Chapter 4: Partial Differentiation
Section 4.9: Constrained Optimization
Essentials
The Lagrange multiplier method solves the constrained optimization problem consisting of an objective function fx1,…,xn, and k constraints of the form gjx1,…,xn=0,j=1,…,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 ∇f=∑j=1kλj gj in the n+k unknowns x1,…,xn,λ1,…,λk.
The geometry of the Lagrange multiplier method is detailed in the Examples, for which n≤3 and k≤2.
Examples
Example 4.9.1
Find the extreme values of fx,y=2 x2+3 y2 subject to the constraint gx,y≡x2+y2−1=0.
Example 4.9.2
Find the extreme values of fx,y=x y2 subject to the constraint 3 x+4 y=12.
Example 4.9.3
Find the (shortest) distance from the origin to the plane 3x+2 y−4 z=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 1,−3,2 to the plane 5x−7 y+2 z=3.
Example 4.9.6
Find the point on the curve x2y−4=1 that is closest to the point 4,5.
Example 4.9.7
Find the point on the surface xy+3−z=0 that is closest to the point 2,7,5.
Example 4.9.8
Find the point(s) on the the curve x2+2x y+3 y2=5 closest to, and farthest from, the point 1,2.
Obtain a graph of the level curves of the objective function fx,y, and on this graph superimpose the graph of the constraint curve.
Example 4.9.9
Find the distance from the point 1,2,3 to the line of intersection of the planes 5 x+4 y−3 z=11 and 7 x−2 y+6 z=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 fx,y,z=2 x+3 y−5 z subject to the constraints 3 x2+2 y2=4 and 4 x+3 z=1.
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