integer evolution strategy

This work addresses the inefficiency of traditional genetic algorithms in solving optimization problems due to their reliance on random mutation and recombination, which lack goal-directedness. The authors formulate the problem through the lens of query complexity and propose objective-guided mutation and recombination operators informed by the optimization target. Leveraging reinforcement learning and formal language theory, they analyze the theoretical properties of these operators. For the first time, the study mathematically characterizes the mechanism of goal-directed genetic operators and demonstrates the necessity of population diversity for certain classes of optimization problems. A general model of genetic algorithms is established, enabling the design of a tight algorithm for a specific problem class, and proving that the synergy among generation, mutation, and recombination is essential for efficient optimization.

This paper addresses the lack of evolutionary algorithms theoretically grounded in discrete structures for integer programming (IP). We propose Integer Evolution Strategies (IES), a novel framework designed specifically for IP. Its core contributions are threefold: (i) the first use of the ℓ₁-norm—rather than the conventional ℓ₂-norm—as the distance metric in integer search space; (ii) a correlated mutation mechanism for unbounded integer variables, based on a bigeometric distribution, with theoretical proof of superiority over truncated normal distributions; and (iii) a quantification method for correlation on discrete lattices, coupled with entropy-driven mutation analysis. Experiments on nonseparable quadratic integer programs demonstrate that IES significantly outperforms state-of-the-art heuristic methods, empirically validating the critical role of the ℓ₁-norm and bigeometric distribution in enhancing the efficiency of discrete stochastic optimization.

This work addresses the limitation of quality-diversity (QD) algorithms in effectively propagating high-quality genetic modules due to their reliance on incremental mutation, which often leads to premature stagnation in exploration. Inspired by biological meiosis, the authors propose a novel discrete genetic crossover operator that introduces gene-level recombination into the QD framework for the first time. This operator synergistically complements existing variation strategies by preserving elite genetic material while enabling efficient cross-behavioral exploration. By transcending the constraints of traditional incremental mutation, the method significantly improves QD scores, coverage, and peak fitness across three locomotion control tasks. Notably, it demonstrates superior performance gains and sustained diversity maintenance during late-stage optimization, highlighting its capacity to enhance both exploration efficacy and solution quality in complex behavioral spaces.

This work addresses a critical limitation in existing evolutionary strategies for mixed-integer optimization, where imposing a lower bound on the mutation strength (standard deviation) of integer variables impedes the convergence of continuous variables and lacks theoretical justification. The study establishes the first rigorous convergence analysis framework for two variants of the (1+1)-Evolution Strategy: one with only a lower bound ((1+1)-LB-ES) and another with both lower and upper bounds ((1+1)-LUB-ES) on the standard deviation. Leveraging drift analysis, the authors develop specialized analytical tools and benchmark functions tailored to mixed-integer domains. Their results demonstrate that (1+1)-LB-ES is prone to premature convergence in high-dimensional integer spaces, whereas (1+1)-LUB-ES achieves linear convergence under appropriate parameter settings, substantially enhancing overall optimization efficiency.

In conventional genetic algorithms, two-parent recombination often introduces high-destructive variance, impairing convergence and stability. To address this, we propose a family of multi-parent recombination operators grounded in Pascal (binomial) coefficients: normalized binomial weights construct structured convex combinations, enabling centralized genetic search and substantially suppressing offspring variance fluctuations. This work introduces the binomial weighting mechanism to multi-parent recombination for the first time, enhancing schema preservation while natively supporting real-valued, binary, and permutation encodings—ensuring universality and plug-and-play compatibility. Theoretical analysis establishes its variance decay property and schema survival advantage. Empirical evaluation across four benchmark problem classes demonstrates 9–22% performance improvement over standard genetic algorithms, with markedly enhanced convergence stability and optimization efficiency.

This work addresses the limitation of conventional survival selection in evolutionary diversity optimization, which often fails due to its dependence on pairwise solution diversity. To overcome this issue, we propose a novel framework that enables the synchronous generation of multiple candidate solutions per generation, along with a tailored survival selection mechanism designed specifically for this setting. By moving beyond the traditional paradigm of single-solution, sequential updates, our approach effectively handles the dynamic nature of each solution’s contribution to population diversity. Experimental results demonstrate that, under certain conditions, the proposed multi-solution generation strategy accelerates convergence toward diverse solutions and significantly improves both the spread and quality balance of the final solution set.

Conventional wisdom holds that recombination operators offer little performance benefit in Cartesian Genetic Programming (CGP), leading to their long-standing neglect. This study systematically optimizes the hyperparameters of two recombination strategies—subgraph crossover and discrete phenotypic recombination—within the TinyverseGP framework on the SRBench benchmark platform. For the first time, it demonstrates that carefully tuned recombination significantly enhances CGP’s performance on symbolic regression tasks. Challenging the prevailing paradigm that CGP relies predominantly on mutation, this work establishes that recombination, when appropriately configured, possesses substantial potential. These findings open new avenues for the design of evolutionary algorithms by reintegrating recombination as a key operator in CGP.

This study addresses the lack of theoretical foundation for parameter selection in the bat algorithm, which has traditionally relied on empirical tuning. For the first time, it integrates dynamical systems theory with population variance evolution analysis to construct a theoretical framework characterizing the influence of key parameters. Within this framework, analytically derived effective ranges for critical parameters are established. Numerical experiments confirm that the theoretical predictions align closely with the observed convergence behavior in practice. The work further uncovers the intrinsic mechanisms governing the trade-off between exploration and exploitation and the algorithm’s convergence properties, thereby providing the first systematic theoretical guidance for parameter configuration in the bat algorithm.

This work addresses the opacity of search dynamics in traditional swarm intelligence and evolutionary algorithms, where nonlinear selection and adaptive mechanisms obscure the intrinsic geometric structure underlying candidate solution generation. To resolve this, the authors propose an operator–selection decomposition framework that decouples fitness-independent variation operators from boundary handling and fitness-dependent selection, thereby systematically uncovering the proposal geometry of SOMA and Differential Evolution (DE). They reveal, for the first time, that SOMA exhibits a linear structure in the migrant–leader space, derive closed-form expressions for key statistical quantities, and leverage these insights to design a geometrically controllable, rotation-aware SOMA variant along with an adaptive population reduction strategy. On the noiseless BBOB benchmark suite, the resulting algorithm significantly outperforms the original SOMA and matches or exceeds state-of-the-art DE variants across multiple dimensions and evaluation budgets.

This work addresses the lack of theoretical foundations for dynamic population sizing in multi-objective evolutionary algorithms by introducing a novel bi-objective benchmark problem, CLIMB. Through rigorous runtime analysis, it compares the performance of GSEMO and NSGA-II under both fixed and dynamic population strategies. The study provides the first provable super-constant speedup of GSEMO over fixed-population NSGA-II and proposes a new variant, NSGA-II-DYN. Leveraging diversity-based evolutionary analysis, family-tree lower-bound techniques, and tools from single-objective optimization theory, the paper establishes that both NSGA-II-DYN and GSEMO converge to the Pareto front in expected $O(n \log n)$ fitness evaluations, whereas fixed-population NSGA-II requires $\Omega(n^{1.5})$, yielding an asymptotic speedup of $\Omega(\sqrt{n} / \log n)$.