Paper P11Theory

MOO Invariance Under NNC Transforms

Multi-Objective Optimization (MOO) Pareto fronts transform predictably under Non-Newtonian Calculus (NNC) coordinate changes. This theoretical framework demonstrates that optimization relationships are preserved under NNC transforms, providing a rigorous foundation for CASCADE's singularity handling.

CASCADE APPLICATION

Optimal k Values

NNC transforms with specific k values preserve Pareto optimality while regularizing singularities.

k = -1.0

Power-law singularities

k = -0.5

Square root singularities

k = 0.0

Classical calculus (smooth)

k = 1.0

Exponential growth

View CASCADE validation results (61.9% win rate, 13/21 simulations) →

Overview

MOO Invariance establishes that Pareto-optimal solutions in multi-objective optimization remain optimal under coordinate transformations induced by Non-Newtonian Calculus. This is not merely a mathematical curiosity - it provides the theoretical foundation for CASCADE's ability to access solution regions near singularities that are unreachable by standard optimization methods.

The key insight is that NNC transforms act as coordinate changesin the objective space. Just as physical laws are invariant under coordinate transformations, Pareto dominance relationships are preserved under NNC transforms. A solution that is non-dominated in one coordinate system remains non-dominated after transformation.

This theoretical guarantee allows CASCADE to perform optimization in NNC-transformed coordinates (where singularities are regularized), then map solutions back to the original space with confidence that optimality is preserved.

Interactive Analysis

Mathematical Formulation

NNC Transform Definition

For a function f(x), the NNC transform with parameter k is:

T_k[f(x)] = exp(k * ln(f(x))) = f(x)^k

Special cases: k = 0 (classical), k = -1 (bigeometric), k = 1 (identity)

Pareto Dominance Preservation

For objectives f_1, f_2 (minimization):

If x dominates y in original space:

f_i(x) <= f_i(y) for all i, with at least one strict inequality

Then x dominates y in transformed space:

T_k[f_i(x)] <= T_k[f_i(y)] for all i (when k > 0 and f_i > 0)

Singularity Regularization

The bigeometric derivative (k = -1) regularizes power-law singularities:

D*[1/r] = d/dr[ln(1/r)] = d/dr[-ln(r)] = -1/r * r = -1

Result: Constant derivative instead of divergence at r = 0

Key Results

Theorem: MOO Invariance

For monotonic NNC transforms T_k with k > 0, the Pareto front topology is preserved. Optimal solutions remain optimal, and dominance relationships are maintained.

- Proven for strictly positive objectives
- Validated empirically on 21 physics simulations
- Extends to constraint handling

Corollary: Singularity Access

When objectives contain singularities, optimal k values enable evaluation in regions where classical calculus fails. The transformed problem is numerically well-behaved.

- k = -1.0 for 1/r^n singularities
- k = -0.5 for crack tip stress fields
- Enables 93.4% improvement in best case

Property: Bijective Mapping

NNC transforms are bijective on positive reals. Every point in transformed space maps uniquely to a point in original space, ensuring no information loss during optimization.

- Inverse: T_k^-1[y] = y^(1/k)
- Domain: (0, infinity)
- Preserves ordering

Validation: Empirical Evidence

21-simulation suite validates MOO invariance on real physics problems with diverse singularity types. CASCADE wins 61.9% of cases by accessing previously unreachable solution regions.

- 13 CASCADE wins, 8 ties/baseline
- Domains: cosmology, fracture, turbulence, QFT
- Best improvement: 93.4% (crack tip)

Practical Implications

For Optimization Engineers

When your MOO problem involves objectives with singularities (1/r potentials, crack tips, black hole horizons), CASCADE with appropriate k values can explore solution regions that NSGA-II/MOEA/D cannot reach. The optimality guarantee ensures transformed solutions are valid.

For Physics Simulation

Many physics problems have natural singularities (point charges, gravitational wells, fracture tips). MOO invariance means you can optimize in regularized coordinates without losing physical validity. Solutions map back to correct physics.

For Theory Development

MOO invariance under NNC suggests a deeper connection between optimization geometry and coordinate systems. Future work may extend this to other transformation groups (conformal, symplectic) and constrained optimization.

Verification

Spectral gap preservation and constancy diagnostic validation (9/9 tests passed)

View verification →

CASCADE Results

21-simulation validation proving singularity access advantage (61.9% win rate)

View results →

Interactive Tools

Explore MOO invariance with interactive simulations and visualizations

Try simulations →