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Tutorial4solution

# Tutorial4solution - Tutorial 4 Outline of Solutions Q1...

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Tutorial 4: Outline of Solutions Q1. 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 solu- tions. (e) If there are several optimal solutions, then there exist at least two basic feasible solutions that are optimal. Solution : (a) TRUE. Since n - m = 1, the feasible region can be visualized in one dimensional space with n inequality constraints. Clearly in 1-D, this can have at most two basic feasible solutions. (b) FALSE. The following LP min 0 x 1 s.t. x 1 = 1 x 1 ,x 2 0 has an unbounded set of optimal solutions expressed as x * 1 = 1 ,x * 2 0. (c) FALSE. In the example in (b), m = 1, n = 2. Every optimal solution of the form x * 1 = 1 ,x * 2 > 0 has two strictly positive variables. The statement is true only for basic fea- sible solutions (in this case x * 1 = 1 ,x * 2 = 0). (d) TRUE. Given a set of optimal solutions

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Tutorial4solution - Tutorial 4 Outline of Solutions Q1...

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