Evolutionary Multiobjective Optimization: A Short Survey of the State-of-the-art

Many real-life problems have a natural representation in the framework of multiobjective optimization. Evolutionary algorithms are generally considered one of the most successful methods for solving the multiobjective optimization problems. In this paper we present state-of-the-art multiobjective evolutionary algorithms and briefly discuss their advantages and disadvantages. In the last section we suggest some possibilities for future research in this area. Introduction Many real life optimization problems require optimizing more than one objective at once. The methods used in multiobjective optimization deal with this kind of problems. There are several multiobjective optimization methods, some of them are purely mathematical [Das and Dennis, 1998], others are based on Particle Swarm Optimization [Kennedy and Eberhart, 1995] or Ant Colony Optimization [Dorigo, 1992]. However, multiobjective evolutionary algorithms (MOGA), seem to be the best method used nowadays. One of their main advantages is that they are population based, thus finding more than one interesting solution in a single run. Another advantage is the lack of assumptions about the problem to be solved. In this paper, we describe some of the most important and widely used MOGAs. It is organized as follows: in the next section, we define the basic terms we will need to talk about multiobjective optimization. Next, we describe the multiobjective evolutionary algorithms and finally, we provide some ideas for future research in this field. Basic definitions In multiobjective optimization the goal is to optimize several functions at once; finding a solution which is optimal in all of them. This cannot generally be achieved by a single solution, more often there is a set of solutions and each of them is better in at least one function compared to each other. The following definition explains formally what a multiobjective optimization problem is. Definition 1. The multiobjective optimization problem (MOP) is a quadruple 〈D,O, f, C〉, where • D is the decision space • O ⊆ R is the objective space • C = {g1, . . . , gm}, where gi : D → R is the set of constraint functions (constraints) defining the feasible space Φ = {x ∈ D|gi(x) ≤ 0} • f : Φ → O is the vector of n objective functions (objectives), f = (f1, . . . , fn), fi : Φ → R x ∈ D is called the decision vector and y ∈ O is denoted as the objective vector. The preference between solutions is given by the following relation. If a solution has a lower value of each objective function than another solution, it is considered to be better. The ordering of solutions is described in the following definition. Definition 2. Given decision vectors x, y ∈ D we say • x weakly dominates y (x y) if ∀i ∈ {1 . . . n} : fi(x) ≤ fi(y). • x and y are incomparable if neither x y nor y x. • x does not dominate y (x y) if y x or x and y are incomparable 13 WDS'10 Proceedings of Contributed Papers, Part I, 13–18, 2010. ISBN 978-80-7378-139-2 © MATFYZPRESS