Midterm 2 - Solutions

Midterm 2 - Solutions - ENGR62/MS&E111 Fall 2006...

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Unformatted text preview: ENGR62/MS&E111 Fall 2006 Introduction to Optimization November, 2006 Prof. Ben Van Roy Midterm Exam Solutions 1: The Geometry of Linear Programming (a) Consider a polyhedron in 2-dimensional Euclidean space defined by 5 inequality con- straints. i) What is the largest possible number of basic solutions? Solution: 10 ii) What is the largest possible number of basic feasible solutions? Solution: 5 iii) What is the smallest possible number of basic solutions? Solution: 0 (b) Consider the following linear program minimize cx 1 + dx 2 subject to x 1 + x 2 ≤ 3 x 1- x 2 ≤ 2 ≤ x 1 x 2- 2 ≥ i) Provide scalars c and d such that the linear program has multiple optimal solutions. Solution: c = d = 0 . ii) Provide scalars c and d such that the vector x = [0 , 3] T is the only optimal solution. Solution: c = 0 ,d =- 1 . iii) Provide scalars c and d such that if the first constraint is relaxed (i.e., the first con- straint is deleted), there is no optimal solution. Solution: c = 0 ,d =- 1 . 2: Super-Replication Recall that the problem of finding the least-expensive super-replicating portfolio for a payoff b ∈ < M with asset prices ρ ∈ < N and a matrix P ∈ < M × N of contingent claim payoffs can be posed as a linear program: minimize ρ T x subject to Px ≥ b. The above model assumes that all portfolios, subject to the payoff constraint, are feasible. In reality, however, both the government and brokerage firms have additional rules that govern trading of securities. For example, when short selling, investors are required to put up collateral to ensure that they have the resources required to buy back the assets being loaned. Depending on the customer’s resources, these kinds of rules restrict what types of trades are possible....
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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 2 - Solutions - ENGR62/MS&E111 Fall 2006...

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