Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming

TL;DR

This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets.

Abstract

This paper presents an algorithm to maximize the volume of an affine slice through a given semialgebraic set. This slice-volume task is formulated as an infinite-dimensional linear program in continuous functions, inspired by prior work in volume computation of semialgebraic sets. A convergent sequence of upper-bounds to the maximal slice volume are computed using the moment-Sum-of-Squares hierarchy of semidefinite programs in increasing size. The computational complexity of this scheme can be reduced by utilizing topological structure (in dimensions 2, 3, 4, 8) and symmetry. This numerical convergence can be accelerated through the introduction of redundant Stokes-based constraints. Demonstrations of slice-volume calculation are performed on example sets.

Figures (5)

• Figure 1: Polynomial over-approximations to $L = [0.1, 0.5] \cup [0.8, 0.9]$
• Figure 2: Slicing geometry and variables from Table \ref{['tab:slice_variables']}.
• Figure 3: Slicing auxiliary function for offset rectangle in Table \ref{['tab:rect_offset_translate']}
• Figure 4: Double-lobe from \ref{['eq:double_lobe']}
• Figure 5: Visualization of double-ellipse-cut region in \ref{['eq:double_ellipse_cut']}

Theorems & Definitions (42)

• Lemma 3.1
• proof
• Definition 3.1
• Theorem 3.2
• proof
• Corollary 1
• proof
• Example 3.1
• Theorem 3.3
• proof