🤖 AI Summary
This work addresses the challenges of convergence in multi-objective optimization arising from non-differentiable objective functions and abrupt structural changes in nondominated fronts. To this end, the authors propose a nonsmooth set-based gradient ascent method that uniquely integrates an amplitude indicator with a hierarchical weighting scheme. They derive its exact gradient and prove its computational complexity is equivalent to that of hypervolume gradient computation. By employing hierarchical aggregation, the method establishes a connection to infinitesimal coding, thereby elucidating the underlying mechanics of nonsmooth optimization. The algorithm further incorporates projected finite differences, repulsion and stagnation-recovery strategies, and coordinate-projected geometric expansion techniques. Experimental results on standard two- and three-objective benchmarks, as well as curved and hyperspherical Pareto fronts, demonstrate that the proposed approach efficiently and robustly approximates the Pareto front.
📝 Abstract
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.