Optimal Configurations in Coverage Control with Polynomial Costs

TL;DR

This paper applies numerical algebraic geometry techniques to identify global optimal configurations in a coverage control problem with polynomial cost functions, surpassing traditional local convergence methods.

Contribution

It introduces a novel application of polynomial homotopy continuation to find all solutions, including global minima, in coverage control problems with polynomial costs.

Findings

Successfully characterized all solutions using algebraic geometry tools.

Demonstrated the superiority of the global solution approach over Lloyd descent.

Provided insights into the structure of optimal vehicle placements.

Abstract

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the \emph{global minima} for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions.