Revision of a Floating-Point Genetic Algorithm GENOCOP V for Nonlinear Programming Problems

In this paper, focusing on general nonlinear programming problems, we attempt to propose a general and highperformance approximate solution method for them. In recent years, S. Koziel et al. have proposed a floating-point genetic algorithm, GENOCOP V, as a general approximate solution method for nonlinear programming problems and showed its efficiency, there are left some shortcomings of the method. In this paper, incorporating ideas to cope with these shortcomings, we propose a revised GENOCOP V (RGENOCOP V). Furthermore, we show the efficiency of the proposed method RGENOCOP V by comparing it with two existing methods, RGENOCOP III and GENOCOP V through the application of them into the numerical examples. INTRODUCTION A nonlinear programming problem is called a convex programming problem when its objective function and its constraint region are convex. For such convex programming problems, there have been proposed many efficient solution methods as the successive quadratic programming method [1], the generalized reduced gradient method [2] and so forth. Unfortunately, no decisive solution method has been proposed for nonconvex programming problems. For such nonconvex programming problems, as practical solution methods, genetic algorithms have been widely used . Genetic algorithms proposed by Holland [3] have attracted considerable attention as global methods for complex function optimization since De Jong considered genetic algorithms in a function optimization setting [4]. However, many of the test function minimization problems solved by a number of researchers during the past 20 years involve only specified domains of variables. Only recently several approaches have been proposed for solving general nonlinear programming problems through genetic algorithms [4-6]. In 1995, Michalewicz et al. [7] proposed a floating-point genetic algorithm GENOCOP III and showed its efficiency for noncovex programming problems. In GENOCOP III, the individual representation using the floating-point format is adopted. Furthermore, in GENOCOP III, two populations are used: one is the search population consisting of individuals which satisfies linear constraints, the other is the reference population consisting of individuals which satisfies all constraints. However, it is known that there exist drawbacks about the generation of initial individuals and the generation of offsprings in the crossover operation. In recent years, Sakawa et al. [8] proposed the revised, *Address correspondence to this author at the Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, Hiroshima, Japan; E-mail: [email protected] GENOCOP III (RGENOCOP III) involving improvements to deal with these drawbacks of GENOCOP III. On the other hand Koziel et al. [9] proposed GENOCOP V which is based on the homomorphous mapping as a new floating-point genetic algorithm and showed its superiority to GENOCOP III in processing time. Unfortunately, some shortcomings exist in GENOCOP V. When the homomorphous mapping is used in GENOCOP V, we need to find at least one feasible solution which is called the basepoint solution. Since GENOCOP V adopts the method to generate the fixed number of solutions in order to find the basepoint solution, it often happens that GENOCOP V cannot start the search because no feasible solution is found even if the problem is feasible. In addition, the precision of (approximate) optimal solutions obtained by GENOCOP V is not good since they greatly depend on the basepoint solution of the homomorphous mapping. Since only one basepoint solution is used in the whole search process in GENOCOP V, the precision of obtained approximate optimal solutions may not be good. In this research, we propose a revised version of GENOCOP V (RGENOCOP V) by considering a new way to find the basepoint solution where a problem to minimize the amount of constraint violation is solved as well as a new method to select and update the basepoint solution. NONLINEAR PROGRAMMING PROBLEMS In this research, we consider general nonlinear programming problem with constraints formulated as: minimize f(x) subject to gi(x) 0 , i = 1, 2, ... , m (1) lj xj uj , j = 1, 2, ... , n x = (x1, x2, ... , xn) T R n where f and gi are convex or nonconvex real-valued functions, lj and uj are the lower bound and the upper bound of each decision variable xj. In the following, we denote the feasible region of (1) by X. Revision of a Floating-Point Genetic Algorithm GENOCOP V The Open Cybernetics and Systemics Journal, 2008, Volume 2 25 GENOCOP V In 1999, Koziel et al. [9] proposed GENOCOP V, a floating-point genetic algorithm incorporating the homomorphous mapping. Homomorphous Mapping In [9], they considered a mapping T: F [-1,1] n , shown in Fig. (1), where X is the feasible region of the problem and [-1,1] n is an n dimensional hypercube. (1) The mapping T maps some point r X to the origin 0 [-1,1] n , i.e., T: r 0. The solution r is called the basepoint solution of T. (2) The mapping T maps each point x X to a point y [-1,1] n , determined as: T: x y = x r tmax max {x1 r1, x2 r2, ... , xn rn} (2) where tmax is a positive real number such that r + (x r) tmax is on the boundary of X. Fig. (1). Homomorphous mapping T. The mapping T can be executed fast and the local relation of points is preserved.. In particular, if we use the n dimensional hypercube [-1,1] n as the search space in place of the feasible region X, we do not have to prepare special genetic operators to keep the feasibility of solutions since [1,1] n is convex. Thereby, the improvement of the search efficiency of the genetic algorithm is expected by the introduction of the homomorpous mapping. Computational Procedures of GENOCOP V Step 1: Calculate the upper bound and the lower bound for each decision variable xj based on linear constraints and upper-lower bound constraints, and generate individuals satisfying these constraints at random. If there is a solution to satisfy all constraints of the problem, i.e., there is a feasible solution, it becomes the basepoint solution r. If not, repeat the generation of individuals satisfying the above constraints at random several times. Go to step 2. If we can find no feasible solution through all trials, it is judged that the problem is infeasible and the procedure is terminated. Step 2: Generate initial individuals in the n dimensional hypercube [-1,1] n randomly. Go to step 3. Step 3: Calculate a solution in the feasible region X corresponding to each individual in the n dimensional hypercube [-1,1] n by using the inverse mapping T -1 of the homomorphous mapping T based on the basepoint solution r, and evaluate all solutions in X. Go to step 4. Step 4: If the termination conditions are satisfied, the procedure is terminated. Otherwise, go to step 5. Step 5: Apply reproduction operator. Go to step 6. Step 6: Select an operator randomly from among seven operators: uniform mutation, boundary mutation, non-uniform mutation, whole arithmetical crossover, simple arithmetical crossover, whole non-uniform mutation, a version of whole arithmetical crossover (some of them are shown in Fig. (2)), and apply the selected operator. Go to step 3. Fig. (2). Typical operators in step 6. However, there exist the following shortcomings in GENOCOP V. (1) Since GENOCOP V adopts the method to generate the fixed number of solutions in order to find the basepoint solution, it often happens that GENOCOP V cannot start the search because no feasible solution is found even if the problem is feasible. In addition, the precision of (approximate) optimal solutions obtained by GENOCOP V is not good since they greatly depend on the basepoint solution of the homomorphous mapping. (2) It is known that approximate optimal solutions obtained by GENOCOP V greatly depend on the basepoint solution of the homomorphous mapping. Since only one basepoint solution is used in the whole search process in GENOCOP V, the precision of obtained approximate optimal solutions may not be good. 26 The Open Cybernetics and Systemics Journal, 2008, Volume 2 Kato et al. REVISION OF GENOCOP V In this section, we propose the revised GENOCOP V which resolves shortcomings of GENOCOP V mentioned in the previous section. Generation of Initial Basepoint Solution In GENOCOP V [9], we have to obtain at least one feasible solution before the first mapping. However, we often cannot find any feasible solutions as initial feasible solutions since they are searched by randomly generating a given number of solutions which satisfy both upper bound constraints and lower ones. Thereby, the shortcoming (1) will be resolved by using the following optimization problem to minimize the degree of violation of constraints like RGENOCOP III [8].