Global Convergence Analysis of the Flower Pollination Algorithm: A Discrete-Time Markov Chain Approach

TL;DR

This paper provides a mathematical proof of the global convergence of the flower pollination algorithm using Markov chain theory, demonstrating its efficiency and optimality in solving nonlinear optimization problems.

Contribution

It introduces a rigorous convergence analysis for the flower pollination algorithm, extending its theoretical foundation and confirming its effectiveness for global optimization.

Findings

Proves convergence to the optimal set under certain conditions

Shows the algorithm converges quickly in practice

Demonstrates global optimality through numerical experiments

Abstract

Flower pollination algorithm is a recent metaheuristic algorithm for solving nonlinear global optimization problems. The algorithm has also been extended to solve multiobjective optimization with promising results. In this work, we analyze this algorithm mathematically and prove its convergence properties by using Markov chain theory. By constructing the appropriate transition probability for a population of flower pollen and using the homogeneity property, it can be shown that the constructed stochastic sequences can converge to the optimal set. Under the two proper conditions for convergence, it is proved that the simplified flower pollination algorithm can indeed satisfy these convergence conditions and thus the global convergence of this algorithm can be guaranteed. Numerical experiments are used to demonstrate that the flower pollination algorithm can converge quickly in practice and can thus achieve global optimality efficiently.