Interpretation of Lagrange multipliers

Background

• Introduction to Lagrange multipliers

Lagrange multipliers technique, quick recap

• Step 1: Introduce a new variable $\greenE{\lambda}$start color #0d923f, lambda, end color #0d923f, and define a new function $\mathcal{L}$L as follows:

  $\mathcal{L}(x, y, \dots, \greenE{\lambda}) = \blueE{f(x, y, \dots)} - \greenE{\lambda} (\redE{g(x, y, \dots)-c})$L, left parenthesis, x, comma, y, comma, dots, comma, start color #0d923f, lambda, end color #0d923f, right parenthesis, equals, start color #0c7f99, f, left parenthesis, x, comma, y, comma, dots, right parenthesis, end color #0c7f99, minus, start color #0d923f, lambda, end color #0d923f, left parenthesis, start color #bc2612, g, left parenthesis, x, comma, y, comma, dots, right parenthesis, minus, c, end color #bc2612, right parenthesis

  This function $\mathcal{L}$L is called the "Lagrangian", and the new variable $\greenE{\lambda}$start color #0d923f, lambda, end color #0d923f is referred to as a "Lagrange multiplier"
• Step 2: Set the gradient of $\mathcal{L}$L equal to the zero vector.

  $\nabla \mathcal{L}(x, y, \dots, \greenE{\lambda}) = \textbf{0} \quad \leftarrow \small{\gray{\text{Zero vector}}}$del, L, left parenthesis, x, comma, y, comma, dots, comma, start color #0d923f, lambda, end color #0d923f, right parenthesis, equals, start bold text, 0, end bold text, left arrow, start color gray, start text, Z, e, r, o, space, v, e, c, t, o, r, end text, end color gray

  In other words, find the critical points of $\mathcal{L}$L.
• Step 3: Consider each solution, which will look something like $(x_0, y_0, \dots, \greenE{\lambda}_0)$left parenthesis, x, start subscript, 0, end subscript, comma, y, start subscript, 0, end subscript, comma, dots, comma, start color #0d923f, lambda, end color #0d923f, start subscript, 0, end subscript, right parenthesis. Plug each one into $f$f. Or rather, first remove the $\greenE{\lambda}_0$start color #0d923f, lambda, end color #0d923f, start subscript, 0, end subscript component, then plug it into $f$f, since $f$f does not have $\greenE{\lambda}$start color #0d923f, lambda, end color #0d923f as an input. Whichever one gives the greatest (or smallest) value is the maximum (or minimum) point your are seeking.

Budgetary constraints, revisited

• Problem: Suppose you are running a factory, producing some sort of widget that requires steel as a raw material. Your costs are predominantly human labor, which is $\$20$dollar sign, 20 per hour for your workers, and the steel itself, which runs for $\$170$dollar sign, 170 per ton. Suppose your revenue $R$R is loosely modeled by the equation
  $\begin{aligned} \quad R(h, s) = 200 h^{2/3} s^{1/3} \end{aligned}$
  Where
  • $h$h represents hours of labor
  • $s$s represents tons of steel
  If your budget is $\$20{,}000$dollar sign, 20, comma, 000, what is the maximum possible revenue?

• We start by writing the Lagrangian $\mathcal{L}(h, s, \lambda)$L, left parenthesis, h, comma, s, comma, lambda, right parenthesis based on the function $\blueE{R(h, s)}$start color #0c7f99, R, left parenthesis, h, comma, s, right parenthesis, end color #0c7f99 and the constraint $\redE{20h + 170s = 20{,}000}$start color #bc2612, 20, h, plus, 170, s, equals, 20, comma, 000, end color #bc2612.
  $\begin{aligned} \quad \mathcal{L}(h, s, \lambda) = \blueE{200h^{2/3}s^{1/3}}-\lambda(\redE{20h+170s-20{,}000}) \end{aligned}$
• Then we find the critical points of $\mathcal{L}$L, meaning the solutions to
  $\begin{aligned} \quad \nabla \mathcal{L}(h, s, \lambda) = 0 \end{aligned}$
• There might be several solutions $(h, s, \lambda)$left parenthesis, h, comma, s, comma, lambda, right parenthesis to this equation,
  $\begin{aligned} \quad (h_0, &s_0, \lambda_0) \\ (h_1, &s_1, \lambda_1) \\ (h_2, &s_2, \lambda_2) \\ &\vdots \end{aligned}$
  so for each one you plug in the $h$h and $s$s components to the revenue function $R(h, s)$R, left parenthesis, h, comma, s, right parenthesis to see which one actually corresponds with the maximum.

Generally speaking

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The insight

The followthrough