# Minimizing sum of functions, while keeping their values non-negative

Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that

$$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} {\mathrm{argmin}} \sum_i f\left(\mathbf{x_i}; \mathbf{\theta} \right)$$ $$\text{s.t.}\quad f\left(\mathbf{x_i}; \mathbf{\theta} \right) \geq 0 \quad \forall i$$

Here $f$ is a scalar valued non-linear function, and all its derivatives can be found analytically. It is also not convex, but there is a way of estimating a good initial guess.

I know very little about non-linear optimization and what exists out there in terms of algorithms for these problems. This, however, looks to me like a problem someone must have tried to solve before.

Do you know of an optimization method that could tackle this?

• In this generality the problem is very hard (ETR hard?) even for a single function. – Igor Rivin Apr 17 '15 at 2:59