The Pareto-Bernoulli Equation: A New Theoretical Framework for Multi-Objective Optimization Based on Energy Conservation Principles

Abstract

This paper introduces a new mathematical concept that bridges two seemingly heterogeneous domains: multi-objective nonlinear programming (MONLP) and Bernoulli's equation in fluid dynamics. Inspired by the fundamental structure of Bernoulli's equation—which describes energy conservation in steady flow—we propose a new model for analyzing Pareto fronts in MONLP problems with non-convex, dynamically coupled functions. We propose a mathematical analogy where the conflicting objectives in MONLP represent the different energy terms in the Bernoulli equation (kinetic energy, pressure energy, and potential energy), and the global "Bernoulli constant" is represented as a Pareto-Sum Constant that indicates the global optimal equilibrium state of the multi-objective system. We present a new mathematical formulation called the Pareto-Bernoulli Equation (PBE) and an associated transformation technique. This research opens new avenues for physics-inspired optimization algorithms and provides a unified theoretical perspective on Pareto optimality through energy conservation principles.