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Chapter 4: Partial Differentiation
Section 4.9: Constrained Optimization
Essentials
The Lagrange multiplier method solves the constrained optimization problem consisting of an objective function $f\left({x}_{1}\,\dots \,{x}_{n}\right)$, and $k$ constraints of the form ${g}_{j}\left({x}_{1}\,\dots \,{x}_{n}\right)\=0\,j\=1\,\dots \,k<n$.
An extreme value of $f$ occurs where a level set of $f$ is tangent to the set of points common to the constraints.
Such points of tangency occur where the gradient of $f$ is a linear combination of the gradients of the constraint functions. Thus, the algorithm requires the solution of the $n\+k$ equations $\nabla f\=\sum _{j\=1}^{k}{\mathrm{\λ}}_{j}{g}_{j}$ in the $n\+k$ unknowns ${x}_{1}\,\dots \,{x}_{n}\,{\mathrm{\λ}}_{1}\,\dots \,{\mathrm{\λ}}_{k}$.
The geometry of the Lagrange multiplier method is detailed in the Examples, for which $n\le 3$ and $k\le 2$.
Examples
Example 4.9.1
Find the extreme values of $f\left(x\,y\right)\=2{x}^{2}plus;3{y}^{2}$ subject to the constraint $g\left(x\,y\right)\equiv {x}^{2}\+{y}^{2}-1\=0$.
Example 4.9.2
Find the extreme values of $f\left(x\,y\right)\=x{y}^{2}$ subject to the constraint $3xplus;4yequals;12$.
Example 4.9.3
Find the (shortest) distance from the origin to the plane $3x\+2y-4zequals;7$.
Example 4.9.4
A pentagon is formed from a rectangle surmounted by an isosceles triangle.
What dimensions give the pentagon least perimeter if the area is fixed at the value 50?
Example 4.9.5
Find the minimum distance from the point $\left(1\,-3\,2\right)$ to the plane $5x-7yplus;2zequals;3$.
Example 4.9.6
Find the point on the curve ${x}^{2}\left(y-4\right)\=1$ that is closest to the point $\left(4\,5\right)$.
Example 4.9.7
Find the point on the surface $xy\+3-z\=0$ that is closest to the point $\left(2\,7\,5\right)$.
Example 4.9.8
Find the point(s) on the the curve ${x}^{2}\+2xyplus;3{y}^{2}equals;5$ closest to, and farthest from, the point $\left(1\,2\right)$.
Obtain a graph of the level curves of the objective function $f\left(x\,y\right)$, and on this graph superimpose the graph of the constraint curve.
Example 4.9.9
Find the distance from the point $\left(1\,2\,3\right)$ to the line of intersection of the planes $5xplus;4y-3zequals;11$ and $7x-2yplus;6zequals;9$.
Use the Lagrange multiplier technique adapted for two constraints.
Solve for the line of intersection and minimize the distance from the given point to this line.
Example 4.9.10
Find the extrema of $f\left(x\,y\,z\right)\=2xplus;3y-5z$ subject to the constraints $3{x}^{2}plus;2{y}^{2}equals;4$ and $4xplus;3zequals;1$.
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