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.

在多目标优化中，考虑到平衡多个目标之间的权衡，解决方案在最优权衡方面被称为帕累托最优；我们研究了解决帕累托集约束优化问题的本地方法，这是一个具有挑战性的问题，因为约束集不仅是隐式定义的，而且通常在目标函数是非凸非光滑的情况下。