Tutorial Overview
This tutorial is divided into 2 parts; they are: 1. Method of Lagrange multipliers with equality constraints 2. Two solved examples
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Prerequisites
For this tutorial, we assume that you already know what are: 1. Derivative of functions 2. Function of several variables, partial derivatives and gradient vectors 3. A gentle introduction to optimization 4. Gradient descent You can review these concepts by clicking on the links given above.
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What Is The Method of Lagrange Multipliers with Equality Constraints?
Suppose we have the following optimization problem: Minimize f(x) Subject to: g_1(x) = 0 g_2(x) = 0 … g_n(x) = 0 The method of Lagrange multipliers first constructs a function called the Lagrange function as given by the following expression. L(x, ????) = f(x) + ????_1 g_1(x) + ????_2 g_2(x) + … + ????_n g_n(x) Here ????represents a vector of Lagra
Solved Examples
This section contains two solved examples. If you solve both of them, you’ll get a pretty good idea on how to apply the method of Lagrange multipliers to functions of more than two variables, and a higher number of equality constraints.
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Relationship to Maximization Problems
If you have a function to maximize, you can solve it in a similar manner, keeping in mind that maximization and minimization are equivalent problems, i.e., maximize f(x) is equivalent to minimize -f(x)
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Importance of The Method of Lagrange Multipliers in Machine Learning
Many well known machine learning algorithms make use of the method of Lagrange multipliers. For example, the theoretical foundations of principal components analysis (PCA) are built using the method of Lagrange multipliers with equality constraints. Similarly, the optimization problem in support vector machines SVMs is also solved using this method
Extensions
This section lists some ideas for extending the tutorial that you may wish to explore. 1. Optimization with inequality constraints 2. KKT conditions 3. Support vector machines If you explore any of these extensions, I’d love to know. Post your findings in the comments below.
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Summary
In this tutorial, you discovered what is the method of Lagrange multipliers. Specifically, you learned: 1. Lagrange multipliers and the Lagrange function 2. How to solve an optimization problem when equality constraints are given
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