MOO Invariance Under NNC Transforms
When is a Pareto front unchanged by transforming the objectives? Exactly when every objective is mapped by a strictly increasing function. Power transforms f^k on positive objectives qualify only for k > 0; for k < 0 they reverse every dominance relation.
Overview
Pareto dominance compares objectives only through the order relation <=. Any strictly increasing map g applied to each objective keeps that order, so the non-dominated set is the same before and after the transform. This is an elementary order-theory fact, not a property specific to non-Newtonian calculus.
A decreasing map flips every inequality: minimising g(f) then maximises f. The power transform f^k is increasing on f > 0 for k > 0 and decreasing for k < 0, so a k = -1 transform (the project default for 1/r cases) does not preserve the front; it turns a minimisation into a maximisation of the original objective.
The stored function-family analysis below classifies 0 of its 22 test functions as invariant (its own summary verdict), so it provides no empirical support for invariance either.
Stored Analysis
Mathematical Formulation
Power Transform
For a positive objective f(x) and k != 0:
T_k[f(x)] = exp(k * ln(f(x))) = f(x)^k
k = 1 is the identity; k > 0 is increasing; k < 0 is decreasing.
Pareto Dominance Preservation (k > 0 only)
For objectives f_i > 0 (minimization), if x dominates y:
f_i(x) <= f_i(y) for all i, with at least one strict inequality
k > 0: T_k[f_i(x)] <= T_k[f_i(y)] for all i, so x still dominates y
k < 0: T_k[f_i(x)] >= T_k[f_i(y)] for all i, so y now dominates x
Bounded Diagnostic (not regularization)
The log-bigeometric elasticity gives a bounded power-law diagnostic:
L_BG[1/r] = r * d/dr[ln(1/r)] = -1; D_BG[1/r] = exp(-1)
Bounded for r > 0, while the original 1/r field remains singular.
Key Results
Proposition (elementary)
For strictly increasing transforms of every objective, including T_k with k > 0 on positive objectives, the Pareto set is unchanged.
Counter-case: k < 0
T_k with k < 0 reverses dominance. Optimising a k = -1 transformed objective is a different problem, so this page gives no optimality guarantee for k < 0 searches.
Bijective on (0, infinity)
T_k is a bijection of the positive reals with inverse y^(1/k). It preserves order for k > 0 and reverses it for k < 0.
Stored Evidence
public/moo_invariance_analysis.json: summary n_invariant = 0 of 22 tested functions. The separate CASCADE toy benchmark is a synthetic distance-to-singularity exercise and says nothing about this proposition.
Related Pages
Spectral-gap boundary map: 79/88 valid cases pass, with nine counterexamples
View verification →Synthetic toy benchmark: reduction in distance to the singular point (not physics results)
View toy benchmark →Browser simulations and visualizations
Try simulations →Technical Notes
- - Objectives must be strictly positive (or shifted to a positive domain) for T_k to be defined
- - Only k > 0 preserves the Pareto set; k < 0 reverses it
- - Constraints are unaffected only if the same transform is not applied to them