Linear Algebra
12.2210 min read

Non-Quadratic Example

Let's apply the complete optimization workflow to a non-quadratic function: f(x,y)=x4+y4−x2−y2+xyf(x,y) = x^4 + y^4 - x^2 - y^2 + xy on the square D=[−2,2]2D = [-2,2]^2.

The interior critical points are found by solving ∇f=(4x3−2x+y,4y3−2y+x)=0\nabla f = (4x^3 - 2x + y, 4y^3 - 2y + x) = \mathbf{0}. This system is nonlinear and may have multiple solutions, found numerically or by exploiting symmetry.

On the boundary (e.g., x=2x = 2, −2≤y≤2-2 \leq y \leq 2), ff reduces to a one-variable function g(y)=16+y4−4−y2+2yg(y) = 16 + y^4 - 4 - y^2 + 2y whose extrema are found by g′(y)=0g'(y) = 0. The global extremum is the best among all interior and boundary candidates.

Formal View

Example 12.3 — Non-Quadratic Optimization on a Compact Domain
For f(x,y)=x4+y4−x2−y2f(x,y) = x^4 + y^4 - x^2 - y^2 on [−1,1]2[-1,1]^2 (simplified): Interior: ∇f=(4x3−2x,4y3−2y)=0\nabla f = (4x^3-2x, 4y^3-2y) = \mathbf{0}. Each equation factors: x(2x2−1)=0x(2x^2-1)=0, giving x∈{0,±1/2}x \in \{0, \pm 1/\sqrt{2}\} similarly for yy. Nine interior critical points. Boundary: check each side. On x=1x=1: minimize g(y)=1+y4−1−y2g(y)=1+y^4-1-y^2 for y∈[−1,1]y\in[-1,1]. Evaluate ff at all candidates to find global extrema.

Interactive Visualization

Local Linear Approximation

Why This Matters

Non-quadratic examples show that the systematic method extends beyond the quadratic case.

  • Nonlinear regression models have non-quadratic objective functions
  • Physical potentials (Lennard-Jones, Morse) are non-quadratic and have multiple local minima
  • Neural network loss functions are highly non-quadratic in their parameters

Quiz

Question 1

For f(x,y)=x4+y4−4x2−4y2f(x,y) = x^4 + y^4 - 4x^2 - 4y^2, how many interior critical points does the gradient equation ∇f=0\nabla f = \mathbf{0} have?

Question 2

For a smooth non-quadratic function, the second derivative test (checking Hessian definiteness) can still classify critical points as local minima, maxima, or saddle points.

Common Mistakes

  • Assuming non-quadratic functions have a unique critical point like positive-definite quadratics.
  • Forgetting that the Hessian changes with location — compute it at each critical point separately.
  • Neglecting the possibility that the global minimum is on the boundary, not at an interior critical point.