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# Tutorial 4 - NATIONAL UNIVERSITY OF SINGAPORE Department of...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 4 1. Consider the standard form polyhedron P = { x R n | Ax = b , x 0 } . Suppose the matrix A has dimensions m × n and its rows are linearly independent. For each of the following statements, state whether it is true or false. If true, provide an argument, else, provide a counter example. (a) If n = m + 1, then P has at most two basic feasible solutions. (b) The set of all optimal solutions is bounded. (c) At every optimal solution, no more than m variables can be positive. (d) If there is more than one optimal solution, then there are uncountably many optimal solutions. (e) If there are several optimal solutions, then there exist at least two basic feasible solutions that are optimal. 2. Consider the following linear programming problem, where b and A i are 3 × 1 column matrices for i = 1 , 2 , 3 , 4. min c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 s.t. A 1 x 1 + A 2 x 2 + A 3 x 3 + A 4 x 4 = b x 1 , x 2 , x 3 , x 4 0 Suppose x = ( x 1 , 0 , x 3 , x 4 ) 0 is a basic feasible solution, where B = ( A 1 , A 3 , A 4 ) is the basis matrix. Let d = ( d 1 , 5 , d 3 , d 4 ) 0 such that x + d is a feasible solution. (a) Prove that d 1 d 3 d 4 =

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