I.3.8.1: Convergence to a Saddle
Convergence, even it occurs, does not need to be towards a minimum. Consider f(x,y)=16y3−12y2x+12yx2−x2+2x. Perspective and contour plots of this function are in figures 1 and 2. Code for the plots is in saddle.R
The derivatives are D1f(x,y)=(y−2)(x−12(y+2)),D2f(x,y)=12(x−y)2, and D2f(x,y)=[y−2x−yx−yy−x]. Start with y(0)>2. Minimizing over x for given y(k) gives x(k)=12(y(k)+2), and minimizing over y for given x(k) gives y(k+1)=x(k). It follows that x(k+1)−2=12(x(k)−2),y(k+1)−2=12(y(k)−2). Thus both x(k) and y(k) decrease to two with linear convergence rate 12. The function f has a saddle point at (2,2), and D2f(2,2)=[0000].