In multi-objective optimization, a single decision vector must balance the trade-offs between many objectives. Solutions achieving an optimal trade-off are said to be Pareto optimal: these are decision vectors for which improving any one objective must come at a cost to another. But as the set of Pareto optimal vectors can be very large, we further consider a more practically significant Pareto-constrained optimization problem, where the goal is to optimize a preference function constrained to the Pareto set. We investigate local methods for solving this constrained optimization problem, which poses significant challenges because the constraint set is (i) implicitly defined, and (ii) generally non-convex and non-smooth, even when the objectives are. We define notions of optimality and stationarity, and provide an algorithm with a last-iterate convergence rate of $O(K^{-1/2})$ to stationarity when the objectives are strongly convex and Lipschitz smooth.

翻译：在多目标优化中，单个决策向量必须在多个目标之间权衡取舍。实现最优权衡的解被称为帕累托最优：这些决策向量在改进任何一个目标时，必须以牺牲另一个目标为代价。但由于帕累托最优向量集可能非常庞大，我们进一步考虑一个更具实际意义的帕累托约束优化问题，其目标是在帕累托集约束下优化偏好函数。我们研究解决这一约束优化问题的局部方法，该问题面临显著挑战，因为约束集（i）是隐式定义的，并且（ii）即使目标函数是凸且光滑的，约束集通常也是非凸且非光滑的。我们定义了最优性和平稳性的概念，并提供了一种算法，当目标函数是强凸且Lipschitz光滑时，该算法具有$O(K^{-1/2})$的末次迭代收敛速率达到平稳性。