Expansion-Trajectory Optimization (ETO): A dual-operator metaheuristic for balanced global and local search

Premature convergence remains a critical limitation in many metaheuristic algorithms, and is often caused by a rapid loss of population diversity as individuals become overly similar early in the search process. To address this challenge, this paper proposes a new metaphor-free algorithm called Expansion-Trajectory Optimization (ETO), which introduces a dual-operator framework designed to maintain diversity and enhance search performance. The ETO algorithm combines two complementary mechanisms: the expansion operator, which leverages collective information from multiple individuals to identify and explore promising regions in the search space; and the trajectory operator, which conducts a guided search following a Fibonacci spiral. This spiral-based path enables a smooth transition from broad exploration to focused exploitation, thereby ensuring a balanced and adaptive search process. The proposed approach was rigorously evaluated against several state-of-the-art metaheuristic algorithms, using a diverse set of benchmark functions. The experimental results confirm that ETO achieves superior performance in terms of both accuracy and robustness, demonstrating its effectiveness in overcoming early convergence and enhancing optimization outcomes.