Lagrange Multipliers | Definition and Examples

In mathematics, a Lagrange multiplier is a potent tool for optimization problems and is applied especially in the cases of constraints. Named after the Italian-French mathematician Joseph-Louis Lagrange, the method provides a strategy to find maximum or minimum values of a function along one or more constraints. It is used in problems of optimization with constraints in economics, engineering, and physics.

Theory Behind Lagrange Multipliers

The theory of Lagrange multipliers was developed by Joseph-Louis Lagrange at the very end of the 18th century. His life bestrode the two independent worlds of mathematics and physics, showcasing profound and seminal work in each. His method was a new, systematic procedure in the solution of previously established ad hoc methods to solve constrained maximization and minimization problems. The importance of the Lagrange multipliers is not of mere historical interest; it is what followed the subject that paved the way for so many optimizations used nowadays. It is a real workhorse of mathematics for analysis and application.

Definition and Mathematical Formulation

Definition

Lagrange multipliers are auxiliary variables, which transform the constrained optimization problem into an unconstrained form in a way that the problem reduces into solving a calculus problem. Conceptually, it gives the constraint or constraints a means by which to be in the process of optimization, through which an optimum solution is to be identified.

The Lagrange Function

The so-called Lagrange function, or just Lagrangian, When we want to maximize or minimize an objective function subject to one or more constraints, the Lagrange function helps us incorporate those constraints directly into the optimization process using Lagrange multipliers.. If we have an objective function f(x,y,z,…) and a constraint g(x,y,z,…)=0, the Lagrange function is defined as:

L(x,y,z,…,λ)=f(x,y,z,…)+λ⋅g(x,y,z,…)

where,

f(x,y,z,…) is the objective function we are optimizing,

g(x,y,z,…)=0 is the constraint we must satisfy,

λ is the Lagrange multiplier. which helps adjust the optimization to respect the constraint.

Solving With Lagrange Multipliers

Construction of Lagrangian

A constrained optimization problem may be solved by defining a Lagrangian that takes into account the constraint or constraints in the objective function. When there are several constraints, additional Lagrange multipliers will be introduced, in which case the Lagrangian will take the form:

L(x,y,z,…,λ1,λ2,…)=f(x,y,z,…)+λ1⋅g1(x,y,z,…)+λ2⋅g2(x,y,z,…)

The Lagrange Multiplier Equations

Next, obtain partial derivatives of the Lagrangian in all variables including the Lagrange multipliers and equate them to zero. To do so will yield a system of equations, known as the Lagrange multiplier equations, which are solved simultaneously for an optimal solution.

To find the optimal solution, you solve the system of equations derived by setting the partial derivatives of the Lagrangian with respect to all variables and Lagrange multipliers equal to zero:

∂L / ∂x = 0 ; ∂L / ∂y= 0 ; ∂L / ∂λ1 = 0; ∂L / ∂λ2 = 0; ....................

These equations ensure that the solution satisfies both the objective function's optimization and the constraints.

Geometric Interpretation of Lagrange Multipliers

Geometrically, this technique of Lagrange multipliers can be seen as the requirement that the gradient of the objective function should be parallel to the gradient of the constraint function(s). That is, at an optimum point, the steepest-ascent or descent direction of the objective function will point in a direction wherein the constraint function doesn't vary. This guarantees avoidance of violations of the constraints while optimizing an objective function.

Example:

Consider a simple case of maximizing a function 𝑓(𝑥,𝑦) subject to a constraint g(x,y)=0. Let’s say the objective function is 𝑓(𝑥,𝑦)=𝑥^2+𝑦^2 , which represents the equation of a circle, and the constraint is g(x,y)=x+y−1=0, which represents a straight line.

The gradient of f(x,y) is ∇f=(2x,2y),

The gradient of g(x,y) is ∇g=(1,1).

At the optimal point, the gradients of f and g must be parallel, meaning:

∇f=λ∇g

This implies that the steepest change in f(x,y) happens along the direction of the constraint, ensuring that the optimization follows the constraint x+y=1.

Lagrange Multipliers Application

The Lagrange multipliers have a lot of applications in most disciplines involved. ;

1. In economics: The Lagrangian multipliers are applied to optimize functions of utility or profit, with restrictions on the resources available or the expenses to expend.
2. Physics serves the purpose to solve fundamental problems, such as the minimization of energy in a system while respecting the conservation laws.
3. Engineering: The department is used primarily in design optimization, where engineers are supposed to optimize a design performance subject to physical or material constraints.
4. Regularization methods in machine learning are used with Lagrange multipliers to put constraints on model complexity.

Solved Examples on Lagrange Multipliers

So how and when to apply? There are certain conditions. Suppose you have the following problem: Find the coordinates of the point on the plane 2x + 3y - 5z = 1 which is at the least distance from the origin. So the function you want to optimize is,

√(x2 + y2 + z2),
Let this be f(x, y, z) 

But we have a constraint;the point should lie on the given plane.Hence this 'constraint function' is generally denoted by g(x, y, z).But before applying Lagrange Multiplier method we should make sure that g(x, y, z) = c where 'c' is a constant. In this situation,

g(x, y, z) = 2x + 3y - 5z

It is indeed equal to a constant that is '1'. Hence we can apply the method. Now the procedure is to solve this equation:

∇f(x, y, z) = λ∇g(x, y, z)

where λ is a real number. This gives us 3 equations and the fourth equation is of course our constraint function g(x, y, z).Solve for x, y, z and λ. An example will make it clear.

Example 1: Find the maximum and minimum values of f(x, y, z) = 3x²+ y subject to the constraint,

4x - 3y = 9 
and x2 + z2 = 9

This example has been deliberately taken to teach you what to do in case of more than one constraint functions. In such cases assume as many arbitrary constants as the number of constraint functions and write the equation in the form:

∇f(x, y, z) = c1∇g(x, y, z) + c2∇h(x, y, z) + c3∇p(x, y, z) ... ...

where c_i for i=1, 2, 3... are just real numbers and g, h, p are constraint functions. Now if you get more than one triplet, figure out which one represents a maximum and which one represents the minimum by satisfying it in the function to be optimized and compare the values.In this question the answer would be:

Maximum for (-2/√13, 3/√13, -2 - 7/√13) and 
Minimum for (2/√13, -3/√13, -2 + 7/√13)

Example 2: Find the point on the plane 4x+5y+z=11 that is nearest to the point (2,0,1)

Solution:

Let the point on the plane be (x,y,z). The distance function to minimize is D(x,y,z)=(x−2)² +y² +(z−1)². The constraint function is g(x,y,z)=4x+5y+z−11=0. Using the method of Lagrange multipliers, we solve for x, y, z, and the multiplier λ. The point that minimizes the distance is found to be (8/3, 1/3, 1).

Example 3: Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in a unit sphere.

Solution:

Let the dimensions of the box be x, y, and z. The volume of the box is V=8xyz, and the constraint is x² +y² +z² =1. Applying the Lagrange multiplier method, the critical points are calculated, leading to the dimensions that maximize the volume.

Practice Problems on Lagrange Multiplier

Problem 1: Maximize the function f(x,y)=x² +y² subject to the constraint x² −y² =1.

Problem 2: Find the points on the surface x² +y² +z² =9 that are closest to the point (1,1,1).

Problem 3: Minimize the function f(x,y,z)=x+2y+3z subject to the constraint x² +y² +z² =1.

Conclusion

Lagrange multipliers provide a powerful methodology for finding extrema of a function subject to constraints. This approach-combining the transformation of a constrained optimization problem to a system of equations-solves easily for optimal solutions satisfying both an objective function and constraints. The method of Lagrange multipliers is one of the most useful tools, extending standard calculus to solve more complex real-world problems in everything from economics models to engineering designs and physics problems.