Sliding Window 3-Objective Pareto Optimization for Problems with Chance Constraints
Frank Neumann, Carsten Witt
TL;DR
This work addresses chance-constrained optimization by extending sliding-window Pareto optimization from two to three objectives. It defines $f_{3D}(x)=(\mu(x),v(x),c(x))$ to capture expected weight, variance, and constraint value, and embeds sliding-window selection into a 3-objective GSEMO framework to efficiently navigate feasible regions. The paper proves improved runtime bounds (showing that a population containing optimal solutions can be obtained within $t_{\max}=O(\max_i P_{\max}^{(i)}\,n^2\log n)$ with high probability) while preserving approximation guarantees, and demonstrates practical scalability through extensive experiments on chance-constrained dominating set problems, achieving large-scale success where prior methods faltered. Overall, the results indicate significant speedups and scalability gains for solving large instances of chance-constrained problems using 3D sliding-window Pareto optimization.
Abstract
Constrained single-objective problems have been frequently tackled by evolutionary multi-objective algorithms where the constraint is relaxed into an additional objective. Recently, it has been shown that Pareto optimization approaches using bi-objective models can be significantly sped up using sliding windows (Neumann and Witt, ECAI 2023). In this paper, we extend the sliding window approach to $3$-objective formulations for tackling chance constrained problems. On the theoretical side, we show that our new sliding window approach improves previous runtime bounds obtained in (Neumann and Witt, GECCO 2023) while maintaining the same approximation guarantees. Our experimental investigations for the chance constrained dominating set problem show that our new sliding window approach allows one to solve much larger instances in a much more efficient way than the 3-objective approach presented in (Neumann and Witt, GECCO 2023).