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Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

Thiago Santos, Sebastiao Xavier

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.

Lyapunov Stability of Stochastic Vector Optimization: Theory and Numerical Implementation

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.
Paper Structure (9 sections, 4 theorems, 29 equations, 1 table, 1 algorithm)

This paper contains 9 sections, 4 theorems, 29 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Rigorous Stability Analysis
  4. Numerical Implementation
  5. Euler--Maruyama discretization
  6. Integration with pymoo
  7. Comparative Analysis with NSGA-II and NSGA-III
  8. Convergence on Convex Benchmark (DTLZ2)
  9. Conclusions and Future Work

Key Result

Lemma 1

Suppose $q$ is locally Lipschitz and satisfies Assumption ass:A. Then there exists $L > 0$ such that for all $x \in \mathbb{R}^n$: Moreover, for $V(x) = \|x\|_2^2$, the infinitesimal generator $\mathscr{L}$ of the diffusion satisfies for constants $C_1, C_2 > 0$.

Theorems & Definitions (12)

  • Definition 1: Wiener Measure
  • Remark 1
  • Lemma 1: Linear Growth Bound
  • proof
  • Theorem 1: Global Existence and Uniqueness
  • proof
  • Theorem 2: Positive Recurrence
  • proof
  • Corollary 1
  • proof
  • ...and 2 more