解约束优化问题的一种新的罚函数模型

New Penalty Model for Constrained Optimization Problems

罚函数法是进化算法中解决约束优化问题最常用的方法之一,它通过对不可行解进行惩罚使得搜索逐步进入可行域。罚函数常定义为目标函数与惩罚项之和,其缺陷一方面在于此模型的罚因子难以控制,另一方面当目标函数值与惩罚项的函数值的差值很大时,此模型不能有效地区分可行解与不可行解,从而不能有效处理约束。为了克服这些缺点,首先引入了目标满意度函数与约束满意度函数,前者是根据目标函数对解的满意度给出的一个度量,而后者是根据约束违反度对解的满意度给出的一个度量。然后将两者有机结合,定义了一种新的罚函数,给出了一种新的罚函数模型。并且设置了自适应动态罚因子,其随着当前种群质量和进化代数的改变而改变。因此它很易于控制。进一步设计了新的杂交和变异算子,在此基础上提出了解决约束优化问题的一种新的进化算法。通过对6个常用标准测试函数所作的数据仿真实验表明,提出的算法是十分有效的。

Penalty function method is one of the most widely used methods for constrained optimization problems in evolutionary algorithms. It makes the search approach the feasible region gradually by the way to punish the infeasible solutions. The penalty functions are usually defined as the sum of the objective function and the penalty terms. The methods will bring two main drawbacks. Firstly, it is difficult to control penalty parameters, secondly, when the difference between the objective function value and the constrained function value is great, the algorithm can not effectively distinguish feasible from infeasible solutions, and thus can not handle the constraints effectively. To overcome the defects, two satisfaction degree functions defined by the objective function and the constraints function were designed, respectively. A new penalty function was constructed by these two satisfaction degree functions. Moreover, we designed an adaptive penalty factor which is varying with the quality of the population and the number of generations. As a result, the penalty factor can be easily controlled. Thus a new penalty function optimization model was proposed. Furthermore, a new crossover operator and a new mutation operator were designed. Based on these, a new evolutionary algorithm for constrained optimization problems was proposed. The simulations are made on six widely used benchmark problems, and the results indicate the proposed algorithm is very effective.