Is it correct to state that if a linear objective function is not in parallel with any of the constraints, than there is a single optimal solution at some vertex of the polytope?
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This is incorrect. To construct a class of simple counterexamples in three dimensions, consider problems of the form $Ax \le b, \, x \ge 0$ where $A$ is any positive $2 \times 3$ matrix of rank 2 and $b$ is a positive 2-vector. This defines a non-empty bounded feasible set in the first octant. Now consider objective functions of the form $x \mapsto cx$ where $c$ is a convex combination of the two rows of $A$. The solution set of the maximization problem is the line segment where $x \ge 0$ and $Ax = b$, but the objective function is not parallel to any of the constraints. You may get multiple solutions in any situation where $c$ is in the space spanned by the rows of $A$. |
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