A linear optimization (also called linear programming) problem has several interesting properties as stated in Wikipedia:
“Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality.”
“In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region’s corners. Further, if an extreme value occurs at two corners, then it must also occur everywhere on the line segment between them.”
Note: the constraint set needs to be bounded (i.e. can fit in a circle for a 2D problem).
Note (2): linear functions are both convex and concave.
Corner solution = basic feasible solution = optimum vertex. “A basic feasible solution (BFS) is a solution with a minimal set of non-zero variables. Geometrically, each BFS corresponds to a corner of the polyhedron of feasible solutions. If there exists an optimal solution, then there exists an optimal BFS. Hence, to find an optimal solution, it is sufficient to consider the BFS-s. This fact is used by the simplex algorithm, which essentially travels from some BFS to another until an optimal one is found.”
