Anytime algorithms allow a practitioner to trade-off runtime for solution quality. This is of particular interest in multi-objective combinatorial optimization since it can be infeasible to identify all efficient solutions in a reasonable amount of time. We present a theoretical model that, under some mild assumptions, characterizes the “optimal” trade-off between runtime and solution quality, measured in terms of relative hypervolume, of anytime algorithms for bi-objective optimization. In particular, we assume that efficient solutions are collected sequentially such that the collected solution at each iteration maximizes the hypervolume indicator, and that the non-dominated set can be well approximated by a quadrant of a superellipse. We validate our model against an “optimal” model that has complete knowledge of the non-dominated set. The empirical results suggest that our theoretical model approximates the behavior of this optimal model quite well. We also analyze the anytime behavior of an \(\varepsilon \)-constraint algorithm, and show that our model can be used to guide the algorithm and improve its anytime behavior.

随时都有算法让从业人员可以权衡运行时间以提高解决方案质量。这在多目标组合优化中特别有意义，因为在合理的时间内确定所有有效的解决方案可能是不可行的。我们提出了一个理论模型，该模型在一些温和的假设下表征了双目标优化随时随地算法的运行时间和解决方案质量之间的“最佳”折衷，以相对超量来衡量。特别地，我们假设有效的解决方案是按顺序收集的，这样每次迭代收集的解决方案都会使超量指标最大化，并且超椭圆形象限可以很好地近似非支配集合。我们根据对非支配集合有完整知识的“最优”模型验证模型。实证结果表明，我们的理论模型可以很好地近似此最优模型的行为。我们还分析了\（\ varepsilon \）-约束算法，并表明我们的模型可用于指导算法并改善其随时行为。