Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

TL;DR

This paper develops a Lyapunov stability analysis for stochastic vector optimization models, providing theoretical guarantees and an accessible implementation, and evaluates its performance against evolutionary algorithms in high-dimensional problems.

Contribution

It offers a novel Lyapunov-based theoretical framework for stochastic vector optimization and implements a practical, reproducible algorithm compatible with existing multi-objective frameworks.

Findings

The method is less competitive in low-dimensional problems but effective in high-dimensional settings.

The proposed approach provides rigorous stability guarantees for stochastic optimization.

Empirical results show the method's viability under limited evaluation budgets.

Abstract

The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion model for unconstrained vector optimization in which the drift is induced by a common descent direction and the diffusion term preserves exploratory behavior. The main theoretical contribution is a self-contained Lyapunov analysis establishing global existence, pathwise uniqueness, and non-explosion under a dissipativity condition, together with positive recurrence under an additional coercivity assumption. We also derive an Euler--Maruyama discretization and implement the resulting iteration as a \emph{pymoo}-compatible algorithm -- \emph{pymoo} being an open-source Python framework for multi-objective optimization -- with an interactive \emph{PymooLab} front-end for reproducible experiments. Empirical results on DTLZ2 with objective counts from three to fifteen indicate a consistent trade-off: compared with established evolutionary baselines, the method is less competitive in low-dimensional regimes but remains a viable option under restricted evaluation budgets in higher-dimensional settings. Taken together, these observations suggest that stochastic drift--diffusion search occupies a mathematically tractable niche alongside population-based heuristics -- not as a replacement, but as an alternative whose favorable properties are amenable to rigorous analysis.