We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:

Minimize $J(x)=\Vert f(x)-z\Vert^2$

subject to box constraints: $l_i \leq x_i \leq u_i$.

This function $f(\cdot)$ has a beautiful property that

$\forall x_1, x_2 \in R^n, x_1\neq x_2$, $(x_1-x_2)^T(f(x_1)-f(x_2))>0$.

If $n=1$, we say $f(\cdot)$ is monotonically increasing. This nonlinear least square problem can be easily solved to global minimum.

But in our real case, $n=10$. Using standard quasi-Newton method together with gradient projection, we are able to solve this problem to local minimum. And usually, they appear to be global minimum. That's why we are into investigating whether they are indeed global optimum.

Global optimality usually requires the objective function $J(x)$ to be convex, or pseudo-convex, even still valid when it's quasi-convex. Since the function $f(\cdot)$ is $10\times 10$ and complicated, we did numerical verifications based on random sampling and discover: $J(x)$ is neither convex nor pseudo-convex, but quasi-convex.

So the questions are: 1) Can we prove from the previously stated property of $f(\cdot)$ that $J(x)$ is quasi-convex? 2) If not, can we solve this problem to global optimality via non-global optimization techniques?

Thanks!!!