An Adaptive KKT-Based Indicator for Convergence Assessment in Multi-Objective Optimization

TL;DR

This paper introduces an adaptive KKT-based convergence indicator for multi-objective optimization that improves robustness and scalability over traditional reference-based metrics, especially in many-objective scenarios.

Contribution

It proposes a robust, adaptive reformulation of a KKT-based convergence indicator using quantile normalization, enhancing performance in complex optimization problems.

Findings

Improved robustness to heterogeneous stationarity residuals.

Enhanced scalability in many-objective optimization scenarios.

Maintains stationarity interpretation without external references.

Abstract

Performance indicators are essential tools for assessing the convergence behavior of multi-objective optimization algorithms, particularly when the true Pareto front is unknown or difficult to approximate. Classical reference-based metrics such as hypervolume and inverted generational distance are widely used, but may suffer from scalability limitations and sensitivity to parameter choices in many-objective scenarios. Indicators derived from Karush--Kuhn--Tucker (KKT) optimality conditions provide an intrinsic alternative by quantifying stationarity without relying on external reference sets. This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization. The proposed indicator preserves the stationarity-based interpretation of the original formulation while improving robustness to heterogeneous distributions of stationarity residuals, a recurring issue in many-objective optimization.