Hamano et al. reveal how bounds on integer variables speed up evolution strategies

Researchers Ryoki Hamano, Kento Uchida, and Shinichi Shirakawa have published a rigorous convergence analysis of evolution strategies (ES) for mixed-integer optimization—problems where some variables are continuous and others integer. The paper, accepted at the Genetic and Evolutionary Computation Conference (GECCO '26), focuses on two (1+1)-ES variants: (1+1)-LB-ES, which enforces a lower bound on the standard deviation of integer variables to prevent them from converging too early, and (1+1)-LUB-ES, which adds an upper bound to accelerate continuous variable convergence. Using drift analysis—a technique borrowed from continuous-domain ES theory—the authors examine behavior on a benchmark mixed-integer function after the integer variables have been optimized.

Their key theoretical finding: (1+1)-LB-ES can exhibit premature convergence when the number of integer variables is large, because the lower bound forces the algorithm to keep exploring integer coordinates even after they are near-optimal, slowing continuous variable progress. In contrast, (1+1)-LUB-ES achieves linear convergence under suitable parameter settings, balancing exploration and exploitation. This is the first theoretical account of how integer-variable handling impacts overall convergence speed, filling a gap in the literature where empirical results had indicated issues but lacked formal proof. The work offers practical design guidelines for mixed-integer evolution strategies, suggesting that both lower and upper bounds on integer step sizes are necessary for reliable performance.