When the implementation matters more than the algorithm: NSGA-II and other multi-objective optimizers in engineering design tools

Architectural and structural design optimization in parametric software affords about a thousand evaluations of an expensive model for problems with tens of variables, a regime for which most metaheuristics were not designed. On the side of artificial intelligence, two hybrid metaheuristics are proposed: a covariance matrix adaptation evolution strategy (CMA-ES) whose sampling is enriched with differential-evolution trial vectors, and a particle swarm with success-history parameter adaptation, differential-evolution injection and local refinement. A budget-aware delay of population-size reduction in adaptive differential evolution is also proposed. Under a hard cap of 1,200 evaluations, all counted, the CMA-ES hybrid ranks first on seven shifted and rotated functions at 10 and 30 variables and beats CMA-ES on six of seven functions at 30 variables. Ablations attribute this gain to the injection and show that a local search pays off only when its result re-enters the population. The delayed schedule improves three adaptive differential-evolution variants in 40 of 51 comparisons, with no loss. On the side of engineering application, in Grasshopper the CMA-ES hybrid and plain CMA-ES are never significantly worse than the optimizers of Galapagos, SilverEye, Opossum and Tunny on travelling-salesman instances and on a built 85-variable free-form shell, where they end 48 % below the best external tool in 15 minutes. On a 22-variable building-performance problem with binary choices, differential evolution leads instead. Shifting and rotating the functions reverses the rank of Harris hawks optimization. Solvers should therefore be selected by problem structure and stopping time, not by a single ranking.

Multiobjective optimization remains challenging for many scientific and engineering problems due to the need to balance convergence, diversity, and computational efficiency across high-dimensional objective landscapes. This work presents the Multiobjective Animorphic Ensemble Optimization (MAEO) framework, a parallelizable ensemble strategy that unifies state-of-the-art evolutionary algorithms within an island-based architecture, overcoming the limitations of relying on a single optimizer, as implied by the No Free Lunch theorem. MAEO uses a parameter-free hypervolume indicator for island performance assessment and a strict Pareto-rank-based individual scoring formulation that incorporates crowding distance and nadir-point proximity to ensure consistent selection pressure within each front. The framework is initiated using four algorithms (NSGA-III, CTAEA, AGEMOEA2, SPEA2) and evaluated through extensive benchmarking on 12 DTLZ/ZDT functions under 36 dimensionality settings using Wilcoxon signed-rank tests with both hypervolume and inverse generational distance metrics. Results show that MAEO achieves balanced convergence-diversity performance, outperforming or matching some of the leading multiobjective optimization algorithms across different benchmark problems. To demonstrate practical applicability, MAEO is applied to the equilibrium-cycle optimization of a small modular nuclear reactor. Eight discrete design variables (and three objectives (levelized cost of electricity, peak soluble boron concentration, fuel cycle length) are optimized under two safety constraints. The algorithm carried out roughly 40000 evaluations using computer simulations. MAEO identifies core designs that lower both the levelized cost of electricity and the peak boron concentration, while preserving fuel cycle length and meeting all safety constraints.

This paper investigates the role of dynamic population sizes in evolutionary multi-objective optimization. Although such approaches are widely used in practice, their benefits remain poorly understood, and rigorous runtime analyses explaining when and why they help are still scarce. To address this, we introduce the bi-objective problem class CLIMB and analyze the runtime of GSEMO and the widely used NSGA-II on this problem. Our results show that allowing a dynamic population size for NSGA-II can lead to a moderate improvement, yielding a speedup of order $Ω(\sqrt{n}/\log n)$. In particular, we prove that GSEMO and NSGA-II-DYN, a version of NSGA-II with dynamic population sizes we propose in this paper, can find the Pareto front of CLIMB in expected $O(n \log n)$ fitness evaluations, whereas NSGA-II with a fixed population size requires $Ω(n^{1.5})$ fitness evaluations in expectation. To the best of our knowledge, this is the first rigorous runtime analysis in multi-objective optimization demonstrating a super-constant speedup of GSEMO over NSGA-II. Our analysis builds on concepts from single-objective optimization, like the evolution of population diversity over time, and employs the well-known family-three method to prove the lower bound.