Overview

We want to optimize (i.e. find the minimum and maximum value of) a function, f(x,y,z)f(x,y,z), subject to the constraint g(x,y,z)=kg(x,y,z)=k. Again, the constraint may be the equation that describes the boundary of a region or it may not be. The process is actually fairly simple, although the work can still be a little overwhelming at times.

Method of Lagrange multipliers

  1. Solve the following system of equations:∇f(x,y,z)=λ∇g(x,y,z)g(x,y,z)=k\begin{aligned}\nabla f\left( {x,y,z} \right) & = \lambda \,\,\nabla g\left( {x,y,z} \right)\\ g\left( {x,y,z} \right) & = k\end{aligned}
  2. Plug in all solutions, (x,y,z)(x,y,z), from the first step into f(x,y,z)f(x,y,z) and identify the minimum and maximum values, provided they exist and ∇g≠0→\nabla g \ne \vec{0} at the point.
    The constant, λλ is called the Lagrange Multiplier.

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Lagrange multiplier theorem

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Notes

Why are Lagrange multipliers chosen as such? This is so that for example, ∂L/∂λ\partial L /\partial \lambda brings back the constraint, and so forth.

maximizex,yf(x,y)subject tog(x,y)=0.\displaystyle {\begin{aligned}{\underset {x,y}{\text{maximize}}}\quad &f(x,y)\\{\text{subject to}}\quad &g(x,y)=0.\end{aligned}}

References:

  1. https://en.wikipedia.org/wiki/Lagrange_multiplier
  2. https://tutorial.math.lamar.edu/classes/calciii/lagrangemultipliers.aspx
  3. https://www.math.cmu.edu/~gautam/sj/teaching/2016-17/269-vector-analysis/pdfs/lagrange.pdf
  4. Gilbert Strang, Linear Algebra and its Applications, ch. 6.4, p. 378, 4th ed., 2006.