  In real-time systems optimization, designers often face a challenging problem
posed by the non-convex and non-continuous schedulability conditions, which may
even lack an analytical form to understand their properties. To tackle this
challenging problem, we treat the schedulability analysis as a black box that
only returns true/false results. We propose a general and scalable framework to
optimize real-time systems, named Numerical Optimizer with Real-Time Highlight
(NORTH). NORTH is built upon the gradient-based active-set methods from the
numerical optimization literature but with new methods to manage active
constraints for the non-differentiable schedulability constraints. In addition,
we also generalize NORTH to NORTH+, to collaboratively optimize certain types
of discrete variables (\eg priority assignments, categorical variables) with
continuous variables based on numerical optimization algorithms. We demonstrate
the algorithm performance with two example applications: energy minimization
based on dynamic voltage and frequency scaling (DVFS), and optimization of
control system performance. In these experiments, NORTH achieved $10^2$ to
$10^5$ times speed improvements over state-of-the-art methods while maintaining
similar or better solution quality. NORTH+ outperforms NORTH by 30\% with
similar algorithm scalability. Both NORTH and NORTH+ support black-box
schedulability analysis, ensuring broad applicability.
