Running-time Analysis of ($μ+λ$) Evolutionary Combinatorial Optimization Based on Multiple-gain Estimation

The running-time analysis of evolutionary combinatorial optimization is a fundamental topic in evolutionary computation. However, theoretical results regarding the $(\mu+\lambda)$ evolutionary algorithm (EA) for combinatorial optimization problems remain relatively scarce compared to those for simple pseudo-Boolean problems. This paper proposes a multiple-gain model to analyze the running time of EAs for combinatorial optimization problems. The proposed model is an improved version of the average gain model, which is a fitness-difference drift approach under the sigma-algebra condition to estimate the running time of evolutionary numerical optimization. The improvement yields a framework for estimating the expected first hitting time of a stochastic process in both average-case and worst-case scenarios. It also introduces novel running-time results of evolutionary combinatorial optimization, including two tighter time complexity upper bounds than the known results in the case of ($\mu+\lambda$) EA for the knapsack problem with favorably correlated weights, a closed-form expression of time complexity upper bound in the case of ($\mu+\lambda$) EA for general $k$-MAX-SAT problems and a tighter time complexity upper bounds than the known results in the case of ($\mu+\lambda$) EA for the traveling salesperson problem. Experimental results indicate that the practical running time aligns with the theoretical results, verifying that the multiple-gain model is an effective tool for running-time analysis of ($\mu+\lambda$) EA for combinatorial optimization problems.

An Experimental Approach for Running-Time Estimation of Multi-objective Evolutionary Algorithms in Numerical Optimization

Multi-objective evolutionary algorithms (MOEAs) have become essential tools for solving multi-objective optimization problems (MOPs), making their running time analysis crucial for assessing algorithmic efficiency and guiding practical applications. While significant theoretical advances have been achieved for combinatorial optimization, existing studies for numerical optimization primarily rely on algorithmic or problem simplifications, limiting their applicability to real-world scenarios. To address this gap, we propose an experimental approach for estimating upper bounds on the running time of MOEAs in numerical optimization without simplification assumptions. Our approach employs an average gain model that characterizes algorithmic progress through the Inverted Generational Distance metric. To handle the stochastic nature of MOEAs, we use statistical methods to estimate the probabilistic distribution of gains. Recognizing that gain distributions in numerical optimization exhibit irregular patterns with varying densities across different regions, we introduce an adaptive sampling method that dynamically adjusts sampling density to ensure accurate surface fitting for running time estimation. We conduct comprehensive experiments on five representative MOEAs (NSGA-II, MOEA/D, AR-MOEA, AGEMOEA-II, and PREA) using the ZDT and DTLZ benchmark suites. The results demonstrate the effectiveness of our approach in estimating upper bounds on the running time without requiring algorithmic or problem simplifications. Additionally, we provide a web-based implementation to facilitate broader adoption of our methodology. This work provides a practical complement to theoretical research on MOEAs in numerical optimization.