MathIsimple

Lesson 6-1: Multivariable Optimization (with Constraints)

Optimize multivariable functions and interpret results in applications.

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Learning Objectives

  • Find stationary points by solving ∇f=0\nabla f=0 and classify via Hessian.
  • Use Lagrange multipliers: ∇f=λ∇g\nabla f=\lambda \nabla g.
  • Check boundaries and interpret units/feasibility.

Unconstrained Optimization

Hessian H=[fxxfxyfyxfyy]H=\begin{bmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy}\end{bmatrix}, determinant D=fxxfyy−fxy2D=f_{xx}f_{yy}-f_{xy}^2
  • D>0,fxx>0D>0, f_{xx}>0: local minimum
  • D>0,fxx<0D>0, f_{xx}<0: local maximum
  • D<0D<0: saddle; D=0D=0: inconclusive

Lagrange Multipliers

L(x,y,λ)=f(x,y)−λ(g(x,y)−c)L(x,y,\lambda)=f(x,y)-\lambda(g(x,y)-c)
Lx=Ly=Lλ=0, g(x,y)=cL_x=L_y=L_\lambda=0,\ g(x,y)=c
Maximize A=xy subject to 2(x+y)=P → x=y=P/4 (square maximizes area)

Worked Examples

Example 1: Classify Stationary Points

For f(x,y)=x3−3x+y2f(x,y)=x^3-3x+y^2, find critical points and classify using Hessian.

Example 2: Constrained Optimum

Maximize xyxy subject to x+2y=12x+2y=12.

Practice Problems

Problem 1

Find and classify stationary points of f(x,y)=x3−3x+y2f(x,y)=x^3-3x+y^2.

Problem 2

Minimize x2+y2x^2+y^2 subject to x+y=10x+y=10.

Problem 3

Maximize xy2x y^2 subject to x+y=12x+ y=12. Use Lagrange multipliers and check boundaries.

Problem 4

Constrained: minimize (x−2)2+(y+1)2(x-2)^2+(y+1)^2 subject to x−2y=3x-2y=3. Interpret geometrically.

Problem 5

Classify critical points of f(x,y)=x2−4xy+5y2f(x,y)=x^2-4xy+5y^2 using the Hessian matrix.

Key Takeaways

  • D=fxxfyy−fxy2D=f_{xx}f_{yy}-f_{xy}^2 determines local type when D>0D>0.
  • Lagrange multipliers handle equality constraints effectively.
  • Always check boundaries and interpretability of solutions.
Continue to Lesson 6-2.
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