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Improved Evolutionary Strategy Genetic Algorithm for Nonlinear Programming Problems Hui-xia Zhu, Fu-lin Wang, Wen-tao Zhang, Qian-ting Li School of Engineering, Northeast Agriculture University, Harbin, China ([email protected]) Abstract - Genetic algorithms have unique advantages in dealing with optimization problems. In this paper the main focus is on the improvement of a genetic algorithm and its application in nonlinear programming problems. In the evolutionary strategy algorithm, the optimal group preserving method was used and individuals with low fitness values were mutated. The crossover operator uses the crossover method according to the segmented mode of decision variables. This strategy ensured that each decision variable had the opportunity to produce offspring by crossover, thus, speeding up evolution. In optimizing the nonlinear programming problem with constraints, the correction operator method was introduced to improve the feasible degree of infeasible individuals. MATLAB simulation results confirmed the validity of the proposed method. The method can effectively solve nonlinear programming problems with greatly improved solution quality and convergence speed, making it an effective, reliable and convenient method. genetic algorithm had been widely used in combinatorial optimization, controller's structural parameters optimization etc. fields, and had become one of the primary methods of solving nonlinear planning problems[1-9]. In this paper, the evolution strategy was improved after analyzing the process of the genetic algorithm and the improved algorithm took full advantages of genetic algorithm to solve unconstrained and constrained nonlinear programming problems. In the MATLAB environment， the numerical example showed that the proposed improved genetic algorithm for solving unconstrained and constrained nonlinear programming was effective, and the experiment proved it was a kind of algorithm with calculation stability and better performance. II. NONLINEAR PROGRAMMING PROBLEMS Keywords - nonlinear programming, genetic algorithm, improved evolutionary strategy, correction operator method I. INTRODUCTION Nonlinear programming problem (NPP) had become an important branch of operations research, and it was the mathematical programming with the objective function or constraints being nonlinear functions. There were a variety of traditional methods to solve nonlinear programming problems such as center method, gradient projection method, the penalty function method, feasible direction method, the multiplier method. But these methods had their specific scope and limitations, the objective function and constraint conditions generally had continuous and differentiable request. The traditional optimization methods were difficult to adopt as the optimized object being more complicated. Genetic algorithm overcame the shortcomings of traditional algorithm, it only required the optimization problem could be calculated, eliminating the limitations of optimization problems having continuous and differentiable request, which was beyond the traditional method. It used the forms of organization search, with parallel global search capability, high robustness and strong adaptability, and it could obtain higher efficiency of optimization. The basic idea was first made by Professor John Holland. The ____________________ Natural Science Foundation of China (31071331) The nonlinear programming problems could be divided into unconstrained problems and constrained problems[1][10].We presented the mathematical model here in its general form. The unconstrained nonlinear programming model: min f(X) X En (1) Where, the independent variable X=(x1,x2,…,xn)T was an n dimensional vector (point) in Euclidean space .It was the unconstrained minimization problem that was for the minimum point of the objective function f(X) in En. The constrained nonlinear programming model: min f(X) X En s.t. hi(X)=0, i=1,2,…,m gj(X)≥0,j=1,2,…,l (2) Where, "min" stood for "minimizing" and symbol "s.t" stood for "subject to". It was the unconstrained minimization problem that was for the minimum point of the objective function f(X) in En. Here, hi(X) =0 and gj(X)≥0 were the constrained conditions. For max f(X) =-min[-f(X)], only the minimization problem of objective function was needed to take into consideration without loss of generality. If some constrained conditions were "≤" inequality, they were needed to be multiplied at both ends of the constraints by "-1". So we could only consider the constraint in the form of "≥". III. ANALYSIS AND DESCRIPTION OF THE IMPROVED GENETIC ALGORITHM Based on the simple genetic algorithm, the following gave the analysis design and description of the algorithm which improved the genetic evolution strategy of genetic algorithm. A. Encoding and Decoding We used the binary encoding and multi-parameter cascade encoding. It meant that we made each parameter encoded by means of the binary method and then connected the binary encoded parameters with each other in a certain order to constitute the final code which represented an individual including all parameters. The bit string length depended on the solving precision of specific problems, the higher precision we required, the longer the bit string. If the interval of someone parameter was [A, B] and the precision was c digits after decimal point, then the calculation formula for bit string length was: (B-A) 10c≤2L (3) Here, L took the smallest integer which made the above equation valid. If the interval of someone parameter was [A, B], the corresponding substring in the individual code was bL bL 1bL 2 b2b1 , then its corresponding decoding formula was: L B A (4) X A ( bi 2i 1 ) L 2 1 i 1 B. Production of the Initial Population There were two conditions when producing initial population. One was to solve the unconstrained problem; the other was to solve the constrained problem. Suppose the number of decision variables was n, the population scale was m, ai and bi were lower limit and upper limit of a decision variable respectively. For the unconstrained problem, binary encoding was adopted to randomly produce initial individuals of the population. For constrained problems, the initial population could be selected at random under certain constraint conditions. It also could be produced in the following manner: First, a known initial feasible individual X1(0) was given artificially. It met the following conditions: g j ( X1(0) ) g j ( X11(0) , X12(0) , X13(0) , , X1(0) n )0 The other individuals were produced in the following way[11]: X 2(0) A r2 ( B A) (5) Here, A=(a1, a2, a3,…,an)T, B=(b1, b2, b3,…,bn)T, r2 =( r21, r22, r23,…, r2n)T, random number rij U(0,1). Then checking whether X 2(0) satisfied the constraints or not. If the constraints were satisfied, another individual would produce as X 2(0) . If the constraints were not satisfied, X 2(0) would be corrected by correction operator. C. Correction Operator Method When genetic algorithm was applied to deal with constrained nonlinear programming problems, the core problem was how to treat constraint conditions. Solving it as unconstrained problems at first, checking whether there were constraint violations in the search process. If there were no violations, it indicated that it was a feasible solution; if not, it meant it was not a feasible solution. The traditional method of dealing with infeasible solutions was to punish those infeasible chromosomes or to discard infeasible chromosome. Its essence was to eliminate infeasible solution to reduce the search space in the evolutionary process [12-17]. The improved evolution strategy genetic algorithm used the correction operator method, which selected certain strategy to fix the infeasible solution. Different from the method of penalty function, the correction operator method only used the transform of objective function as the measure of the adaptability with no additional items, and it always returned feasible solution. It had broken the traditional idea, avoided the problem of low searching efficiency because of refusing infeasible solutions and avoided early convergence due to the introduction of punishment factor, and also avoided some problems such as the result considerably deviating constraint area after mutation operation. If there were r linear equations of constraints, and the linear equations’ rank was r < n, all decision variables could be expressed by n-r decision variables. Taking them into inequality group and the objective function, the original n decision variables problem became n-r decision variables problem with only inequality group constraints. So we could only consider problems with only inequality group constraints. The production of initial individuals, offspring produced by crossover operation and individuals after mutation, all were needed to be judged whether they met the constraints, if not, fixed them in time. Such design of genetic operation made solution vectors always bounded in the feasible region. The concrete realization way of correction operator was: Each individual was tested whether it satisfied the constraints. If so, continued the genetic operation; if not, let it approach the former feasible individual (assumed X 1(0) and the former feasible individual should be an inner point). The approaching was an iterated process according to the following formula: X 2(0) X 1(0) ( X 2(0) X 1(0) ) (6) Where, α was step length factor. If it still didn't satisfy the constraint, then the accelerated contraction step length was used, that was α=(1/2)n, here, n was search times. Big step length factor could affect the constraint satisfaction and reduce the repairing effect and even affect the search efficiency and speed, whereas, too small step length factor couldn't play the role of proper correction. So the method of gradually reducing the step length factor could both protect the previous correction result and give full play to correction strategy. Thus X 2(0) was made to feasible individual after some times of iteration, then X 3(0) was produced as X 2(0) and become feasible individual. In the same way, all the needed feasible individuals were produced. For binary genetic algorithm, these feasible individuals were phenotype form of binary genetic algorithm. Real coding individuals were converted into binary string according to the mapping relationship between genotype and phenotype. Then the feasible individuals of binary genetic algorithm were obtained. This kind of linear search way of infeasible individual moving to the direction of feasible individual had the advantage of improving infeasible individual, initiative guiding infeasible individuals to extreme point of population, making the algorithm realize optimization in global space. This paper introduced the correction operator to improve the feasibility of infeasible individuals. This method was simple and feasible. And the treatment on infeasible individuals was also one novelty of improving evolution strategy of genetic algorithm. m ps fi / fi (9) i 1 F. Crossover Operator The number of decision variables might be more in practical problems. Because binary encoding and multiparameter cascade encoding were adopted, the one point crossover would make only one decision variable cross in a certain position in this encoding mode, leaving no crossover for other variables. So the segmented crossover mode of decision variables was used, giving each decision variable the probability pc of single point crossover. Each decision variable had a cross opportunity to produce offspring. This improvement was also another novelty of evolutionary strategy of genetic algorithm. G. Mutation Operator Alleles of some genes were randomly reversed according to mutation probability pm. Parent population individuals and child population individuals after crossover were sorted together according to their fitness values before mutation and only individuals with low fitness values were mutated. Thus not only good schema could avoid being destroyed, but also mutation probability could be appropriately increased, so generating more new individuals. It was good to increase the population's diversity, to traverse all of the state, and to jump out local optimum. H. Population Evolution D. Fitness functions If the objective function was for minimal optimization, the following transformation was applied[18]: c f ( x) Fit ( f ( x)) max 0 f ( x) cmax other (7) Here, cmax was a estimated value which was enough large for the problem. If the objective function was for maximal optimization, the following transformation was applied: f ( x) cmin Fit ( f ( x)) 0 f ( x) cmin other (8) Here, cmin was an estimated value which was enough small for the problem. E. Selection Operator Selection operator used the roulette selection method. The selection probability of individual i: In the process of population evolution, parent population individuals and child population individuals after crossover were put together to form a new temporary population, and the fitness value of each individual in the new temporary population was calculated, m individuals with high fitness values were preserved, then m individuals with low fitness values were mutated and the mutated m individuals and the m previous preserved individuals were put together to form a new temporary population. Thereafter individuals in the new temporary population were sorted according their fitness values and m individuals with high fitness value were selected as the next generation to accomplish the population evolution. The evolution method was based on the traditional elite preserving method, realizing preserving optimal group. The advantage of this method was to reduce possibilities of optimal solution being destroyed by crossover or mutation in the process of evolution. Moreover, premature convergence was avoided which might be present in traditional elite preserving method because all individuals approached one or two individuals with high fitness values quickly. This was another novel place of improving the evolution strategy of genetic algorithm. I. Algorithm Stopping Criteria was Microsoft Windows XP, compile environment was MATLAB 7.11.0 (R2010b). Two criteria were adopted to terminate algorithm: (1)The number of generations was more than a preset value; (2)The difference of fitness value between two successive evolutions was less than or equal to a given precision, namely to meet the condition: | Fitmax - Fitmin |≤ε (10) B. Experimental Results and Analysis In the below table, the interval lower bound was a, the interval upper bound was b, the precision was c digits after decimal point, the population size was m, the maximum evolution generation was T. Example 1: n Here, Fitmax was the individual's maximum fitness value of a population; Fitmin was the individual's minimum fitness value of a population. IV. EXPERIMENTAL DATA AND RESULTS A. Experimental Data and Parameters min f1(x)= xi2 (11) i 1 f1(x) was a continuous, convex, single peak function. It was an unconstrained optimization problem. Only one global minimum in the 0, the minimum was 0. We selected n=2, n=5, n=10 in the simulation experiment to verify the correctness of Improved Evolutionary Strategy Genetic Algorithm (IESGA). 100 times were executed for f1(x) with crossover probability 0.75, mutation probability 0.05, the end precision 0. All runs converged to the optimal solution. Parameter settings and calculation results were shown in Table 1: In the experiment, simulations of two examples were used to validate the correctness of the algorithm and to test the performance of the algorithm. The hardware environment in the experiment were Intel Pentium DualCore [email protected], 2GB RAM. The operating system TABLE I PARAMETER SETTINGS AND CALCULATION RESULTS f1(x) a b c m T Number of generation to obtain the optimal solution for the first time Variable values Optimal solution n=2 -5.12 5.12 6 80 100 45 (0, 0) 0 n=5 -5.12 5.12 6 80 200 145 (0, 0, 0, 0, 0) 0 n=10 -5.12 5.12 6 80 500 350 (-0.000023, 0, -0.000030, 0, 0, 0 , -0.000396, -0.000396, 0, 0.000010) 0 Observing the optimization results in Table 1, Improved Evolutionary Strategy Genetic Algorithm had faster computing speed and higher accuracy, and could robustly convergence to global optimal solution. With the increment of the number of decision variables, the number of generation to obtain the optimal solution for the first time also increased. This accorded with the objective law, and was also correct. Example 2: max f2(x)= -2 x12 +2x1 x2-2 x22 +4x1+6x2 (12) s.t. 2 x12 -2x2≤0 x1 +5x2≤5 x1 , x2≥0 The objective function f2(x) was a quadratic, polynomial function. Under the conditions of inequality, linear and nonlinear constraints, the theoretical optimal value f2(0.658,0.868)=6.613. In the simulation experiment, the crossover probability was 0.75, mutation probability was 0.05, the end of precision was 0, the maximum number of evolution generation was 70, 100 times were executed for f2(x), and all converged to the optimal solution. The comparison of simulation results which used methods of Feasible Direction (FD), Penalty Function (PF)[19] and Improved Evolutionary Strategy Genetic Algorithm (IESGA) was shown in Table 2. TABLE II THE COMPARISION OF SIMULATION RESULTS OF FD, PF AND IESGA f2(x) x1 x2 Optimal solution FD 0.630 0.874 6.544 PF 0.645 0.869 6.566 IESGA 0.658872 0.868225 6.613083 Observing the optimization results in Table 2, the result of using Improved Evolutionary Strategy Genetic Algorithms was better than the two others, and the optimal solution got the theoretical value. This showed that using Improved Evolutionary Strategy Genetic Algorithms to optimize constrained nonlinear programming was correct and effective and it was a reliable and efficient global optimization algorithm. V. CONCLUSIONS (1) The Improved Evolutionary Strategy Genetic Algorithm preserves the optimal groups based on the traditional elite preservation method. The advantage of this method is that it reduces the possibility of optimal solutions being destroyed by crossovers or mutations in the process of evolution. Premature convergence, which may be present in the traditional elite preservation method, is avoided because all individuals quickly converge to one or two individuals with high fitness values. (2) The correction operator breaks the traditional idea and avoids some problems such as low searching efficiency by refusing infeasible solutions, early convergence by introducing a punishment factor and deviation from the constraint area considerably after mutation operation. (3) The combination of the improved evolutionary strategy and the method of correction operator can effectively solve many nonlinear programming problems, greatly improve solution quality and convergence speed, realize the linear search method of moving infeasible individuals towards feasible individuals, and effectively guide infeasible individuals. The disposal of infeasible individuals by the correction operator is simple and effective. It is proved to be an effective, reliable, and convenient method. 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