Global minima and local minima of a function:
One of the major difficulties one has to face in solving an NLPP is the determination of the solution point, which gives not only optimal solution for the objective function at the point but also optimizes the function over the complete solution space.
Definition of Global Minimum:
A function f (x) has a global minimum at a point xº of a set of points K if an only if f(x°) ≤ f(x) for all x in K.
Definition of Local Minimum:
A function f (x) has the local minimum point xº of a set of points K if and only if there exists a positive number such that f (xº) ≤ f (x) for all x in K at which || x0 – x || < ⊂-
Lagrange multiplier:
L (x1, x2, ……xn, λ1, λ2 , …. λn ) – Σ λi [gi (x1, x2, ….xn) = bi where i = 1, 2, …m and λ1, λ2 ,…λn are called as Lagrange Multipliers. The optimal solution to the Lagrange function is determined by taking partial derivatives of the function L with respect to each variable (including Lagrange multipliers and setting each partial derivative to zero and finding the values that make the partial derivatives zero. Then the solution will turn out to be the solution to the original problem.
Example: Find the extreme value of Z = f (x1, x2) = 2 x1x2Subject to x12 x22 = 1