Hybrid Algorithm Blends Multiverse Physics and Pyramid Building to Crack Tough Optimization Problems

A new open-access study published in Discover Computing on 28 October 2026 introduces MV-GPCO, a hybrid metaheuristic optimization algorithm that fuses two very different search strategies into a single framework. Developed by Nikita, Jyoti Yadav, Vijander Singh and Vijay Mohan, the method combines the Multiverse Optimization (MVO) algorithm, inspired by white holes, black holes and wormholes, with the Giza Pyramid Construction (GPC) algorithm, which mimics the step-wise stacking of pyramid blocks. The authors report that the hybrid achieves faster convergence, higher solution accuracy and improved robustness across a standard benchmark suite and three classical engineering design problems.

The central problem the researchers set out to address is a long-standing tension in metaheuristic optimization: the balance between global exploration and local exploitation. Algorithms that search broadly, like MVO, can cover high-dimensional spaces well but often converge slowly or stagnate in later stages. Algorithms that refine locally, like GPC, can converge precisely but risk premature convergence in complex, multimodal landscapes. MV-GPCO integrates the stochastic exploration of MVO with the deterministic local refinement of GPC within a unified framework, aiming to capture the strengths of both parents while avoiding their individual weaknesses.

Technically, the algorithm operates on a population of candidate solutions, each treated as a universe in the MVO phase. Solution quality is evaluated with a cost function, and information transfer between universes is governed by white-hole and wormhole mechanisms, with probabilities that adapt over iterations. The wormhole update concentrates exploitation around the best solution found so far, controlled by a Wormhole Existence Probability and a Traveling Distance Rate that shrink as iterations proceed. Feasibility is enforced by clamping out-of-bounds variables to their valid bounds.

A distinctive feature of the design is that the two phases are not executed sequentially. Instead, the authors employ what they call a probabilistic blended hybridization mechanism: for each search agent at each iteration, a random variable is compared against a dynamically decreasing threshold proportional to the ratio of current to maximum iterations. Early iterations favor the MVO phase for broad exploration, while later iterations increasingly activate the GPC phase for focused local refinement. The two phases share a common population matrix and a global best solution, and a greedy selection rule retains only beneficial updates. An anti-stagnation perturbation, triggered when the distance to the global best falls below a microscopic threshold, guards against premature convergence.

For constrained engineering problems, the algorithm applies Deb’s feasibility rules uniformly across both phases: feasible solutions always beat infeasible ones, feasible candidates are compared by objective value, and infeasible candidates are compared by the magnitude of constraint violation. This allows the search to pass through slightly infeasible transitional states without penalty factors while guaranteeing that the final optimum is strictly feasible.

The evaluation used the CEC 2022 single-objective benchmark suite of 12 shifted and rotated functions with unimodal, multimodal, hybrid and composition landscapes, tested at dimensionalities of 10 and 20 over a uniform search domain. All 15 compared algorithms, including MV-GPCO and 14 competitors such as PSO, GA, GWO, MVO, GPC, Firefly and Symbiotic Organism Search, ran under identical settings: a population of 50, a maximum of 1,000 iterations, 30 independent runs per function, and the same hardware, an 11th-generation Intel Core i7 with 16 GB of RAM.

The reported results show MV-GPCO achieving the best or highly competitive average fitness values across diverse landscape types, with notably small standard deviations and coefficients of variation on difficult functions such as F3, F9 and F12 at 10 dimensions. At 20 dimensions, the authors report that MV-GPCO obtained the best average fitness on functions including F4, F5, F7, F8 and F12. Some competitors, including EMSO, MVO and SOS, recorded lower absolute minima on individual functions such as F1, F3 and F10, but the authors argue the hybrid’s advantage lies in its consistency across the entire suite rather than isolated wins.

Statistical validation relied on two non-parametric tests at a significance level of 0.05. The Friedman test, treating the 12 benchmark functions as blocks and the 15 algorithms as treatments, gave MV-GPCO the lowest mean rank of 2.5833 at 10 dimensions and 3.0833 at 20 dimensions, the best overall ranking among the 15 algorithms. Pairwise Wilcoxon rank-sum tests showed more statistically significant wins than losses against all 14 competitors at both dimensionalities, although ties and some losses occurred against EMSO, MVO, SOS and HBO on certain functions. The authors note that the reported pairwise p-values are raw and not corrected for multiple comparisons.

Sensitivity analyses suggested the method is robust to its internal parameters. Varying the minimum Wormhole Existence Probability across five values and the GPC substitution probability across five values produced negligible variance in final fitness on representative functions, supporting the default configuration without manual tuning. Runtime analysis showed MV-GPCO averaging 0.4634 seconds per run, faster than standalone GPC at 0.9706 seconds and only about 0.11 seconds slower than standalone MVO at 0.3525 seconds, with an asymptotic complexity matching baseline metaheuristics. The authors also candidly document limitations: on highly deceptive functions such as F9 and F12, the GPC phase can sample a misleading local optimum if the true global basin is not found early, and the micro-perturbation radius is too small to escape large deceptive basins.

On three constrained engineering design problems, the hybrid delivered strictly feasible solutions with favorable statistics. For the welded beam design, MV-GPCO obtained a minimum fabrication cost of 1.7248, slightly better than MVO’s 1.7254, with a mean cost of 1.7262 and a standard deviation of 0.0029 versus MVO’s 0.0048. For the pressure vessel problem, it produced a cost of 5886.47 with discrete thicknesses satisfying manufacturing constraints and all constraint functions satisfied. For the speed reducer problem, it achieved a minimum weight of 2995.8312 with the pinion tooth count respecting the discrete requirement, whereas a competing RDO result of 2855.1910 was deemed mechanically infeasible due to a non-integer tooth count. The authors conclude that MV-GPCO offers a practical framework for nonlinear and constrained engineering optimization, while noting that future work will extend it to large-scale, multi-objective and dynamic problems.

Subject of Research: A hybrid metaheuristic optimization algorithm combining Multiverse Optimization and Giza Pyramid Construction for engineering design problems

Article Title: A hybrid multiverse giza pyramid construction metaheuristic for robust engineering optimization

Article References: Nikita, Yadav, J., Singh, V., & Mohan, V. (2026). A hybrid multiverse giza pyramid construction metaheuristic for robust engineering optimization. Discover Computing, 29(1), Article 821. https://doi.org/10.1007/s10791-026-10665-1

Image Credits: AI Generated

DOI: 10.1007/s10791-026-10665-1

Keywords: metaheuristic optimization, MV-GPCO, Multiverse Optimization, Giza Pyramid Construction, CEC 2022 benchmark, exploration-exploitation balance, constrained engineering design, welded beam design, pressure vessel design, speed reducer problem, Friedman test, Wilcoxon rank-sum test