Tiered neighborhood-exchange differential evolution for budget-constrained multi-root localization of nonlinear equation systems

Budget-constrained localization of multiple roots of nonlinear equation systems requires both broad coverage of different attraction basins and rapid refinement of promising candidates when the number of residual evaluations is limited. Many niching variants of differential evolution perform replacement within local neighborhoods, but overly local mating can reduce basin coverage and cause premature stagnation. This paper introduces Tiered Neighborhood-Exchange Differential Evolution, a crowding-based solver that preserves neighborhood replacement while injecting controlled global information. The method uses a residual-gated dual mutation that switches between neighborhood exploitation and a global anchor, and a tiered neighborhood-exchange crossover that couples individuals across three fitness strata to counteract diversity loss. An archive of verified roots and distance-based duplicate filtering are employed to maintain a set of distinct solutions. Experiments on six benchmark systems show that, under identical evaluation budgets, the proposed method improves the recovered-root proportion and the probability of finding all distinct roots compared with representative niching differential-evolution baselines.

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