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

Thiago Santos; Sebastiao Xavier

math.OCMar 4, 2026

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

Authors: Thiago Santos, Sebastiao Xavier

Organizations: Department of Mathematics, Federal University of Ouro Preto, UFOP, Ouro Preto, Brazil

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.

Figures & tables

Algorithm	$\mathcal{H}_{adap}\downarrow$	$\mathcal{H}_{old}\downarrow$	$\Delta_{p}\downarrow$	HV $\uparrow$
NSGA-III	$7.01\text{e-2}\;(8.19\text{e-2})$	$3.50\text{e-2}\;(2.93\text{e-2})$	$1.17\text{e+0}\;(1.94\text{e+0})$	$5.56\text{e-1}\;(3.81\text{e-1})$
NRVMOEA	$6.16\text{e-2}\;(8.88\text{e-2})$	$3.46\text{e-2}\;(1.62\text{e-2})$	$1.43\text{e+2}\;(1.59\text{e+1})$	$0.00\text{e+0}\;(0.00\text{e+0})$
CMOEA-CD	$2.08\text{e-1}\;(9.88\text{e-3})$	$2.32\text{e-3}\;(3.46\text{e-3})$	$6.27\text{e+1}\;(5.07\text{e+1})$	$3.80\text{e-3}\;(2.08\text{e-2})$

TABLE I: Performance comparison on DTLZ1 (

m=12

Algorithm	$\mathcal{H}_{adap}\downarrow$	$\mathcal{H}_{old}\downarrow$	$\Delta_{p}\downarrow$	HV $\uparrow$
NSGA-III	$5.81\text{e-2}\;(3.76\text{e-2})$	$2.28\text{e-2}\;(2.02\text{e-2})$	$6.34\text{e-1}\;(5.43\text{e-2})$	$9.14\text{e-1}\;(1.55\text{e-1})$
NRVMOEA	$6.39\text{e-2}\;(1.60\text{e-2})$	$1.33\text{e-2}\;(3.16\text{e-3})$	$5.74\text{e-1}\;(5.27\text{e-3})$	$9.67\text{e-1}\;(1.88\text{e-3})$
CMOEA-CD	$1.36\text{e-1}\;(1.43\text{e-2})$	$1.98\text{e-1}\;(2.19\text{e-2})$	$8.64\text{e-1}\;(1.25\text{e-1})$	$4.57\text{e-1}\;(1.12\text{e-1})$

TABLE II: Performance comparison on DTLZ2 (

m=12

Algorithm	$\mathcal{H}_{adap}\downarrow$	$\mathcal{H}_{old}\downarrow$	$\Delta_{p}\downarrow$	HV $\uparrow$
NSGA-III	$4.95\text{e-2}\;(2.57\text{e-2})$	$1.46\text{e-1}\;(5.20\text{e-2})$	$5.18\text{e+1}\;(2.72\text{e+1})$	$0.00\text{e+0}\;(0.00\text{e+0})$
NRVMOEA	$9.92\text{e-2}\;(1.84\text{e-2})$	$2.45\text{e-1}\;(2.40\text{e-2})$	$6.41\text{e+2}\;(4.51\text{e+1})$	$0.00\text{e+0}\;(0.00\text{e+0})$
CMOEA-CD	$1.29\text{e-1}\;(2.96\text{e-2})$	$2.33\text{e-1}\;(3.15\text{e-2})$	$3.00\text{e+2}\;(1.46\text{e+2})$	$0.00\text{e+0}\;(0.00\text{e+0})$

TABLE III: Performance comparison on DTLZ3 (

m=12

Algorithm	$\mathcal{H}_{adap}\downarrow$	$\mathcal{H}_{old}\downarrow$	$\Delta_{p}\downarrow$	HV $\uparrow$
NSGA-III	$1.95\text{e-1}\;(1.35\text{e-2})$	$8.37\text{e-8}\;(1.00\text{e-8})$	$6.32\text{e-1}\;(3.01\text{e-2})$	$9.46\text{e-1}\;(2.95\text{e-2})$
NRVMOEA	$1.88\text{e-1}\;(1.26\text{e-2})$	$9.37\text{e-8}\;(7.08\text{e-9})$	$5.72\text{e-1}\;(2.70\text{e-3})$	$9.72\text{e-1}\;(3.87\text{e-4})$
CMOEA-CD	$2.19\text{e-1}\;(9.75\text{e-3})$	$1.80\text{e-7}\;(1.23\text{e-8})$	$6.96\text{e-1}\;(1.41\text{e-2})$	$7.14\text{e-1}\;(4.22\text{e-2})$

TABLE IV: Performance comparison on DTLZ4 (

m=12

Algorithm	$\mathcal{H}_{adap}\downarrow$	$\mathcal{H}_{old}\downarrow$	$\Delta_{p}\downarrow$	HV $\uparrow$
NSGA-III	$1.05\text{e-1}\;(2.76\text{e-2})$	$2.87\text{e-2}\;(1.25\text{e-2})$	$8.00\text{e-1}\;(7.57\text{e-2})$	$7.33\text{e-2}\;(1.18\text{e-2})$
NRVMOEA	$9.26\text{e-2}\;(1.74\text{e-2})$	$3.36\text{e-2}\;(1.05\text{e-2})$	$1.36\text{e+0}\;(7.90\text{e-2})$	$2.65\text{e-2}\;(2.45\text{e-2})$
CMOEA-CD	$8.97\text{e-2}\;(1.42\text{e-2})$	$4.22\text{e-2}\;(6.46\text{e-3})$	$2.62\text{e+0}\;(4.95\text{e-2})$	$1.91\text{e-2}\;(2.61\text{e-2})$

TABLE V: Performance comparison on DTLZ5 (

m=12

Multi-Objective Optimization

Multi-objective optimization studies how to find and choose solutions that balance competing goals, constraints, and preferences.

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Nonsmooth Set-Gradient Ascent to the Pareto Front via Layered Hypervolume and Magnitude Indicators

Michael T. M. Emmerich

Faculty of Information Technology, University of Jyväskylä, Finland

Multi-Objective Optimization Pareto Optimization Nonsmooth Convex Optimization

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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.

Abstract

Multi-Objective Optimization

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