Optimizing reinforced concrete cantilever retaining walls for cost and embodied carbon under reliability constraints using surrogate models and hybrid metaheuristics

Retaining walls are designed today with two objectives in mind: construction cost and embodied carbon, while the ground properties that govern their safety are known only in distribution. Treating both requirements at once yields a reliability-constrained multi-objective problem whose nested Monte Carlo evaluation is expensive, and the surrogate models typically used to reduce that cost are validated on limit-state responses rather than on the reliability index they are ultimately tasked with delivering. This study develops a multi-objective, reliability-based design optimization framework for reinforced concrete cantilever retaining walls and validates each component. The four governing limit states, overturning, sliding, bearing capacity and stem flexure, are written explicitly and verified against a documented worked example and against the first-order reliability method, which agrees to 0.051 in the reliability index. Three surrogate families are then compared on the same Latin hypercube design: extreme gradient boosting, ordinary Kriging with an anisotropic Matérn kernel, and sparse polynomial chaos expansion fitted by least-angle regression. Kriging is the most accurate but costs 472 s per million evaluations, which excludes it from a loop requiring $$\:7.5\times\:{10}^{8}$$ ; the polynomial expansion is exact on the three smooth limit states and saturates at 0.9013 on the bearing limit state, whose eccentricity term is not smooth, and the tree ensemble reaches 0.9544 there. Propagated to the reliability index over 40 designs, the mean absolute errors are 0.106, 0.205 and 0.356, respectively, all between three and ten times the Monte Carlo sampling error, so the analytical limit states are retained for the optimization, and a break-even condition is derived that states when a surrogate is worth building. The hybrid Grey Wolf Optimizer–Differential Evolution recovers a reliability-consistent optimum of 873 USD and 2,602 kg CO₂ per metre run, governed jointly by sliding, bearing capacity and stem flexure and significantly outperforms the standalone Grey Wolf Optimizer and NSGA-II ( $$\:p=0.00029$$ and $$\:p=0.00029$$ ) while being indistinguishable from differential evolution ( $$\:p=0.18588$$ ). Cost and embodied carbon are aligned rather than conflicting over the feasible design space (Pearson $$\:r=0.9957$$ ), so the Pareto front collapses to a narrow minimum-material region and the decisive design drivers lie in the description of the ground: introducing the cross-correlation reported for shear-strength parameters lowers the optimum by 25%, spatial averaging of the soil properties over the mobilized dimension of each limit state lowers it by 28%, and raising the coefficients of variation by 10% above the baseline removes the feasible set entirely. Estimating the failure probability on the same Monte Carlo sample that drives the search biases the reliability index upward by 0.047, which a 0.05 buffer removes at an 8.5% cost penalty.