Midterm Solutions

Midterm Solutions - ENGR62/MS&E111 Fall 2008...

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Unformatted text preview: ENGR62/MS&E111 Fall 2008 Introduction to Optimization May, 2008 Prof. Ben Van Roy Midterm Exam Instructions • This is a take-home, open book/notes exam. • There are 4 questions. Each is worth a total of 25 points. Partial credit is possible for each question provided that what you hand in is relevant and partially solves a problem. • The exam is due at Terman 381 by 7:30PM. A TA will be there from 7:00-7:30PM to collect your exams. The entrance on the east side of Terman (i.e., the one closest to the Quad) should be open- if not your Stanford ID can get you in. • Collaboration is not permitted on this exam. Exam-related questions will not be answered by the teaching staff (unless you think there is a serious exam typo). If you are unsure about something in a question, do your best to answer it with the information given. • Questions 3 and 4 require the use of Excel. Be sure to check off “Assume Linear Model”. Please print out and hand in with your exam the worksheets you use to solve the problems (no solution reports are necessary). • Also, print out and attach to the front of your exam this cover sheet with the honor code statement signed. • GOOD LUCK! Honor Code In recognition of and in the spirit of the Honor Code, I certify that I will neither give nor receive unpermitted aid on this examination and that I will report, to the best of my ability, all Honor Code violations observed by me. Name: Signature: 1 LINEAR PROGRAMMING THEORY A True/False (15 points) State whether each of the following statements is true or false. For each part, your answer should be one of “True” or “False.” Any explanations will not be read and graded. For each correct answer, you will receive three points. For each incorrect answer, you will lose two points. No points are assigned or lost for a part unanswered. (a) For any linear program in which all constraints are linearly independent, there exists exactly one basic feasible solution. (b) If a market is incomplete, then there exists an arbitrage opportunity. (c) If the feasible region of a linear program contains a line, then it must have infinitely many optimal solutions. (Note: A set U is said to contain a line if there exists a vector x ∈ U and a vector d 6 = 0 such that x + αd ∈ U for all α ∈ < .) (d) If a linear program has at least one optimal solution, then the set of all optimal solutions is convex. (e) If a payoff matrix has full rank, any structured products can be replicated by contingent claims available in the market. Solution (a) FALSE. The statement doesn’t say anything about the numbers of variables or constraints. So the number of basic feasible solutions can vary. Counter example: min x 1 s . t . x 1 + x 2 = 1 It has only one (linearly independent) constraint, but there is no basic feasible solution....
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This note was uploaded on 10/12/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.

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Midterm Solutions - ENGR62/MS&E111 Fall 2008...

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