Joint Optimization of Continuous Variables and Priority Assignments for Real-Time Systems with Black-box Schedulability Constraints

TL;DR

This work tackles real-time system optimization when schedulability constraints are non-differentiable or available only as a binary black-box. It introduces NORTH, a gradient-based active-set framework that avoids differentiating schedulability constraints and uses variable elimination to improve efficiency, achieving large speedups while maintaining solution quality. Building on NORTH, NORTH+ jointly optimizes continuous variables and discrete priority assignments via an iterative, hybrid scheme guided by response-time changes, delivering additional performance improvements. The framework is validated on DVFS energy minimization, DAG-based energy optimization, and control-performance optimization, showing substantial speedups ($10^2$–$10^5$) and competitive or superior results compared with state-of-the-art methods. Overall, NORTH/NORTH+ offer a general, scalable methodology for real-time design under black-box schedulability analysis with broad applicability.

Abstract

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 (e.g., 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.

Figures (4)

• Figure 1: NORTH framework and components overview. NMBO: Beginning with a feasible solution $\textbf{x}$, NORTH first utilizes trust-region methods to find an update direction $\boldsymbol{\Delta}$ for the optimization problem without the schedulability constraints. If $\boldsymbol{\Delta}$ leads $\textbf{x}$ into an infeasible region, then $\boldsymbol{\Delta}$ is kept decreased until $\textbf{x}+\boldsymbol{\Delta}$ is feasible. When $\boldsymbol{\Delta}$ becomes small enough such as $\| \boldsymbol{\Delta}\| < 10^{-5}$, performing further iterations only brings small performance improvements, and therefore we terminate the iterations and move on to the next step. VE: After NMBO terminates, we check if there are close active constraints. If so, the involved variables are transformed into constants in future iterations. The algorithm terminates when there are no more variables to optimize.
• Figure 2: Variable elimination motivating example. Consider the problem in Example \ref{['example_nmbo']} and assume NMBO terminates at $(5.999, 1.499)$. The red arrow shows the update step $\boldsymbol{\Delta}$ from classical unconstrained optimizers such as equation \ref{['lm_update']}. Moving toward the red arrow will make task 1 miss its deadline. However, we can improve $\textbf{c}^{(k)}$ without violating the schedulability constraints by only updating $\textbf{x}_2$ following $\boldsymbol{\Delta}$ while leaving $\textbf{x}_1$ unchanged (moving towards the blue arrow).
• Figure 3: Overall optimization framework NORTH+. In each iteration, the NMBO and VE components are shown in Figure \ref{['fig:main_framework_fig']}, and the other two steps are introduced in this section. The execution order of each step within an iteration is based on the following considerations: (1) variable elimination (VE) should be the last step because it reduces the variable space; (2) Numerical method-based optimization (NMBO) and priority assignments (RTA optimization and priority updates) optimize different types of variables, and so their execution order can be interchanged.
• Figure 4: The performance and run-time speed of the experiments.

Theorems & Definitions (24)

• Definition 3.1: Differentiable point
• Definition 3.2: Descent vector
• Definition 3.3: Descent direction
• Example 1
• Definition 5.1: Active schedulability constraint
• Definition 5.2: Dimension feasibility test
• Example 2
• Definition 5.3: Strict descent step
• Theorem 1
• proof