Chapter 4: Partial Differentiation
Section 4.9: Constrained Optimization
Example 4.9.5
Find the minimum distance from the point 1,−3,2 to the plane 5x−7 y+2 z=3.
Solution
Mathematical Solution
The objective function should be
x−12+y+32+z−22
but taking the objective function f as the square of this leads to much simpler algebra.
The constraint is
gx,y,z≡5x−7 y+2 z−3=0
In Figure 4.9.5(a), the sphere of radius 243/26, the level surface f=243/26, is tangent to the constraint plane g=0, drawn in red.
use plots,plottools in module() local p1,p2,p3; p1:=sphere([1,-3,2],sqrt(243/26)); p2:=plot3d(-(5/2)*x+(7/2)*y+3/2,x=-2..2,y=-2..2,color=red); p3:=display(p1,p2,scaling=constrained,labels=[x,y,z],axes=frame,orientation=[15,75,0],tickmarks=[4,5,5],view=-1..5); print(p3); end module: end use:
Figure 4.9.5(a) Level surface of f and g=0
To implement the Lagrange multiplier technique, solve ∇f=λ ∇g, and the constraint g=0, for the four unknowns x,y,z,λ.
The resulting equations are
−5x+7y−2z+3,−5λ+2x−2,7λ+2y+6,−2λ+2z−4
with solution P:x,y,z=−1926,−1526,1713 when λ=−913, so that fP=24326.
The minimum distance from 1,−3,2 to the plane is therefore 243/26=978/26.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign Name
f=x−12+y+32+z−22→assign
g=5 x−7 y+2 z−3→assign
Apply the LagrangeMultipliers command in the Student MultivariateCalculus package, captured in the task template. See Table 4.9.3(a).
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Optimization≻Lagrange Multiplier Method
Method of Lagrange Multipliers
Enter objective function f
Enter constraints gk=0,k=1,…,entered as functions g1,g2,…
Enter coordinate variables, separated by commas:
Table 4.9.3(a) The Lagrange Multiplier Method task template
Implement the Lagrange multiplier method via first principles
F=f−λ g→assign
Write F and press the Enter key.
Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient
Context Panel: Conversions≻To List
Context Panel: Solve≻Solve
F
−5x−7y+2z−3λ+x−12+y+32+z−22
→gradient
→to list
−5x+7y−2z+3,−5λ+2x−2,7λ+2y+6,−2λ+2z−4
→solve
λ=−913,x=−1926,y=−1526,z=1713
Optimization Assistant
The Optimization Assistant is no longer listed in Tools≻Assistants. It is now found in Tools≻Tutors≻Optimization
A numeric solution is available via the , launched from the Context Panel on the sequence f,g=0.
Figure 4.9.5(b) shows the Optimization Assistant finding the minimum of 243/26≐9.35 at
−1926,−1526,1713
Figure 4.9.5(b) Constrained maximum
Solution via the Lines & Planes tools in the Student MultivariateCalculus package
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻L
5 x−7 y+2 z=3→make plane<< Plane 1 >>→assign to a nameL
Write a sequence of the point and L, the name of the plane.
Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
1,−3,2,L = 1,−3,2,<< Plane 1 >>→distance92678
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define f, the objective function.
f≔x−12+y+32+z−22:
Define g, the constraint function.
g≔5 x−7 y+2 z−3:
Implement the Lagrange multiplier method
Invoke the LagrangeMultipliers command from the Student MultivariateCalculus package.
LagrangeMultipliersf,g,x,y,z
Add the "detailed" option to the LagrangeMultipliers command.
LagrangeMultipliersf,g,x,y,z,output=detailed
x=−1926,y=−1526,z=1713,λ1=−913,x−12+y+32+z−22=24326
Implement the Lagrange multiplier method from first principles
Define F.
F≔f− λ g:
Use the Gradient command to obtain ∇f−λ ∇g.
Use the Equate command to equate each component of ∇f−λ ∇g to zero.
Use the solve command to obtain the solutions of the equations in ∇f−λ ∇g=0.
solveEquateGradientF,x,y,z,λ,0,0,0,0,x,y,z,λ
x=−1926,y=−1526,z=1713,λ=−913
Define the plane with the Plane command and obtain the distance with the Distance command.
Distance1,−3,2,Planeg = 92678
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