The lagrangian method

Lagrangian method is one of the optimization methods and is an extened form of Classic method of simplifying the formulae and methods. This method helps in finding the maxima i.e. greatest possible amount, and minima i.e. minimum possible degree, of a function depending on the constraints.

Following steps are usually followed in the langragian method:

1. Objective function is determined:
Objective function is a function in optimization theory. This function is to be maximized or minimized i.e. to be optimized. Here function is a quantity depending on another quantity and it can be changed by changing the values of the other quantity.
We can express objective funcion with expression like "Z(X')"
Where X' is the decision variable representing variable for which we can make decisions or which can be changed i.e. X' = (X1, X2,...Xn)

2. Constraints are determined
Constraints are the factors that causes limitations to the freedom of something. Problems related to pharmaceutical product and process desgin are considered as constrained optimization problems.

3. Inequality constraints are converted into equality constraints

4. From the lagrange function, assign

    a. One lagrange multiplier i.e. lambda for each constraint

    b. One slack variable i.e q for each inequaltiy constraint

5. The lagrange function for each variable, is differentiated partially, and setting the derivative equal to zero.

6. The set of simultaneous equations are solved

7. Resulting values are substituted into the objective functions

Provision for this method is the completion of the experiment before the optimization so that the mathematical models can be generated. Moreover, in lagrangian method several responses or dependent variables can be handled but work can be done on only two independent variables.