HW 2 - ENGR62/MS&E111 Introduction to Optimization Prof Ben...

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Spring 2008 Introduction to Optimization April 18, 2008 Prof. Ben Van Roy Homework Assignment 2 : Solutions Solve Questions 1, 2, 3, 8 and 10 from Chapter 3. 3.1. Adding x + y 0 reduces the feasible region to { [0 , 0] } . 3.2. The polyhedron is the unit simplex, which has one face and three vertices. It is a triangle with vertices (1 , 0 , 0), (0 , 1 , 0) and (0 , 0 , 1). The maximum of x + 2 y + 3 z is 3 and is attained in the vertex (0 , 0 , 1). 3.3 Infeasibility is evident from sketching the feasible region. Here we show this algebraically: Let x,y be feasible. Then, from the third constraint, we have that x,y must satisfy 3 2 (2 x + 5 y ) 9 2 , and from the last constraint, x,y must satisfy - 3 x + 8 y ≤ - 5. Adding these constraints, we have that x,y must satisfy 31 2 y ≤ - 1 2 . But the second constraint requires y 0. Since we cannot simultaneously have y 0 and 31 2 y ≤ - 1 2 , the feasible region must be empty. 3.8

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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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HW 2 - ENGR62/MS&E111 Introduction to Optimization Prof Ben...

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