Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables

Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have recently 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. By generalizing results for the class of linear functions to the sum of transformed linear functions, we show that the (1+1)~EA can optimize the chance constrained setting without additional constraints in time O(n log n). However, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time for the (1+1)~EA. We therefore propose 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 show that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem.