Bacterial Foraging Optimization Algorithm: Theoretical Foundations, Analysis, and Applications Swagatam Das 1 , Arijit Biswas 1 , Sambarta Dasgupta 1 , and Ajith Abraham 2 1 Dept. of Electronics and Telecommunication Engg, Jadavpur University, Kolkata, India 2 Norwegian University of Science and Technology, Norway swagatamdas19@yahoo.co.in, arijitbiswas87@gmail.com, sambartadg@gmail.com, ajith.abraham@ieee.org Abstract. Bacterial foraging optimization algorithm (BFOA) has been widely accepted as a global optimization algorithm of current interest for distributed optimization and control. BFOA is inspired by the social foraging behavior of Escherichia coli. BFOA has already drawn the attention of researchers because of its efficiency in solving real-world optimization problems arising in several application domains. The underlying biology behind the foraging strategy of E.coli is emulated in an extraordinary manner and used as a simple optimization algorithm. This chapter starts with a lucid outline of the classical BFOA. It then analyses the dynamics of the simulated chemotaxis step in BFOA with the help of a simple mathematical model. Taking a cue from the analysis, it presents a new adaptive variant of BFOA, where the chemotactic step size is adjusted on the run according to the current fitness of a virtual bacterium. Nest, an analysis of the dynamics of reproduction operator in BFOA is also discussed. The chapter discusses the hybridization of BFOA with other optimization techniques and also provides an account of most of the significant applications of BFOA until date. 1. Introduction Bacteria Foraging Optimization Algorithm (BFOA), proposed by Passino [1], is a new comer to the family of nature-inspired optimization algorithms. For over the last five decades, optimization algorithms like Genetic Algorithms (GAs) [2], Evolutionary Programming (EP) [3], Evolutionary Strategies (ES) [4], which draw their inspiration from evolution and natural genetics, have been dominating the realm of optimization algorithms. Recently natural swarm inspired algorithms like Particle Swarm Optimization (PSO) [5], Ant Colony Optimization (ACO) [6] have found their way into this domain and proved their effectiveness. Following the same trend of swarm-based algorithms, Passino proposed the BFOA in [1]. Application of group foraging strategy of a swarm of E.coli bacteria in multi-optimal function optimization is the key idea of the new algorithm. Bacteria search for nutrients in a manner to maximize energy obtained per unit time. Individual bacterium also communicates with others by sending signals. A bacterium takes foraging decisions after considering two previous factors. The process, in which a bacterium moves by taking small steps while searching for nutrients, is called chemotaxis and key idea of BFOA is mimicking chemotactic movement of virtual bacteria in the problem search space. Since its inception, BFOA has drawn the attention of researchers from diverse fields of knowledge especially due to its biological motivation and graceful structure. Researchers are trying to hybridize BFOA with different other algorithms in order to explore its local and global search properties separately. It has already been applied to many real world problems and proved its effectiveness over many variants of GA and PSO. Mathematical modeling, adaptation, and modification of the algorithm might be a major part of the research on BFOA in future. This chapter is organized as follows: Section 2 provides the biological motivation behind the BFOA algorithm and outlines the algorithm itself in a comprehensive manner. Section 3 provides a simple mathematical analysis of the computational chemotaxis of BFOA in the framework of the classical