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:

•      As the non-linear programming problem is composed of some differentiable objective function and equality side constraints, the optimization may be achieved by the use of Lagrange multipliers (a way of generating the necessary condition for a stationary point).
•        A Lagrange multiplier measures the sensitivity of the optimal value of the objective function to change in the given constraints bi in the problem. Consider the problem of determining the global optimum of
•       Z = f (x₁, x₂,…. x_n) subject to the ‘m’ constraints gi (x₁, x₂,…x_n) = bi , i = 1, 2, …m.
•        Let us first formulate the Lagrange function L defined by:

L (x₁, x₂, ……x_n, λ₁, λ₂ , …. λn ) – Σ λi [g_i (x₁, x₂, ….x_n) = bi where i = 1, 2, …m and λ₁, λ₂ ,…λ_n 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