Constrained Real-Parameter Optimization with Generalized Differential Evolution Saku Kukkonen and Jouni Lampinen Abstract— This paper presents results for the CEC 2006 Special Session on Constrained Real-Parameter Optimization where the Generalized Differential Evolution (GDE) has been used to solve given test problems. The given problems consist of 24 problems having one objective function and one or more in-/equality constraints. Almost all the problems were solvable in a given maximum number of solution candidate evaluations. The paper also shows how GDE actually needs lower number of function evaluations than usually required. I. I NTRODUCTION Many practical problems have multiple objectives and sev- eral aspects cause multiple constraints in the problems. For example, mechanical design problems have several objectives such as obtained performance and manufacturing costs, and available resources may cause limitations. Constraints can be divided into boundary constraints and constraint functions. Boundary constraints are used when the value of a decision variable is limited to some range, and constraint functions represent more complicated constraints, which are expressed as functions. A mathematically constrained multi-objective optimization can be presented in the form: minimize {f 1 (x),f 2 (x),...,f M (x)} subject to (g 1 (x),g 2 (x),...,g K (x)) T ≤  0. Thus, there are M functions to be optimized and K constraint functions. Maximization problems can be easily transformed to minimization problems and different constraints can be converted into form g j (x) ≤ 0, thereby the formulation above is without loss of generality. Problems used in this paper are single-objective problems with constraints, and they have been defined in [1] for the CEC 2006 Special Session on Constrained Real-Parameter Optimization. The problems have different number of vari- ables and in-/equality constraints, and difficulties of functions vary from linear to non-linear. Also evaluation criteria are given in [1]. This paper continues describing the basic Differential Evolution (DE) in Section II and its extension Generalized Differential Evolution (GDE) in Section III. Section IV describes experiments and finally conclusions are given in Section V. The authors are with the Department of Information Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeen- ranta, Finland; email: saku.kukkonen@lut.fi. II. DIFFERENTIAL EVOLUTION The DE algorithm [2], [3] was introduced by Storn and Price in 1995 and it belongs to the family of Evolutionary Algorithms (EAs). The design principles of DE are simplic- ity, efficiency, and the use of floating-point encoding instead of binary numbers. As a typical EA, DE has a random initial population that is then improved using selection, mutation, and crossover operations. Several ways exist to determine a stopping criterion for EAs but usually a predefined upper limit G max for the number of generations to be computed provides an appropriate stopping condition. Other control parameters for DE are the crossover control parameter CR, the mutation factor F , and the population size NP . In each generation G, DE goes through each D dimen- sional decision vector x i,G of the population and creates the corresponding trial vector u i,G as follows in the most common DE version, DE/rand/1/bin [4]: r 1 ,r 2 ,r 3 ∈{1, 2, . . . , NP } , (randomly selected, except mutually different and different from i) j rand = floor (rand i [0, 1) · D)+1 for(j = 1; j ≤ D; j = j + 1) { if(rand j [0, 1) < CR ∨ j = j rand ) u j,i,G = x j,r3,G + F · (x j,r1,G - x j,r2,G ) else u j,i,G = x j,i,G } In this DE version, NP must be at least four and it remains fixed along CR and F during the whole execution of the al- gorithm. Parameter CR ∈ [0, 1], which controls the crossover operation, represents the probability that an element for the trial vector is chosen from a linear combination of three randomly chosen vectors and not from the old vector x i,G . The condition “j = j rand ” is to make sure that at least one element is different compared to the elements of the old vector. The parameter F is a scaling factor for mutation and its value is typically (0, 1+] 1 . In practice, CR controls the rotational invariance of the search, and its small value (e.g., 0.1) is practicable with separable problems while larger values (e.g., 0.9) are for non-separable problems. The control parameter F controls the speed and robustness of the search, i.e., a lower value for F increases the convergence rate but it also adds the risk of getting stuck into a local optimum. Parameters CR and NP have the same kind of effect on the convergence rate as F has. 1 Notation means that the upper limit is about 1 but not strictly defined. 0-7803-9487-9/06/$20.00/©2006 IEEE 2006 IEEE Congress on Evolutionary Computation Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006 207 Authorized licensed use limited to: Nanyang Technological University. Downloaded on March 24,2010 at 21:48:14 EDT from IEEE Xplore. Restrictions apply.