Problem 3.2

Examples where $0<c_2<c_1<1$ and there are no step lengths that satisfy the Wolfe conditions can be found, e.g. the line search results in minimizing $\phi(\alpha)=(\alpha -2)^2$.

However, there are examples where all choices of $0<c_2<c_1<1$ still result in step lengths that satisfy the Wolfe conditions, e.g. consider a a line search on $\phi(\alpha)$ where the tangent line at 0, $\ell(\alpha)=\phi(0)+ \alpha \phi^\prime(0)$, cuts through the graph leaving a strict local minimum below the line. Therefore, any value of $c_1$ close to 1 still cuts of this local minimum. And, any value of $c_2$ will not eliminate the points with a positive slope that follows the local minimum. (See Figure 1.)

Figure 1: Plot for Problem 3.2: graph with tangent line

The graph was generated by the following matlab program.

y=[1  .9  .8 .7  .3  -.1 -.5 -.1 .3 .7];
x=[0:9];
xx = 0:.25:9;
yy = spline(x,y,xx);
plot(x,y,'o',xx,yy)
slope=-.1;
y=slope*x+1;
hold on
plot(x,y,'x:')