标题： 可扩展的极小-极大优化通过原始-对偶精确帕累托优化 显示英文标题

标题： Scalable Min-Max Optimization via Primal-Dual Exact Pareto Optimization

摘要： 在多目标优化中，最小化最差的目标函数值可能比最小化平均目标函数值更可取，因为这能更好地确保各目标之间的公平性。由于所得的最小-最大优化问题具有非光滑特性，经典的基于次梯度的方法通常表现出收敛速度较慢的问题。受多智能体优化和学习中的原始-对偶一致性技术启发，我们基于增广拉格朗日法提出了一个平滑的最小-最大问题变体。所得到的通过增广拉格朗日实现的精确帕累托优化（EPO-AL）算法在目标数量较多时比基于次梯度的策略更具扩展性，同时相较于最近基于平滑化的同类方法，在每次迭代的复杂度上更低。我们证明了在适度假设下，所提出算法的每个不动点既是帕累托最优的又是最小-最大最优的，并通过数值模拟展示了其有效性。 显示英文摘要

摘要： In multi-objective optimization, minimizing the worst objective can be preferable to minimizing the average objective, as this ensures improved fairness across objectives. Due to the non-smooth nature of the resultant min-max optimization problem, classical subgradient-based approaches typically exhibit slow convergence. Motivated by primal-dual consensus techniques in multi-agent optimization and learning, we formulate a smooth variant of the min-max problem based on the augmented Lagrangian. The resultant Exact Pareto Optimization via Augmented Lagrangian (EPO-AL) algorithm scales better with the number of objectives than subgradient-based strategies, while exhibiting lower per-iteration complexity than recent smoothing-based counterparts. We establish that every fixed-point of the proposed algorithm is both Pareto and min-max optimal under mild assumptions and demonstrate its effectiveness in numerical simulations.