A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion over coordinate projections contains extent, projected-area, and volume terms. The scalar objectives are nonsmooth because nondomination layers change combinatorially and the active orthogonal-union geometry changes piecewise. On fixed strata, where layer assignments and active geometry remain unchanged, the indicators are piecewise smooth and chamberwise continuous. For the magnitude indicator, an exact gradient formula is derived as a linear combination of hypervolume gradients of projected shadow sets. Thus, for fixed objective dimension, magnitude gradients have the same asymptotic time complexity as hypervolume gradients. Lexicographic layer aggregation is related to a unary infinitesimal encoding. For finite-$ε$ surrogates, the main nonsmoothness mechanisms are isolated and chamberwise Lipschitz continuity on bounded sets is proved; a two-point counterexample shows that hard-layer scalarization is not globally continuous across layer switches. The theory motivates a projected finite-difference implementation with repulsion and recovery from stagnation. Numerical examples and reproducible code cover two- and three-objective settings, including objective-space tests, curved fronts, a supersphere benchmark, and traces comparing layered magnitude and hypervolume ascent.

翻译：针对多目标优化中有限逼近集合向Pareto前沿移动的问题，本文提出了一种非光滑集合梯度上升方法。该方法优化分层集合指标：对连续非支配层评估基础指标，并将各层值与快速递减权重结合，从而为非支配点和支配点提供上升方向，同时防止深层补偿第一前沿的退化。分别处理超体积指标和支配集幅度指标两种基础指标——后者在坐标投影上的展开包含范围、投影面积和体积项。由于非支配层发生组合性变化且活跃正交并集几何结构呈现分段特性，标量目标函数具有非光滑性。在层分配与活跃几何结构保持不变的固定分层上，指标呈分段光滑且分室连续特性。针对幅度指标，推导出其精确梯度公式为投影影子集超体积梯度的线性组合。因此，在固定目标维度下，幅度梯度与超体积梯度具有相同的渐近时间复杂度。词汇分层聚合与一元无穷小编码存在关联。针对有限ε逼近，分离了主要非光滑机制并证明了有界集上的分室Lipschitz连续性；通过两点反例表明硬分层标量化在层切换时不具备全局连续性。该理论支撑了具有排斥与停滞恢复机制的投影有限差分实现。数值实验与可复现代码涵盖双目标与三目标场景，包括目标空间测试、曲面前沿、超球基准测试以及分层幅度与超体积上升轨迹对比。