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Chance constrained optimization problems allow to model problems whereconstraints involving stochastic components should only be violated with asmall probability. Evolutionary algorithms have been applied to this scenarioand shown to achieve high quality results. With this paper, we contribute tothe theoretical understanding of evolutionary algorithms for chance constrainedoptimization. We study the scenario of stochastic components that areindependent and Normally distributed. Considering the simple single-objective(1+1)~EA, we show that imposing an additional uniform constraint already leadsto local optima for very restricted scenarios and an exponential optimizationtime. We therefore introduce a multi-objective formulation of the problem whichtrades off the expected cost and its variance. We show that multi-objectiveevolutionary algorithms are highly effective when using this formulation andobtain a set of solutions that contains an optimal solution for any possibleconfidence level imposed on the constraint. Furthermore, we prove that thisapproach can also be used to compute a set of optimal solutions for the chanceconstrained minimum spanning tree problem. In order to deal with potentiallyexponentially many trade-offs in the multi-objective formulation, we proposeand analyze improved convex multi-objective approaches. Experimentalinvestigations on instances of the NP-hard stochastic minimum weight dominatingset problem confirm the benefit of the multi-objective and the improved convexmulti-objective approach in practice.

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