Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixed-integer problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

可以有效地计算双准则连续凸问题的帕累托边界，并建立了最佳理论性能界限。然而，在双标准混合整数问题的情况下，帕累托边界的近似变得更加困难。在本文中，我们提出了一种新的算法来逼近具有凸约束的双准则混合整数程序的 Pareto 边界。这样的 Pareto 边界由具有离散变量共享分配的解决方案的补丁组成。通过自适应地创建这样的拼凑，我们的算法能够创建快速收敛到真实帕累托边界的近似值。作为质量度量，我们使用近似和真实 Pareto 边界之间的超体积差异。至少需要一定数量的补丁才能获得具有给定质量的近似值。这种补丁的复杂性给出了所需计算数量的下限。我们表明，我们的算法执行了许多优化步骤，这些步骤与此下限具有相似的顺序。我们提供了该算法的基于 MIP 的高效实现。数值结果说明了我们算法的效率，表明我们的算法具有强大的理论性能保证，同时在实践中与其他最先进的方法具有竞争力。我们提供了该算法的基于 MIP 的高效实现。数值结果说明了我们算法的效率，表明我们的算法具有强大的理论性能保证，同时在实践中与其他最先进的方法具有竞争力。我们提供了该算法的基于 MIP 的高效实现。数值结果说明了我们算法的效率，表明我们的算法具有强大的理论性能保证，同时在实践中与其他最先进的方法具有竞争力。