3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.