# Minimize interesting objective function with knowledge of gradient nonlinearity?

I plan on using a Quasi-Newton method (L-BFGS) to minimize a non-linear objective function.

$$f: \mathbb{R}^n \rightarrow \mathbb{R}$$

The gradient is kind of interesting: as the values of the first set of parameters approach infinity, the gradient of the objective function with respect to the remaining parameters will approach zero. In other words,

$$\lim_{x_i\to\infty} \frac{\partial f}{\partial x_j} = 0$$ $$i < j$$

The important concept: this relationship has no bearing on whether the objective function is being minimized or not. I fear the L-BFGS algorithm will follow a path where the gradient is minimized by this relationship instead of being minimized appropriately.

Can I use this knowledge to aid in my optimization strategy?