Linear Optimization – Spring 2020

Course: Math 464 (CAPS)Website: http://www.math.wsu.edu/faculty/tasaki/welcome.html#page-3Times: TTh 9:10–10:25Location: Pullman (FSHN T101) and Vancouver (VECS 125)Instructor: Tom Asaki (tasaki@wsu.edu), Neill 228Office Hours: MTW 11:00-12:30 (Neill 228 or https://wsu.zoom.us/j/952281142)Text: Introduction to Linear Optimization (1997)

by Dimitris Bertsimas and John Tsitsiklis

Description

This course explores the theory and application of linear programming – a very importantand practical subfield of mathematical optimization. We will learn how mathematiciansexpress optimization problems, model real-world decision-making problems, understand andpractice solution methods, employ software for linear programs. This course is a mathematicsCapstone course which is accessible to students from a variety of majors, provided thattheir mathematical background is solid. The course prerequisite is Math 273 or Math 283.However, completion of Math 301, Math 364 and/or Math 220/230/420 will be very helpful.

Course Materials

In this course, we will work through several chapters in the textbook Introduction to LinearProgramming by Dimitris Bertsimas and John Tsitsiklis. This (1997) edition can be rentedfrom Amazon for about $30 or purchased for about $75. Some homework assignments willbe completed using Matlab (or Octave) – this software is available through WSU andcan also be accessed online at no cost (details will be provided when needed). You will alsocomplete the writeup of the course project using the online service Overleaf for typesettingdocuments using LATEX. So, you will need productive internet access.

Course Goals

Math 464 is a Mathematics Capstone course. As such it is “Intended to be taken in thefinal year of a student’s degree...” and “...serve as a culminating experience for students todemonstrate achievement of the university’s undergraduate learning goals.” Some aspects ofCapstone course requirements include

• “demonstrate a depth of knowledge within their chosen academic field of study thatintegrates its history, core methods, techniques, vocabulary, and unsolved problems.”

• ”apply concepts from their general and specialized studies to personal, academic, ser-vice learning, professional, and/or community activities.”

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• “demonstrate how the methods and concepts of a chosen discipline relate to those ofother disciplines through engaging in cross-disciplinary activities.”

In this couse, you will realize some or all of these aspects through homework, classwork,and a project based on your modeling and assessment of a decision-making process from anactual business of your choosing. In addition, this couse serves as a venue for developingyour proof consruction and writing skills.

Course Elements

Participation. Some class activities involve group work and short presentations to theclass.

Homework. Regular homework is intended to assist your study of various topics andextend your knowledge. Homework solutions are not scratchwork nor do they consist onlyof answers to assigned problems. Homework solutions should be examples of self-containedmathematical documents. They should exhibit good grammar, appropriate mathematicallanguage and notation, explanations of the key ideas addressed by the question. Examplesof good homework solutions are included at the end of this syllabus.

Project. The course project involves several aspects of real-world implementation of math-ematical optimization. Individual projects are expected, though a strong proposal for a smallgroup project will be considered. Any such a proposal should be submitted (or presented)before Tuesday, January 21.

A project will include several elements (details are provided later in the syllabus): openinga conversation with a business or other agency, interviewing key staff with regard to oneor more optimization problems of relevance, modeling at least one question of importance,demonstrating a solution to the problem, and writing a report of the process and results. Thisproject is intended to give you the experience of approaching a potential client, understandingtheir decision-making needs, developing a mathematical optimization model of relevance, andproviding a solution. The goal is to have a complete consulting-like experience in which yourmathematical skills play an integral part.

Exams. There will be no exams for this course. The knowledge and skill typically probedby exams will be manifested and measured in homework and the project.

Evaluation Scale

A course percentage will be determined by a weighted combination of the course requirementelements: participation (10%), homework (60%), project (30%). The course grade will thenbe no lower than provided by the standard scale:

Course Percent 90 87 80 77 70 60 <60Course Grade A B+ B C+ C D F

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WSU Reasonable Accomodation

Students with Disabilities: Reasonable accommodations are available for students with adocumented disability. If you have a disability and need accommodations to fully participatein this class, please either visit or call the Access Center to schedule an appointment withan Access Advisor. All accommodations MUST be approved through the Access Center.

Academic Integrity

Academic integrity is the cornerstone of higher education. As such, all members of theuniversity community share responsibility for maintaining and promoting the principles ofintegrity in all activities, including academic integrity and honest scholarship. Academicintegrity will be strongly enforced in this course. Students who violate WSU’s AcademicIntegrity Policy (identified in Washington Administrative Code (WAC) 504-26-010(3) and-404) will receive a failing grade on any relevant assignment (and possbily for the course),will not have the option to withdraw from the course pending an appeal, and will be reportedto the Office of Student Conduct.

Cheating includes, but is not limited to, plagiarism and unauthorized collaboration asdefined in the Standards of Conduct for Students, WAC 504-26-010(3). You need to readand understand all of the definitions of cheating. If you have any questions about what isand is not allowed in this course, you should ask course instructor before proceeding.

If you wish to appeal a faculty member’s decision relating to academic integrity, pleaseuse the form available at conduct.wsu.edu.

Safety and Emergency

Classroom and campus safety are of paramount importance at Washington State University,and are the shared responsibility of the entire campus population. WSU urges students tofollow the “Alert, Assess, Act,” protocol for all types of emergencies and the “Run, Hide,Fight” response for an active shooter incident. Remain ALERT (through direct observationor emergency notification), ASSESS your specific situation, and ACT in the most appropriateway to assure your own safety (and the safety of others if you are able).

Please sign up for emergency alerts on your account at MyWSU. For more informationon this subject, campus safety, and related topics, please view the FBI’s Run, Hide, Fightvideo and visit the WSU safety portal.

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Schedule

The following table shows the classroom topics, homework due dates and project statusdue dates. Classroom topics and days are tentative and subject to change. Homeworkassignments and project updates are due at the beginning of class on the stated day.

Date Class Topics Homework Project [pts]

Tu Jan 14 Introduction/NotationTh Jan 16 LP Examples Overleaf Document Link [5]Tu Jan 21 LP Examples 1,2Th Jan 23 LP Examples 3 Initial Contact List [5]Tu Jan 28 LP Solution Concepts 4,5Th Jan 30 Convexity/ Polyhedra Initial Contact Plan [10]

Tu Feb 4 6Th Feb 6 Basic Feasible Solutions 7Tu Feb 11 Standard Form Polyhedra 8,9Th Feb 13 Geometric Theorems 10Tu Feb 18 Optimality and Simplex Method 11Th Feb 20 Simplex Method Examples 12,13Tu Feb 25 Simplex Method Examples 14Th Feb 27 15

Tu Mar 3 Big-M and Two-Step Methods 16Th Mar 5 Cycling and Degeneracy 17 Initial Meeting Synopsis [10]Tu Mar 10 Duality 18Th Mar 12 Duality 19,20 Problem Statement [10]Tu Mar 17 Spring BreakTh Mar 19 Spring BreakTu Mar 24 LP Solution EfficiencyTh Mar 26 Ellipsoid Method ConceptTu Mar 31 Ellipsoid Method 21 Optimization Model [10]

Th Apr 2 Interior Point Method ConceptsTu Apr 7 Affine Scaling 22Th Apr 9 Barrier MethodsTu Apr 14 Primal-Dual Path Following Solution [10]Th Apr 16 Integer Programming Concepts 23Tu Apr 21 Cutting Planes MethodsTh Apr 23 Branch and Bound Methods Reflection [10]Tu Apr 28 Dynammic Programming 24Th Apr 30 Report [30]

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Project Details

All project updates are to be recorded in a single document created using the online LATEX type-setting service at overleaf.com. If you are not familiar with basic LATEX, then please let meknow. Each element has a specific due date given in the schedule.

Overleaf Document Creation

The specific procedures for creating the report in Overleaf are given below. You shouldcomplete these steps. I will receive an automated email with the read/write link to yourdocument. No additional action is needed on your part.

1. Download the Project template file (OptimizationProject.tex) and the figure speed.pngfrom the Math 464 course website to your computer.

2. Navigate to https://www.overleaf.com.

3. Set up a free account and log in.

4. At left, click on “New Project.”

5. Select “Blank Project” and name your project.

6. Right click the file main.tex and delete it.

7. Use the “upload” icon in the upper left to upload OptimizationProject.tex and speed.pngfrom your computer.

8. Click on the uploaded file in the file list at left.

9. Click on the “Recompile” button at top-center.

After following these instructions, the formatted project document should appear in thewindow at right. You do not need to supply a paper or electronic copy of this document tome at any time. Instead you will give me read/write access to the document so that I cansupply feedback online. To give me access, do the following.

1. Click the “Share” icon at upper right.

2. Enter my wsu email address in the collaborator box.

3. Make sure that “Can Edit” is selected.

4. Click “Share.”

I will then receive an email with an edit link to your project. I will not edit your project,but I can use the edit ability to add comments at the end of your document.

Contact List

Provide an initial list of several potential contacts for mathematical optimization problems.Do not initiate contact yet! The contacts should be a business, a government agency, anon-profit orginization, an academic office, not particular people. Keep the following generalcriteria in mind.

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1. Smaller businesses and offices are more likely to be willing to meet quickly.

2. Having several options is important because not each of your choices will be willing toassist in your project.

3. Focus your choices according to your interests.

4. Local businesses and offices are easier to work with in terms of personal communication.

What reasons do you have for your choices? What types of optimization problems mightyou expect?

Initial Contact Plan

Begin this step only after your Contact List has been approved.

Devise an initial contact plan based on your contact list. Your plan should include a strategyfor working through your list and examples of email introductions. Some considerations:

1. Initial contact by phone or in person is not prefered in this case because of the antici-pated length of conversation.

2. Contacting one or two at a time to gauge interest and willingness.

3. Begin with your top choices.

4. Attempt to direct your email/call appropriately.

5. Be clear about your project needs.

6. Do not offer mathematical help.

7. Do not ask to see data.

8. Be clear about the course and instructor.

9. Ask for 30 minutes of time or less.

10. Give specific potential meeting times.

Here is an example intial contact email:

Dear Dr. Sarah Blastik,

I understand that you are the scheduling manager for Opticorp. I am a

student in Dr. Tom Asaki’s Mathematical Optimization Course at

Washington State University. As part of my class project, I am

researching how mathematics can be used to assist decision-making

processes in business situations. One of my interests is in scheduling

events and personnel.

It would very helpful if we could have a short meeting (20-30 minutes)

to discuss scheduling decisions related to companies such as yours and

what challenges you face. I could meet next week any day after 3:00 or

on Monday or Wednesday mornings before 10:00.

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Please let me know. If you have any question, please feel free to

contact my professor at tasaki@wsu.edu.

Sincerely,

I.M. Student

Here is another example email:

Sam Sneed,

I a student in a Mathematical Optimization course at WSU. As part of

my course project, I am considering modeling how product ordering is

handled based on inventory, suppliers, consumer demand, etc. I was

hoping that you, as the inventory manager, would have some time to help

me understand more about the actual process details. I could learn much

in a short 20-30 minute meeting.

I could meet with you any time Monday or Friday. I am also free some

afternoons.

Please let me know if you can help, and please feel free to contact my

professor at tasaki@wsu.edu if you have any questions.

I.M. Student

Initial Meeting Synopsis

Meet with your contacts as soon as possible after your contact plan has been approved.

Your synopsis should include only your experiences with people with whom you actuallymet. You can briefly mention failed contacts if you wish. The synopsis should summarizemeeting converstations. Here are some things to keep in mind:

1. You can meet by phone conversation, but you can lose context and detail. Some timesthere is no substitute for experiencing the work place.

2. Try and be flexible with scheduling. But, don’t compromise your other obligationsunnecessarily.

3. Aim for a first meeting as soon as possible. Remember that the meeting synopsis alsohas a due date.

4. Do not be late for a meeting.

5. Be prepared with questions. Take notes. Listen.

6. As you meet, think about potential optimization problems.

7. Be willing to change the expected direction of your conversation.

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8. Request that you can ask follow-up questions at a later date.

Being prepared with good questions is very important. Keep in mind that as you formulatean optimization problem you will need to understand your decision variables, your objectivefunction, and your constraints. You should not use this language in your meeting. Instead,consider questions such as the following (You will need to tailor these to your specific needs).

1. How many different products do you manufacture?

2. What specific choices do you make? How often?

3. Can be manufactured in any quantity or in specfic batch quantities?

4. How do costs change with specific strategies?

5. How is profit affected by ?

6. How do you decide which is a better strategy, or ?

7. Do you consider it better to or ?

8. How does one choice affect another?

9. Are there limits on the number of you can make?

10. Can you have both and ?

11. Are some limitations soft or are they all very strict?

It can be difficult to predict what questions you should ask, but you should come as preparedas possible. Feel free to ask for additional guidance as you think about questions.

Problem Statement

Now that your initial meeting(s) is complete, you can write up a formal problem statement.This statement is from the perspective of the client. It should resemble a (possibly complex)word problem, similar to our class examples. It should exhibit good grammar, completedetails and a specific optimization goal.

You may or may not have specific data to work with – but in either case, it should notbe included here. Instead, given data should be represented as parameter vector quantities.For example, the manufacturing cost vector for n production items can be represented asd ∈ Rn with entries d1, d2, · · · , dn. Specific values are not needed in the problem statement,except to say that d is given.

Optimization Model

Now that your problem statement is well-formulated, you can provide an optimization modelwhose solution answers the problem statement. You should explain all aspects of yourmodeling process and justify every model item. Your model should be expressed as compactlyas possible – probably in matrix form. Specific values for problem variables (e.g. c, A, b) arenot needed, but should be expressed in terms of parameter variables. In this sense, the modelis not specific to any particular scenario you have discussed with your contact. Instead, yourcontact’s question is one example of a problem your model can solve.

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Solution

Now that your model is complete, you can work on answering specific questions. If you havedata provided by your contact, then you can answer this specific question. If you do nothave data, then you can create “synthetic data” (data which is realistic and tests the abilityof your analysis to provide a clear answer).

Solutions should be provided for several scenarios. It may be necessary to ask clarifyingquestions of your contact and modify your model accordingly. This is quite typical in appliedoptimization. If such changes are necessary, feel free to return to previous sections and modifyaccordingly.

A brief discussion should be included which demonstrates that you understand the results andsome of the implications for the contact. That is, relate your numerical results to the Prob-lem Statement and give some discussion from both a mathematical and business/industryperspective.

Reflection

The final aspect of the Project Report is a personal reflection statement. This is a keypiece of the report and of the course. Your response should be significant, truthful and notsuperficial. You should write in your own style, following any reasonable format. But insome way, you should address at least the following questions which I will ask myself.

1. Does this student use correct mathematical and English grammar?

2. What went well for this student? What not so well?

3. What might this student do differently in a similar future situation?

4. Does the student fully understand their model and why it will answer the problemstatement?

5. Can the student explain any unusual/unexpected solution results?

6. Was this student effective at communicating with the contact?

7. Was this experience valuable for the student?

8. How comfortable was the student in completing the assigned tasks?

Report

The final project report requires no additional specific content besides the tasks alreadycompleted. The final report should be provided to the instructor as a pdf document attachedto an email. You should take the time to carefully check your report for appropriate grammar,spelling, mathematical notation and rich content.

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Procedure for Turning in Homework Assignments

All homework assignments are to be submitted electronically by email to my WSU emailaddress (tasaki@wsu.edu). Each submission must conform to the following criteria.

1. Each assignment should be submitted separately, even if due on the same day.

2. The format is to be PDF only.

3. The file name format is to be FirstLast N.pdf where First and Last are your given nameand family name with first letter capitalized, is the underscore character, and N isthe assignment number. You may use a shortened version of your name if you prefer,but please be consistent. For example, my submission for assignment seven would beTomAsaki 7.pdf.

4. Submitted files should not be larger than 1 MB. There are good online tools for reducingthe size of PDF documents (e.g. https://pdfcompressor.com).

5. Submitted work can be derived from pictures of hand-written notes or typeset docu-ments (i.e. LATEX or Word). There are good online tools for collecting pictures intoPDF documents (e.g. https://jpg2pdf.com).

The next few pages show examples of well-written homework assignments (without theusual header information). Explanatory comments about these examples are shown in bluetext.

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Figure 1: Example #1 of a well-written homework solution. The solution is in black andcomments are in blue.

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Figure 2: Example #2 of a well-written homework solution. The solution is in black andgreen and comments are in blue.

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Figure 3: Example #3 of a well-written homework solution. The solution is in black andcomments are in blue.

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Homework Problems

1. Pots-R-Us obtains its stock of outdoor decorative pots from three suppliers. Eachsupplier sells pots in four sizes (large, medium, small and tiny) in batches of 20 pots.The table shows the distrubution of pots in each batch.

Supplier Batch Cost Large Medium Small Tiny1 $80 7 8 3 22 $65 6 5 7 23 $45 2 2 10 6

Each quarter, Pots-R-Us places and order with each supplier. At least 100 large, 60medium, 50 small and 40 tiny pots must be purchased each quarter in order to meetdemand. Because the supplier facilities have limited production, at most 120 pots canbe purchased from any supplier in any quarter. Formulate an integer program thatcan be used to minimize the cost of acquiring the needed pots for one quarter.

2. A chemical company manufactures two products from four raw materials. The companycan use three different batch processes in any combination to meet consumer demand.The table below shows, for each process, the required raw materials, the amount ofoutput products and the manufacturing cost. The table also includes the consumerdemand for each product. Formulate an integer program that can be used to findthe processes to employ that minimizes cost to the company while meeting consumerdemand. The available raw material quantities are 125, 175, 225 and 150, repectively.

Process 1 Process 2 Process 3 Consumer DemandRaw Material 1 1 2 0Raw Material 2 2 2 1Raw Material 3 4 1 2Raw Material 4 0 3 4

Product 1 3 2 1 217Product 2 1 3 1 167

Cost $44 $32 $29

3. A company produces two different products, say item #1 and item #2. Each of item#1 requires 1

4hours of assembly labor, 1

8hour of testing labor and $1.20 worth of raw

materials. Each of item #2 requires 13

hours of assembly labor, 13

hour of testing laborand $0.90 worth of raw materials. The company currently has 90 hours of assemblylabor and 80 hours of testing labor available each day. The products have marketvalues of $9 and $8, respectively. Formulate an integer program that can be used tomaximize the daily profit of the company.

Next, suppose that the company can schedule up to 50 hours of overtime assemblylabor at an additional cost of $7 per hour. Provide a modified integer program.

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4. (Exercise 1.8) Consider a road divided into n segments which is illuminated by mlamps. Let pj be the power of the jth lamp. Suppose also that the illumination of

the ith road segment is Ii =n∑

j=1

aijpj, where the aij are known constants (determined

by the relative positions of the road segments and lamps). The road department hasspecified a desired illumination Ji for each road segment. How can we choose lamppowers so that the road is well-illuminated? Provide a reasonable linear programmingformulation.

5. CopyMaster is a company that specializes in making lots of copies. The company hasm copy machines of varying capabilities and speed. The time required to complete anyof n types of jobs varies with machine choice. Let cij be the machine time required tocomplete a job of type j = 1, 2, . . . , n using machine i = 1, 2, . . . ,m. Suppose Copy-Master has Nj jobs of each type j to complete. The machines can run concurrently.This week begins with the maching data and job requirements listed in the followingtables. Times cij are in minutes.

cij j = 1 j = 2 j = 3 j = 4 j = 5 j = 6

i = 1 6 12 8 12 10 3i = 2 8 6 12 7 6 10i = 3 8 11 6 6 7 4i = 4 3 9 8 10 3 10

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6Nj 641 190 386 469 395 482

In order to improve the logistics of dividing and re-combining job types, also considerconstraints to enforce a lower bound on the size of subjobs at 24. That is, if anymachine is used to make a batch of copies, then that batch must be at least 24 copies.Formulate a mixed-integer program that can be used to decide how to assign jobs tomachines in order to complete the jobs as quickly as possible. Remember, machinesrun concurrently.

6. In this exercise, you will solve the CopyMaster problem using a software solver.

(a) Obtain access to Matlab or Octave software. There are many options:

• WSU students have access to a student version of Matlab which can bedownloaded onto a personal computer.

• Octave can be downloaded on a personal computer (no cost).

• Sign up for a free account at octave-online.net and complete exercises online.

(b) Complete the tutorial exercises given in the document MatlabOctaveTutorial.pdffor either Matlab or Octave.

(c) Use either Matlab or Octave to solve the CopyMaster problem.

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7. Let f : Rn → R be a convex function and let c ∈ R. Prove that the set X ={x ∈ Rn | f(x) ≤ c} is convex.

8. For each of the following sets, determine whether it is a polyhedron. Carefully justifyyour answers.

(a) W = ∅ (empty set).

(b) X = {x ∈ R | x2 − 8x + 15 ≤ 0}.(c) Y = {x ∈ R2 | x2

1 + x22 ≤ 1}.

(d) The set of all optimal solutions to a linear program.

9. (see Exercise 2.4) We know that every linear program can be converted into an equiva-lent standard form linear program. We also know that nonempty polyhedra in standardform have at least on extreme point. We might then incorrectly conclude that everynonempty polyhedron has at least on extreme point. Explain why this conclusion isfalse and provide some clarifying examples.

10. (see Exercise 2.16) Consider the set

{x ∈ Rn | x1 = x2 = · · · = xm = 0, 0 ≤ xm+1, · · · , xn ≤ 1}.

Could this set be the feasible region of a standard form linear program?

11. (Exercise 2.10) Consider the standard form polyhedron P = {x ∈ Rn | Ax = b, x ≥ 0},where A is m×n with linearly independent rows. For each of the following statements,state whether it is true or false. If true, provide a formal proof. If false, provide a clearcounterexample.

(a) If n = m + 1, then P has at most two basic feasible solutions.

(b) The set of all optimal solutions is bounded.

(c) At every optimal solution, no more than m variables can be positive.

(d) If there is more than one optimal solution, then there are infinitely many optimalsolutions.

(e) If there are several optimal solutions, then there exist at least two basic feasiblesolutions which are optimal.

(f) Consider the problem of minimizing f(x) = max{cTx, dTx} over the set P . If thisproblem has an optimal solution, then it must have an optimal solution which isan extreme point of P .

12. (Exercise 2.9) Consider the standard form polyhedron P = {x ∈ Rn | Ax = b, x ≥ 0},where the rows of A are linearly independent.

(a) Suppose that two differnt bases lead to the same basic solution. Show that thebasic solution is degenerate.

(b) Consider a degenerate basic solution. Is it true that it corresponds to two distinctbases? Prove or give a counterexample.

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(c) Suppose that a basic solution is degenerate. Is it true that there exists an adjacentbasic solution which is degenerate? Prove or give a counterexample.

13. Let P1 and P2 be polyhedra in Rn. Prove that the set Q = {x1 + x2 | x1 ∈ P1, x2 ∈ P2}is a polyhedron.

14. (Exericse 3.3) Consider the standard form polyhedron P = {x ∈ Rn | Ax = b, x ≥ 0}and y ∈ P . Prove that d ∈ Rn is a feasible direction at y if and only if Ad = 0 anddi ≥ 0 whenever xi = 0.

15. Let P = {x ∈ R3 | x1 + 2x2 + 3x3 = 6, x ≥ 0}. Find the set of feasible directions at

y =(0 3 0

)Tand at w =

(1 1 1

)T.

16. (Exercise 3.12) Consider the linear program

maxx

z = 2x1 + x2

s.t. x1 − x2 ≤ 2

x1 + x2 ≤ 6

x ≥ 0

x ∈ R2

(a) Convert the LP into standard form and construct the basic feasible solution forwhich (x1, x2) = (0, 0).

(b) Carry out the full tableau implementation of the Simplex Method, starting withthe basic feasible solution of part (a).

(c) Sketch a graphical representation of the problem in terms of the original variablesx1 and x2, and indicate the path taken by the Simplex Algorithm.

17. Solve the following optimization problem using the Simplex Method.

maxx

z = x1 + x2 + x3

s.t. x1 + x2 − 2x3 ≤ 2

x1 + x2 − x3 ≤ 3

− x1 + 2x2 + x3 ≤ 4

x ≥ 0

x ∈ R3

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18. Solve the following optimization problem using the Simplex Method.

maxx

z = 2x1 + 3x2

s.t. 3x1 + 4x2 ≤ 12

4x1 + 3x2 ≤ 12

− x1 + x2 ≤ 3

x1 − x2 ≤ 3

x ≥ 0

x ∈ R2

19. Solve the following optimization problem using the Simplex Method.

maxx

x1 − x2 + x3

s.t x1 + x2 + x3 ≤ 2

2x1 + 2x2 − 6x3 ≥ 6

x1 − x2 − x3 ≥ 0

x ≥ 0

x ∈ R3

20. (Exercise 3.17) Solve the following linear program using the Two-Phase Simplex Method.

minx

z = 2x1 + 3x2 + 3x3 + x4 − 2x5

s.t. x1 + 3x2 + 4x4 + x5 = 2

x1 + 2x2 − 3x4 + x5 = 2

− x1 − 4x2 + 3x3 = 1

x ≥ 0

x ∈ R5

21. Consider the Ellipsoid Interior Point Method for solving linear programs. It wouldseem that a sequence of interior spheroids would simplify the computational task offinding the optimal point on the boundary of the spheroid. Carefully explain whysuch a sequence cannot satisfy all of the key requirements of this type of interior pointmethod.

22. (to be added)

23. (to be added)

24. (to be added)

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