When Interval Analysis Helps Inter-block Backtracking

Abstract

Inter-block backtracking (IBB) computes all the solutions of sparse systems of non-linear equations over the reals. This algorithm, introduced in 1998 by Bliek et al., handles a system of equations previously decomposed into a set of (small) k ×k sub-systems, called blocks. Partial solutions are computed in the different blocks and combined together to obtain the set of global solutions. When solutions inside blocks are computed with interval-based techniques, IBB can be viewed as a new interval-based algorithm for solving decomposed equation systems. Previous implementations used Ilog Solver and its IlcInterval library. The fact that this interval-based solver was more or less a black box implied several strong limitations. The new results described in this paper come from the integration of IBB with the interval-based library developed by the second author. This new library allows IBB to become reliable (no solution is lost) while still gaining several orders of magnitude w.r.t. solving the whole system. We compare several variants of IBB on a sample of benchmarks, which allows us to better understand the behavior of IBB. The main conclusion is that the use of an interval Newton operator inside blocks has the most positive impact on the robustness and performance of IBB. This modifies the influence of other features, such as intelligent backtracking and filtering strategies.