assn10_1

assn10_1 - ORIE 3300/5300 Individual work ASSIGNMENT 10...

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ORIE 3300/5300 ASSIGNMENT 10 Fall 2010 Individual work. Due: 3 pm, Friday November 19. 1. We have spent much time on the idea of a “certificate of optimality,” which can be used to show that a proposed solution to a linear pro- gramming problem is optimal. In this question you will see how to produce a certificate that a linear programming problem is infeasible. Suppose the constraints of a linear programming problem in standard equality form are Ax = b, x 0 , where A is an m × n matrix. (a) Suppose there is a vector y m satisfying A T y 0 , b T y < 0 . Show that the linear programming problem has no feasible solu- tion. (Hint: consider y T Ax = ( A T y ) T x for a feasible x .) Hence such a y can be viewed as a certificate of infeasibility. As with optimality, we want to show that whenever a linear program- ming problem is infeasible, such a certificate of infeasibility exists. (b) Suppose for simplicity that b 0. Show that if the linear pro- gramming problem is infeasible, then the auxiliary problem max ( - e ) T v subject to Ax

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