Surrogate-Assisted Artificial Immune Systems for Expensive Optimization Problems
Introduction When an animal is exposed to antigens an efficient immune response is developed in order to defend the organism where specific antibodies are produced to combat them. The best antibodies multiply (cloning) and are improved (hypermutation and replacement) while new antibodies, produced by the bone marrow, are generated. Thus, if the organism is again attacked by the same antigen a quicker immune response takes place. This scheme of adaptation is known as clonal selection and
... maturation by hypermutation or, more simply, clonal selection (Garrett, 2004) . Computational methods inspired by the biological immune system are called Artificial Immune Systems (AISs). Immune-inspired algorithms have found applications in many domains. One of the most important area, the optimization, is a mathematical principle largely applied to design and operational problems in all types of engineering, as well as a tool for formulating and solving inverse problems such as parameter identification in scientific and engineering situations. When applied to optimization problems, the AISs are stochastic populational search methods which do not require a continuous, differentiable, or explicit objective function, and do not get easily trapped in local optima. However, the AISs, as well as other nature-inspired techniques, usually require a large number of objective function evaluations in order to reach a satisfactory solution. As modern problems have lead to the development of increasingly complex and computationally expensive simulation models, this becomes a serious drawback to their application in several areas such as Computational Structural Mechanics, Reservoir Simulation, Environmental Modeling, and Molecular Dynamics. Thus, a good compromise between the number of calls to the expensive simulation model and the quality of the final solutions must often be established. A solution to this problem is to modify the search process in order to obtain either a reduction on the total computational cost or an increase in the efficiency of the optimization procedure. The solution considered here is the use of a surrogate model (or metamodel), which provides an approximation of the objective function, replacing the computationally intensive original simulator evaluation. Additionally, the surrogate model can help to smooth out the objective function landscape, and facilitate the optimization process. The idea of reducing the computation time or improving the solutions performing less computationally expensive function evaluations can be found in the evolutionary computation literature . In addition, several surrogates may be derived from physical or numerical simplifications of the original simulation model. In this paper we propose an artificial immune system assisted by a Similarity-Based Surrogate Model (SBSM) in which the objective is to allow the AIS to evolve for a larger number of generations, but still using a fixed number of expensive evaluations, in order to obtain improved final solutions. This chapter is organized as follows. Section 2 gives a formulation for the optimization problems considered here. AISs are presented in Section 3. Sections 4 and 5 present the Surrogate Models and the surrogate-assisted AIS, respectively. The computational experiments and a discussion of the results obtained can be found in Section 6. The concluding remarks are given in Section 7.