Discussion
For an LP problem which of the following statements is false ?
Answer: B
The false statement is if an optimal solution exists, then the feasible region is bounded.
An LP can have an optimal solution on an unbounded feasible region. For example, minimise \(x_{1} + x_{2}\) subject to \(x_{1}, x_{2} \ge 0\): the region is the whole first quadrant, which is unbounded, yet the minimum 0 is attained at the origin.
The other statements are true. If the feasible polyhedron has an optimum, at least one vertex (basic feasible solution) is optimal. If the half-spaces of the constraints have empty intersection, there is no feasible point. And an intersection of half-spaces is convex, so the feasible set is convex.
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