Linear Program:Multiple Optima for multivariable Obj. Func.?

[StevenJacobs990]

I know there can be an infinite number of solutions when the objective function with 2 variables has an equal slope as a constraint's slope (assuming the constraint is affecting the feasible region and not a redundant constraint).

How can you know there are multiple optimal solutions for multi-variable object function?

 

[andrewkirk]

In two variables the condition you describe can occur when the objective function is $a_1x_1+a_2x_2$ and one of the non-redundant constraints can be written as $a_1x_1+a_2x_2\leq b$ or $a_1x_1+a_2x_2\geq b$.

Similarly, with $n$ variables. if the objective function is $\sum_{k=1}^n a_kx_k$ and one of the non-redundant constraints can be written as $\sum_{k=1}^n a_kx_k\leq b$ or $\sum_{k=1}^n a_kx_k\geq b$ then there can be infinitely many solutions.