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

Thiago Santos, Sebastiao Xavier

TL;DR

This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization, which 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.

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.

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

TL;DR

This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization, which 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.

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.
Paper Structure (5 sections, 2 theorems, 14 equations, 5 tables)

This paper contains 5 sections, 2 theorems, 14 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Adaptive $\mathcal{H}$ Indicator
  4. Simulation Results
  5. Conclusion

Key Result

Theorem 1

Let $X=\{x_1,\dots,x_N\}$ be a finite approximation set and let $s_i=s(x_i)$ be the KKT stationarity residuals defined by where $\lambda^\star(x)$ solves the quadratic program in eq:qpp. Fix $0<\alpha<\beta<1$ and assume $Q_\beta>Q_\alpha$. Define $\mathcal{H}_{adap}(X)$ by eq:Hadap with $\varepsilon=0$. Then:

Theorems & Definitions (5)

  • Theorem 1: Boundedness and scale invariance of the adaptive $\mathcal{H}$ indicator
  • proof
  • Proposition 1: Computational complexity of the adaptive $\mathcal{H}$ indicator
  • proof
  • Remark 1: Implications for ultra-many-objective optimization