Embedded Optimization Methods for Industrial Automatic Control**Support by the EU via ERC-HIGHWIND (259 166), ITN-TEMPO (607 957), and ITN-AWESCO (642 682) and by the DFG within Reseach Unit FOR 2401 is gratefully acknowledged.

Starting in the late 1970s, optimization-based control has built up an impressive track record of successful industrial applications, in particular in the petrochemical and process industries. More recently, optimization methods for automatic control are more and more deployed on so-called embedded hardware to cater for application-specific needs such as guaranteed communication latency, low energy consumption or cost effectiveness. This development greatly broadens the scope of applications to which optimization methods can be applied to sectors such as robotics, automotive, aerospace or power electronics. However, it also poses additional challenges regarding both the algorithmic concepts and their actual implementations for a given computing hardware. This survey paper discusses key challenges for using embedded optimization methods and summarizes their main use cases in current industrial practice. Motivated by this discussion, a number of dedicated embedded optimization algorithms and their actual implementations are reviewed. The presentation is organized according to the mathematical structure of the embedded optimization problem, ranging from convex quadratic programming over more general convex and nonconvex problems to formulations comprising discrete optimization variables.