The Contraction Mapping Optimizer: A Fixed-Point-Theoretic Metaheuristic for Global and Engineering Optimization

This paper introduces the Contraction Mapping Optimizer (CMO), a population-based optimizer whose update rule is derived directly from the Banach fixed-point theorem rather than from a biological or physical metaphor, and for which global convergence is proved. Most metaheuristics are built on such metaphors and offer little formal insight into why their update rules converge; CMO instead makes the mechanism explicit. Each candidate solution is moved by a damped contraction map toward an attractor formed from the best solutions found so far, perturbed by a Gaussian exploration term whose amplitude vanishes over the run. Because the contraction factor is kept strictly below one by construction, the deterministic part of every update is provably a contraction, making the exploration-exploitation balance explicit and schedulable. We further prove that the full stochastic, population-based iteration converges to the global optimum, and we demonstrate CMO on a nonlinear integral equation whose solution is itself a fixed point. CMO is compared with nine established metaheuristics (GA, PSO, DE, GWO, HHO, RUN, EO, FGO and WOA) on CEC2017 (D = 30, 50, 100) and CEC2022 (D = 10, 20), on three constrained engineering design problems, and on a six-unit economic load dispatch problem, over thirty runs at a common budget. Pooled over the 111 benchmark instances, CMO attains a Friedman mean rank of 3.14, statistically inseparable from the best competitor (FGO, 2.20) and significantly ahead of the remaining seven baselines; it ranks first on both CEC2022 suites and, on the constrained applications, is second only to differential evolution. Sensitivity and ablation studies identify the vanishing exploration schedule and greedy selection as the decisive components of the design.