# Minimizing a sum of thresholded quadratics

Let $W_1, \ldots, W_k$ be positive semi-definite matrices, $b_1, \ldots, b_k$ be vectors, and $a_1, \ldots a_k$, $c_1, \ldots, c_k$ be scalars. How difficult is it to find an approximate minimum of $$f(x) = \sum_{i=1}^k \min(x^T W_i x + b_i^T x + a_i, c_i)$$

The difficulty is that the individual functions $\min(x^T W_i x + b_i^T x + a_i, c_i)$ are not necessarily convex. Moreover, although the individual functions $\min(x^T W_i x + b_i^T x + a_i, c_i)$ are quasiconvex, a numerical experiment suggests that their sum may not be quasiconvex.