I.3.8.1: Convergence to a Saddle

Convergence, even it occurs, does not need to be towards a minimum. Consider f(x,y)=16y312y2x+12yx2x2+2x. Perspective and contour plots of this function are in figures 1 and 2. Code for the plots is in saddle.R


plot of chunk saddle_contour

Figure 1: Contour Plot Bivariate Cubic



plot of chunk saddle_persp

Figure 2: Perspective Plot Bivariate Cubic


The derivatives are D1f(x,y)=(y2)(x12(y+2)),D2f(x,y)=12(xy)2, and D2f(x,y)=[y2xyxyyx]. 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].