Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables
Frank Neumann, Carsten Witt
TL;DR
This paper addresses chance-constrained optimization with normally distributed stochastic components, introducing both a single-objective analysis and a multi-objective Pareto approach. It shows that the classic (1+1) EA can have exponential runtime under simple uniform constraints, motivating a bi-objective formulation that minimizes the mean $\mu$ and the variance $v$, with GSEMO guaranteeing a Pareto set that contains optimal solutions for all $\alpha \in [1/2,1)$. The authors prove that the extreme points of the Pareto front correspond to solutions optimal for some $\lambda$-weighted objective, enabling polynomial-time computation of all relevant trade-offs; they extend these results to chance-constrained minimum spanning trees and present convex hull-based MOEAs that bound population size while preserving guarantees. Empirical results on NP-hard stochastic dominating-set instances validate the theoretical advantages of multi-objective and convex MOEAs, particularly in settings with strong trade-offs or negative correlation between cost and variance. Overall, the work provides both theoretical foundations and practical algorithms for efficiently solving chance-constrained combinatorial problems under normal uncertainty and demonstrates broad applicability to classic problems such as minimum spanning trees and dominating sets.
Abstract
Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and normally distributed. Considering the simple single-objective (1+1) EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. In order to deal with potentially exponentially many trade-offs in the multi-objective formulation, we propose and analyze improved convex multi-objective approaches. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem confirm the benefit of the multi-objective and the improved convex multi-objective approach in practice.