Chapter 4: Partial Differentiation
Section 4.9: Constrained Optimization
Example 4.9.3
Find the (shortest) distance from the origin to the plane 3x+2 y−4 z=7.
Solution
Mathematical Solution
The objective function should be x2+y2+z2, but taking fx,y,z=x2+y2+z2 as the objective function leads to much simpler algebra.
The constraint is
gx,y,z≡3x+2 y−4 z− 7=0
In Figure 4.9.3(a), the sphere of radius 7/29, the level surface f=49/29, is tangent to the constraint plane g=0, drawn in red.
To implement the Lagrange multiplier technique, solve ∇f=λ ∇g, and the constraint g=0, for the four unknowns x,y,z,λ.
use plots in module() local p1,p2,p3; p1:=plot3d([7/sqrt(29),t,s],t=0..2*Pi,s=0..Pi,coords=spherical); p2:=plot3d(-7/4+(3/4)*x+(1/2)*y,x=-1..2,y=-1.5..2,color=red); p3:=display(p1,p2,scaling=constrained,labels=[x,y,z],axes=frame,orientation=[-70,75,0],tickmarks=[4,5,5],view=-2..1.5); print(p3); end module: end use:
Figure 4.9.3(a) Level surface of f and g=0
The resulting equations are
−3x−2y+4z+7,−3λ+2x,−2λ+2y,4λ+2z
with solution P:x,y,z=2129,1429,−2829 when λ=1429, so that fP=4929.
The minimum distance from the origin to the plane is therefore 7/29.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign Name
f=x2+y2+z2→assign
g=3 x+2 y−4 z−7→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
−3x+2y−4z−7λ+x2+y2+z2
→gradient
→to list
−3x−2y+4z+7,−3λ+2x,−2λ+2y,4λ+2z
→solve
λ=1429,x=2129,y=1429,z=−2829
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 4.9.3(b) shows the Optimization Assistant finding the minimum of 49/29≐1.69 at
2129,1429,−2829
Figure 4.9.3(b) Constrained maximum
Solution via the Lines & Planes tools in the Student MultivariateCalculus package
Write g, the name of the constraint function. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Assign to a Name≻L
g = 3x+2y−4z−7→make plane<< Plane 1 >>→assign to a nameL
Write a sequence of the origin (as a point) and L, the name of the plane. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Distance
0,0,0,L = 0,0,0,<< Plane 1 >>→distance72929
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define f, the objective function.
f≔x2+y2+z2:
Define g, the constraint function.
g≔3 x+2 y−4 z−7:
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=2129,y=1429,z=−2829,λ1=1429,x2+y2+z2=4929
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=2129,y=1429,z=−2829,λ=1429
Define the plane with the Plane command and obtain the distance with the Distance command.
Distance0,0,0,Planeg = 72929
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