[PDF] optimal solution

What happens if the optimal solution lies at the intersection point?

An optimal solution that lies at the intersection point of two constraints causes both of those constraints to be considered active. If any of the constraint lines do not pass through the optimal point, those constraints are called inactive. In general, we ignore the constraints at 0 and focus on the constraints generated by limits on resources.

What is the first order necessary optimality condition?

The above corollary is a ?rst order necessary optimality condition for an unconstrained minimization problem. The following theorem is a second order necessary optimality condition Theorem 5 Suppose that f (x) is twice continuously di?erentiable at x¯ ? X. If x¯ is a local minimum, then ?f (¯x)=0and H (¯x) is positive semide?nite.

Where should a solution lie to a problem?

solution to the problem must lie in the region in order to obey both x2 of the constraints. x1minimum must lie on the boundary. In fact, it is most likely that the optimum occurs at one of the corner points.

What is a sufficient condition for local optimality?

su?cient condition for local optimality is a statement of the form: “iif ¯ satis?es . . . , then x¯ is a local minimum of (P).” Such a condition allows us to automatically declare that x¯ is indeed a local minimum.