105_Lagrange Multiplier Theorem

The Lagrange Multiplier Theorem is a technique used to solve the provlem of finding the extrema of multivariable functions under specific constraints. This method is very effective for solving optimization problems with constrains.

In this post, I will explain using an example of an optimization problem with equality constraints.

 

 

Assuming that $x$ is a regular point where it is a local minimum, there exists a Lagrange multiplier $v_j^*$ as follows.

 

 

The Lagrange function can be defined as follows.

 

 

Through the Lagrange function, the first equation can be represented as follows.

 

 

Differentiating $L$ with respect to $v_j$ tields the equality constraint.

 

 

Using the above two equations, we can see that the Lagrange function is stationary with respect to ${\bf{x}}$ and ${\bf{v}}$, allowing the problem to be transformed into an unconstrained problem for ${\bf{x}}$ and ${\bf{v}}$, thereby simplifying the solution to the minimization problem. Additionally, the Lagrange multiplier can be either positive or negative.

Solve a simple example.

 

Assume we neet to design a cylindrical storage tank at minimum cost. The total surface area of the storage tank, composed of a closed end and a cylinder, is as follows.

 

 

If the cost per unit area of the metal used is denoted as $c$, then the cost function for the problem is as follows.

 

 

Additionally, the volume constraints for the tnak is as follows.

 

 

To solve the problem using the Lagrange Muliplier Method, the given conditions can be organized as follows.

 

 

The equation, when expressed in the form of the Lagrange Muliplier Method, is as follows.

 

 

The necessary conditions of the Lagrange Multiplier Theorem are as follows.

 

 

Therefore, three unknowns and three equations are derived, allowing for the solution to be found.

 

 

We have solved a multivariable problem with equality constraints using the Lagrange Multiplier Method. Although it may seem complex at first glance, applying the method to a problem allows for a simple, intuitive, and quick solution.