When solving large-scale multiobjective optimization problems, solvers can get stuck because of memory and/or time limitations. In such cases, one is left with no information on the distance to the best feasible solution, found before the optimization process has stopped, to the true Pareto optimal solution. In this work, we show how to provide such information. To this aim we make use of the concept of lower shells and upper shells, developed in our earlier works. No specific assumptions about the problems to be solved are made. We illustrate the proposed approach on biobjective multidimensional knapsack problems derived from single-objective multidimensional knapsack problems in the Beasley OR Library. We address cases when a top-class commercial mixed-integer linear solver fails to provide Pareto optimal solutions attempted to be derived by scalarization.

解决大规模多目标优化问题时，由于内存和/或时间限制，求解器可能会卡住。在这种情况下，在优化过程停止之前，到真正的帕累托最优解为止的最佳可行解的距离就没有任何信息。在这项工作中，我们展示了如何提供此类信息。为了达到这个目的，我们利用了在早期工作中发展起来的下壳和上壳的概念。没有关于要解决的问题的特定假设。我们在Beasley OR Library中说明了从单目标多维背包问题导出的双目标多维背包问题的建议方法。