I want to minimize a twice-continuously differentiable strictly pseudoconvex function (which is not convex) from $\mathbb{R}_{++}$ to $\mathbb{R}_{++}$ (strictly positive reals)
I know that I can use gradient descent and find the only global minimum when the gradient is zero.
Can I also use Newton's method to find this minimum? In other words, is Newton's method guaranteed to converge for pseudoconvex functions?
A related question: can I use Nesterov acceleration technique to get faster convergence of gradient descent for pseudoconvex functions?
It's possible to construct a function function $f:R \rightarrow R$ that is strictly pseudoconvex but which has points where $f''(x)=0$. If you applied Newton's method to such a function starting at a point where $f''(x)=0$, it would fail.
Here's a specific example. We'll construct the function by starting with a second derivative that is non-negative but has some roots. Let
$g(x)=(x-1)^{2}(x+1)^{2}x^{2}$.
Then integrate $g(x)$ to get the first derivative,
$h(x)=x^{7}/7-2x^5/5+x^3/3$
We'll integrate this one more time to get $f(x)$
$f(x)=x^{8}/56-x^{6}/15+x^{4}/12$
If you start Newton's method at $x_{0}=1$, then it will fail immediately due to the fact that $f''(x_{0})=0.$ It's easy to check that this function is strictly pseudoconvex (In fact, it's strictly convex.)
There are of course many ways of making Newton's method more robust- you could for example simply take a gradient step whenever the Hessian matrix is nearly singular. Almost any general purpose nonlinear optimization routine should practically be able to handle this problem, particularly if the dimension is small.