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?


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