HW 4 - ENGR62/MS&E111 Introduction to Optimization Prof Ben...

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ENGR62/MS&E111 Spring 2008 Introduction to Optimization May 2, 2008 Prof. Ben Van Roy Homework 4 Solutions Part A True or false: a) TRUE. Let x 1 be a solution to Ax = b . If x 2 is a second distinct solution, then it must be that x 2 - x 1 N ( A ). But here dim ( N ( A )) = 1. Let { ¯ x } be a basis for N ( A ). It follows that we can write x 2 - x 1 as λ ¯ x for a suitable scalar λ . We may thus conclude that every solution of Ax = b is of the form x 1 + λ ¯ x . Thus, P is constrained to a line. Recall that every basic feasible solution is also a vertex. Thus to have more than two basic feasible solutions, we would need to have more than two vertices on the line given by x 1 + λ ¯ x . But it is impossible to have more than two vertices on a line (say we have three, then it is clear that one can be written as a convex combination of the remaining two). Hence, we cannot have more than two basic feasible solutions. b) FALSE. Consider min x 1 , s.t. x 1 = 1, ( x 1 ; x 2 ) 0. The optimal solution set is unbounded.
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