[论文解读] Simple Approximations of Semialgebraic Sets and their Applications to Control

Fabrizio Dabbene, Didier Henrion+1 more|arXiv (Cornell University)|Sep 14, 2015

Advanced Optimization Algorithms Research参考文献 42被引用 33

一句话总结

本文提出多项式超水平集（PSS）作为控制系统中复杂半代数集的简单且计算可行的近似方法。通过在半代数集上对正多项式进行 $L^1$-范数最小化，并施加正性约束，该方法生成的外近似在体积、几乎处处以及几乎一致的意义下收敛于原集合，从而实现高效的采样与集合重构。

ABSTRACT

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples.

研究动机与目标

• 解决控制系统中常见但形状复杂、非凸的半代数集近似问题，其形状给分析与设计带来困难。
• 开发一种计算上可行的方法，生成此类集合的外近似，兼具简洁性与准确性。
• 通过多项式密度近似，实现从有限采样中重构集合，并在半代数集上实现均匀采样。
• 通过允许非凸、高阶多项式超水平集，推广经典椭球近似方法。
• 为近似层次结构提供理论收敛保证，涵盖体积、$L^1$-范数以及几乎处处收敛。

提出的方法

• 将近似问题表述为在半代数集 $\mathcal{K}$ 上最小化正多项式 $p(x)$ 的 $L^1$-范数，同时满足正性约束。
• 构建多项式超水平集（PSS）为 $\mathcal{U}(p) = \{x \in \mathcal{B} : p(x) \geq 1\}$，作为 $\mathcal{K}$ 的外近似。
• 利用线性矩阵不等式（LMI）问题的层次结构，计算随次数增加的近似多项式 $p$。
• 借鉴迹最小化启发式方法，类似最小体积椭球方法，但适配于多项式超水平集。
• 从多项式 $p$ 导出边缘与条件概率密度，以通过算法1实现拒绝采样，通过算法2实现逆累积分布函数（CDF）采样。
• 确保所有边缘与累积密度的积分均可解析计算，避免数值积分。

实验结果

研究问题

• RQ1 多项式超水平集能否作为椭球近似的推广，以低复杂度捕捉非凸与非连通的半代数集？
• RQ2 为何对正多项式进行 $L^1$-范数最小化能产生在体积和几乎处处意义下收敛于原始半代数集的近似？
• RQ3 多项式超水平集在多大程度上可实现从有限采样中对集合的精确重构？
• RQ4 所提出的多项式密度近似能否在无需数值积分的情况下生成复杂半代数集上的均匀采样？
• RQ5 随着多项式次数增加，PSS近似层次结构的理论收敛行为如何？

主要发现

• 多项式超水平集的层次结构在体积上收敛于原始半代数集 $\mathcal{K}$，且在 $L^1$-范数、几乎处处以及几乎一致的意义下收敛。
• 当次数 $d=20$ 时，最优 PSS $p_{d,d}^*$ 对非凸集合 $\mathcal{K}$ 提供了紧密的外近似，数值示例已验证此结果。
• 该方法通过拒绝采样（算法1）和逆 CDF 采样（算法2）实现了在半代数集上的均匀采样，两者均避免了数值积分。
• 从多项式 $p$ 导出的边缘与条件密度可精确解析计算，支持无数值积分的精确采样。
• 该方法推广了经典椭球近似：二次 PSS 可恢复著名的最小体积椭球结果。
• 该方法同时支持外近似（用于体积最小化）与内近似（用于最大 $L^1$-体积内接 PSS），实现灵活的集合表征。