几类动态与静态优化问题的进化算法
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摘要
进化算法的出现为许多复杂优化问题的求解提供了新的思路,由于进化算法具有的智能性、通用性、稳健性、本质并行性和全局搜索能力,已在各个静态优化领域得到了成功的应用。近几年来,利用进化算法求解动态优化问题已成为进化计算领域一个新的研究方向。本文主要对几类动态和静态优化问题进行了系统深入的研究,针对不同类型提出了不同的进化算法进行求解,主要工作如下:
1.在目标函数随时间连续变化的假设下,建立了解动态无约束多目标优化问题(DUMOP)的一种静态优化模型,同时给出了解新模型的进化算法。该方法首先将DUMOP的时间变量区间进行了等间隔分割,在每个子区间上把DUMOP近似为静态多目标优化问题。其次,为了提高算法的有效性,进一步将每个静态多目标优化问题转化成了双目标优化问题。这样,原来的DUMOP就被近似地转化成了一系列两个目标的静态优化问题。理论分析和计算机仿真表明新算法在不同环境下能够找到一组范围广、分布均匀且数量充足的Pareto最优解。
2.构建了包含任意个子目标函数的动态约束多目标优化问题的一种动态双目标优化模型,同时给出了求解模型的一种进化算法。数值实验结果表明新算法对动态约束多目标优化问题的求解是有效的。
3.提出了一种基于核分布估计的动态多目标优化进化算法。当探测到问题环境发生改变时,算法利用以前搜索到的有用解信息对下一环境进化种群中的个体进行近似估计,产生新的进化种群。仿真结果表明新算法能有效快速跟踪并以较小的计算量求出动态多目标优化问题质量较好的Pareto最优解。
4.研究了一类时间取值于离散空间,自变量的维数随时间的变化而发生变化的动态多目标优化问题的PSO算法。
5.给出了解带约束的静态多目标优化问题的一种新进化算法。该方法定义了个体的Pareto累积序值和个体的约束度,利用这两个定义给出了一种新的适应度函数和带偏好的选择算子,从而对种群中的个体进行评估或排序时无需特别关心个体是否可行,避免了罚函数选择参数的困难,最后用标准的Benchmark函数对算法的性能进行了测试,与目前公认的有效算法的比较结果表明所给算法是有效可行的。
6.研究了一类动态非线性约束优化问题的新解法。该方法将动态非线性约束优化问题的约束条件引入到问题的目标中来,从而将原问题转化成了无约束的动态多目标优化问题,针对转化后的优化问题提出了一种新的进化算法,同时给出了算法的收敛性证明。最后的数值仿真结果表明新算法能够有效跟踪并求出动态非线性约束优化问题的最优解或近似最优解。
Because of its intelligence, wide applicability, robustness, global search ability and parallelism, evolutionary algorithm provides a new tool for complex optimization problems and has been widely used in many static optimization fields. In recent years, using evolutionary algorithm to solve dynamic optimization problems has become a new research field. In this dissertation, studies are mainly focused on several kinds of complex dynamic and static optimization problems, and new evolutionary algorithms are proposed for these problems. The main contributions of this thesis can be summarized as follows:
1. When the objective functions of dynamic unconstrained multi-objective optimization problems (DUMOP) are continuously changing with time, a static optimization model and a new evolutionary algorithm for DUMOP are proposed. Firstly, the time variable period of DUMOP is divided into several equal subperiods. In each subperiod, the DUMOP is seen as a static multi-objective optimization problem (SMOP) by taking the time parameter fixed. Second, to decrease the amount of computation and efficiently solve the SMOP, each SMOP is transformed into a two-objective optimization problem. Thus, the original DUMOP is approximately transformed into several two-objective optimization problems. The theoretic analysis and the simulation results show that the proposed algorithm is effective and can find high quality solution set in varying-environment in terms of convergence, diversity, and the distribution of the obtained Pareto optimal solutions.
2. A dynamic bi-objective optimization model for dynamic constrained multiobjective optimization problems with any number of objective functions is given, and a new evolutionary algorithm for it is proposed. The simulation results indicate that the proposed algorithm is effective for solving dynamic constrained multi-objective optimization problems.
3. A dynamic multiobjective optimization evolutionary algorithm based on core estimation of distribution is presented. When a change in the environment is detected, the method uses the collected information from the previous search to predict the location of individuals in the next environment and an initial population in the new environment is generated. The simulation results show that the proposed algorithm can effectively track and quickly obtain the Pareto optimal solutions with smaller amount of computation.
4. For a special class of dynamic multiobjective optimization problems, in which the time variable is defined on discrete space and the dimension of independent variable changes with the time, a new dynamic multiobjective optimization PSO algorithm is proposed.
5. For static constrained multi-objective optimization problems, a new evolutionary algorithm is proposed. The Pareto summation rank value and the scalar constraint violation of the individual are firstly defined. Then, based on these two definitions, a new fitness function and a preference selection operator are presented with following properties: when individuals are evaluated or ranked, it is unnecessary to care about the feasibility of the individuals. It is a penalty-parameterless constraint-handling approach. Furthermore, the convergence of the proposed algorithm is proved, and the computer simulations are made and the results demonstrate the effectiveness of the proposed algorithm.
6. A new method for dynamic nonlinear constrained optimization problems (DNCOP) is presented. First, inspired from the idea of multiobjective optimization, the constraints of DNCOP are transformed into one of the objective functions and thus DNCOP is transformed into unconstrained dynamic multiobjective optimization problems. For the transformed problem, a new convergent multiobjective evolutionary algorithm is proposed. The simulation results indicate that the proposed algorithm can effectively track and obtain the optimal solutions or approximately optimal solutions of DNCOP.
1.在目标函数随时间连续变化的假设下,建立了解动态无约束多目标优化问题(DUMOP)的一种静态优化模型,同时给出了解新模型的进化算法。该方法首先将DUMOP的时间变量区间进行了等间隔分割,在每个子区间上把DUMOP近似为静态多目标优化问题。其次,为了提高算法的有效性,进一步将每个静态多目标优化问题转化成了双目标优化问题。这样,原来的DUMOP就被近似地转化成了一系列两个目标的静态优化问题。理论分析和计算机仿真表明新算法在不同环境下能够找到一组范围广、分布均匀且数量充足的Pareto最优解。
2.构建了包含任意个子目标函数的动态约束多目标优化问题的一种动态双目标优化模型,同时给出了求解模型的一种进化算法。数值实验结果表明新算法对动态约束多目标优化问题的求解是有效的。
3.提出了一种基于核分布估计的动态多目标优化进化算法。当探测到问题环境发生改变时,算法利用以前搜索到的有用解信息对下一环境进化种群中的个体进行近似估计,产生新的进化种群。仿真结果表明新算法能有效快速跟踪并以较小的计算量求出动态多目标优化问题质量较好的Pareto最优解。
4.研究了一类时间取值于离散空间,自变量的维数随时间的变化而发生变化的动态多目标优化问题的PSO算法。
5.给出了解带约束的静态多目标优化问题的一种新进化算法。该方法定义了个体的Pareto累积序值和个体的约束度,利用这两个定义给出了一种新的适应度函数和带偏好的选择算子,从而对种群中的个体进行评估或排序时无需特别关心个体是否可行,避免了罚函数选择参数的困难,最后用标准的Benchmark函数对算法的性能进行了测试,与目前公认的有效算法的比较结果表明所给算法是有效可行的。
6.研究了一类动态非线性约束优化问题的新解法。该方法将动态非线性约束优化问题的约束条件引入到问题的目标中来,从而将原问题转化成了无约束的动态多目标优化问题,针对转化后的优化问题提出了一种新的进化算法,同时给出了算法的收敛性证明。最后的数值仿真结果表明新算法能够有效跟踪并求出动态非线性约束优化问题的最优解或近似最优解。
Because of its intelligence, wide applicability, robustness, global search ability and parallelism, evolutionary algorithm provides a new tool for complex optimization problems and has been widely used in many static optimization fields. In recent years, using evolutionary algorithm to solve dynamic optimization problems has become a new research field. In this dissertation, studies are mainly focused on several kinds of complex dynamic and static optimization problems, and new evolutionary algorithms are proposed for these problems. The main contributions of this thesis can be summarized as follows:
1. When the objective functions of dynamic unconstrained multi-objective optimization problems (DUMOP) are continuously changing with time, a static optimization model and a new evolutionary algorithm for DUMOP are proposed. Firstly, the time variable period of DUMOP is divided into several equal subperiods. In each subperiod, the DUMOP is seen as a static multi-objective optimization problem (SMOP) by taking the time parameter fixed. Second, to decrease the amount of computation and efficiently solve the SMOP, each SMOP is transformed into a two-objective optimization problem. Thus, the original DUMOP is approximately transformed into several two-objective optimization problems. The theoretic analysis and the simulation results show that the proposed algorithm is effective and can find high quality solution set in varying-environment in terms of convergence, diversity, and the distribution of the obtained Pareto optimal solutions.
2. A dynamic bi-objective optimization model for dynamic constrained multiobjective optimization problems with any number of objective functions is given, and a new evolutionary algorithm for it is proposed. The simulation results indicate that the proposed algorithm is effective for solving dynamic constrained multi-objective optimization problems.
3. A dynamic multiobjective optimization evolutionary algorithm based on core estimation of distribution is presented. When a change in the environment is detected, the method uses the collected information from the previous search to predict the location of individuals in the next environment and an initial population in the new environment is generated. The simulation results show that the proposed algorithm can effectively track and quickly obtain the Pareto optimal solutions with smaller amount of computation.
4. For a special class of dynamic multiobjective optimization problems, in which the time variable is defined on discrete space and the dimension of independent variable changes with the time, a new dynamic multiobjective optimization PSO algorithm is proposed.
5. For static constrained multi-objective optimization problems, a new evolutionary algorithm is proposed. The Pareto summation rank value and the scalar constraint violation of the individual are firstly defined. Then, based on these two definitions, a new fitness function and a preference selection operator are presented with following properties: when individuals are evaluated or ranked, it is unnecessary to care about the feasibility of the individuals. It is a penalty-parameterless constraint-handling approach. Furthermore, the convergence of the proposed algorithm is proved, and the computer simulations are made and the results demonstrate the effectiveness of the proposed algorithm.
6. A new method for dynamic nonlinear constrained optimization problems (DNCOP) is presented. First, inspired from the idea of multiobjective optimization, the constraints of DNCOP are transformed into one of the objective functions and thus DNCOP is transformed into unconstrained dynamic multiobjective optimization problems. For the transformed problem, a new convergent multiobjective evolutionary algorithm is proposed. The simulation results indicate that the proposed algorithm can effectively track and obtain the optimal solutions or approximately optimal solutions of DNCOP.
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