All directions ascent directions does not imply local minimiser
- Peter J.C. Dickinson
The green 3d curve plots the function f(x_1,x_2)=(x_1^2-2x_2)(x_1^2-x_2). The orange point is the origin. The violet curve gives a curve from the origin in which the function is strictly decreasing, and thus the origin is not a local minimiser. This is parameterised as (2t,3t^2) for t>=0. The blue zone in the middle graph plots when the function is strictly positive, and thus strictly greater than f(0,0). The magenta line segment gives a segment from the origin in the direction of h where the function is strictly positive, and thus strictly greater than f(0,0). The ends are excluded in this line segment. We thus see that every direction from the origin is a strict ascent direction. The red plot in the right hand graph plots how the function changes along the parameterised curve.