

(b) Are the KKT conditions satisfied at the point x∗? Justify the answer.

(c) Write down the sets F(x∗), and C (x∗,λ∗ ) if appropriate.

(d) Are the second-order necessary conditions satisfied? Are the second-order sufficient conditions satisfied? Justify your answers.

(e) Solve this problem using appropriate solver in MATLAB and SAS.

4. (a) Consider the following quadratic programming problem:

i. Write this quadratic programming problem in the standard form.

ii. Solve this problem using an appropriate solver in MATLAB.

iii. Solve this problem using an appropriate solver in SAS.

(b) Applying the sequential quadratic programming with the initial guess x0 = [−1, 1]^T, solve the constrained optimization problem

with µ0 = 1 in 2 iterations.

5. A trust fund manager in New Zealand intends to build up an investment portfolio, which consists of 6 stocks listed on the New Zealand Stock Exchange (NZX). He wishes to know what proportion of the portfolio should be invested in each selected stock, based upon the stock returns over the period from 1 August 2021 to 15 September 2021. So, he needs the advice from his Analytics Intelligence Team. Assume that you just joined this team and are asked to help.

(a) Choose 6 New Zealand stocks. Download .csv or Excel files containing the stock prices for your chosen stocks for the period from 1 August 2021 to 15 September 2021 from a website, such as yahoo finance: https://nz.finance.yahoo.com/q/hp?s=NZX.NZ. Create a single datafile containing the date and the closing stock prices for each of your stocks using a software package of your choice (e.g. SAS, Excel or R). Sort in ascending date order. Show the top 10 rows of your dataset in your PDF file. You should also submit the data file.

(b) Create a data file containing the date and the log return for each stock. Using SAS package or otherwise, compute some summary statistics of the log returns (including the mean and covariance matrix) and write 1 paragraph interpreting these results. If si is the stock price at time i, i ≥ 0, then the log return is:

(c) The manager asks you to analyse what percentage of his portfolio to invest in each of the selected stocks, in order to maximize the portfolio’s return. Assume that you formulated the following model for this task:

where r is the mean of log returns, Σ is the covariance matrix, and x is the proportion of the portfolio invested in each stock. Set up an appropriate risk level R and solve the problem with:

i. SAS using proc optmodel.

ii. MATLAB.

For each solution method, write a brief report in 100 to 200 words on your findings.

(d) The manager asks you to analyse what percentage of his portfolio to invest in each of the selected stocks, in order to maximize the portfolio’s return and minimize the portfolio’s risk simultaneously. You modified your model in (c) and formulated the following model for this task: