I'm in high-school and I'm told that the maximum/minimum of a linear programming occurs at the vertex.For more info see the chapter here. For convinience I'm putting relevant excerpt here:
Now, we see that every point in the feasible region satisfies all the constraints, and since there are infinitely many points, it is not evident how we should go about finding a point that gives a maximum value of the objective function. To handle this situation, we use the following theorems which are fundamental in solving linear programming problems. The proofs of these theorems are beyond the scope of the book.
Theorem 1 Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
Theorem 2 Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R
I searched wikipedia here to get under Optimal vertices (and rays) of polyhedra:
"..then the optimum value is always attained on ... This principle underlies the simplex algorithm for solving linear programs.."
But I couldn't understand it. Can someone explain?It might be helful to add that I have studied basic Calculus.