A provably convergent heuristic for stochastic bicriteria integer programming

We propose a general-purpose algorithm APS (Adaptive Pareto-Sampling) for determining the set of Pareto-optimal solutions of bicriteria combinatorial optimization (CO) problems under uncertainty, where the objective functions are expectations of random variables depending on a decision from a finite feasible set. APS is iterative and population-based and combines random sampling with the solution of corresponding deterministic bicriteria CO problem instances. Special attention is given to the case where the corresponding deterministic bicriteria CO problem can be formulated as a bicriteria integer linear program (ILP). In this case, well-known solution techniques such as the algorithm by Chalmet et al. can be applied for solving the deterministic subproblem. If the execution of APS is terminated after a given number of iterations, only an approximate solution is obtained in general, such that APS must be considered a metaheuristic. Nevertheless, a strict mathematical result is shown that ensures, under rather mild conditions, convergence of the current solution set to the set of Pareto-optimal solutions. A modification replacing or supporting the bicriteria ILP solver by some metaheuristic for multicriteria CO problems is discussed. As an illustration, we outline the application of the method to stochastic bicriteria knapsack problems by specializing the general framework to this particular case and by providing computational examples.