Convergence Analysis of Evolution Strategies for Mixed-Integer Optimization

We introduce a new simple model to study the fitness progress of Evolution Strategies (ES) in generic problems. In this model, we bypass the underlying fitness landscape and assume that the mutation of any individual produces an offspring whose fitness relative to the parent is given by an invariant distribution $Z$, such as a mean-shifted Gaussian. This serves as a prototypical model for the optimisation landscape when an evolution algorithm operates far from the global optimum. This simple model can be used to approximate the optimisation process for problems where it is intractable to model the exact fitness function, including tasks such as hyperparameter tuning in machine learning models. We rigorously analyse the expected growth rate $\mathcal{R}_μ$ of the continuous steady-state $(μ+1)$-ES in this model. Unlike comma-selection strategies, the steady-state $(μ+1)$-ES maintains overlapping generations, introducing complex mathematical dependencies among surviving parents that make it harder to analyse. We give a general technique to analyse the the $(μ+ 1)$-ES by constructing modified processes whose growth rates provably sandwich that of the original process. These modified processes are then easier to analyse but still close enough to the true process to give a tight bound on the expected growth rate. When $Z = \mathcal{N}(-δ, 1)$ and $μ\le e^δ$, we show that $\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$.

We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schr{ö}dinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ''survival of the flattest'' phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

Mixed categorical-continuous optimization arises in many practical domains, yet remains challenging. In the black-box setting, evolution strategy-based approaches have shown promise in extending the efficiency and robustness of the CMA-ES to mixed-variable spaces. However, these methods exhibit worsened performance when strong categorical-continuous interactions are present, as their underlying search distributions assume independence between categorical and continuous variables. To address this limitation, we propose a bilevel optimization framework that explicitly captures such interactions by optimizing over categorical variables in an outer loop, and over continuous variables conditioned on each categorical configuration in an inner loop. We formulate each level of the bilevel problem as a stochastic relaxation under information-geometric optimization. To mitigate the high computational cost inherent to bilevel optimization, we introduce a warm-starting strategy that accelerates the lower-level search by selecting the best among multiple cached configurations and updating the cache after each iteration. Experimental results on binary-continuous domain demonstrate that the proposed method outperforms existing state-of-the-art approaches in interaction-handling capability while also being more computationally efficient across benchmarks encompassing both previously reported and newly proposed types of interaction.