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PolyAR: A Highly Parallelizable Solver For Polynomial Inequality Constraints Using Convex Abstraction Refinement

机译：Polyar：使用凸抽象细化的多项式不等式限制的高度平行化的求解器

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摘要

Numerical tools for constraints solving are a cornerstone to control verification problems. This is evident by the plethora of research that uses tools like linear and convex programming for the design of control systems. Nevertheless, the capability of linear and convex programming is limited and is not adequate to reason about general nonlinear polynomials constraints that arise naturally in the design of nonlinear systems. This limitation calls for new solvers that are capable of utilizing the power of linear and convex programming to reason about general multivariate polynomials. In this paper, we propose PolyAR, a highly parallelizable solver for polynomial inequality constraints. PolyAR provides several key contributions. First, it uses convex relaxations of the problem to accelerate the process of finding a solution to the set of the non-convex multivariate polynomials. Second, it utilizes an iterative convex abstraction refinement process which aims to prune the search space and identify regions for which the convex relaxation fails to solve the problem. Third, it allows for a highly parallelizable usage of off-the-shelf solvers to analyze the regions in which the convex relaxation failed to provide solutions. We compared the scalability of PolyAR against Z3 8.9 and Yices 2.6 on control designing problems. Finally, we demonstrate the performance of PolyAR on designing switching signals for continuous-time linear switching systems.
机译：解决求解的数值工具是控制验证问题的基石。这是通过普遍的研究如此明显，该研究使用了类似于线性和凸编程的工具，用于设计控制系统。然而，线性和凸编程的能力是有限的，并且不充分地推理在非线性系统设计中自然出现的一般非线性多项式约束。这种限制呼吁能够利用线性和凸编程的力量来推崇通用多变量多项式的新求解器。在本文中，我们提出了多项式不平等约束的PolyAR，一种高度平行化的求解器。 Polyar提供了几个关键贡献。首先，它使用问题的凸起放松来加速找到对非凸多变量多项式的组件的解决方案的过程。其次，它利用迭代凸抽象的细化过程，该过程旨在修剪搜索空间，并识别凸松弛无法解决问题的区域。第三，它允许高度平行化的离心求解器来分析凸松弛未能提供解决方案的区域。我们将多元与Z3 8.9和YICE 2.6的可扩展性进行了比较控制设计问题。最后，我们展示了PolyAR在用于连续时间线性开关系统的设计开关信号的性能。

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