We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the gradients and to add the inertial force (we will empirically show that it is more efficient than using a single extrapolation point), (2) they allow to randomly select the block of variables to update, and (3) they do not require a restarting step. We prove the subsequential convergence of the generated sequence under mild assumptions, prove the global convergence under some additional assumptions, and provide convergence rates. We deploy the proposed methods to solve non-negative matrix factorization (NMF) and show that they compete favorably with the state-of-the-art NMF algorithms. Additional experiments on non-negative approximate canonical polyadic decomposition, also known as nonnegative tensor factorization, are also provided.

SPACE is a model-driven engineering technique for reactive distributed systems. It enables to develop system models from reusable building blocks, formal analysis by model checking as well as automated transformation to executable code. In this paper, we describe an extension of the SPACE formalism which allows to model and verify also real-time behavior. In particular, one specifies real-time constraints in the interface descriptions of the building blocks, so-called Real-Time External State-Machines (RTESMs). The RTESMs are translated to guards, clocks and invariants of Timed Automata which can be analyzed by means of the model checker UPPAAL. The approach is explained by a component protecting an electrical motor controller system against overspeed. In particular, we prove that by keeping certain maximum response times, this system guarantees that the speed of the motor stays within certain limits.