ITM: Induced Tchebycheff Measures for Weight Design

Uniform Tchebycheff weights need not produce uniform coverage of a Pareto front. We analyse the distribution induced by the weight-to-optimum map and its relation to front coverage. We introduce ITM (Induced Tchebycheff Measures), which estimates the solutions selected by candidate weights from a non-dominated archive. Three alternative constructions use isotonic inversion, pullback $k$-center, or density flattening to propose new weights, with updates screened by an archive-based covering score. Across 13 analytically specified fronts, covering gains from isotonic inversion and density flattening correlate with departure from uniform arclength, with Spearman coefficients of 0.846 and 0.739, respectively. For isotonic weights, a similar association appears across seven front settings under preference-conditioned network training. Oracle comparisons in MOEA/D show that coverage at subproblem optima and performance after search can respond differently to the same weight set.