区间自适应遗传算法优化无约束非线性规划问题

针对无约束非线性规划传统优化方法存在的问题,将区间自适应遗传算法引入无约束非线性规划优化中,算法可以利用当前进化信息,自适应移动搜索区间,找到全局最优解,故可缩短搜索区间长度,提高编码精度,降低算法计算量,解决了传统遗传算法处理优化问题时,给定区间必须包含最优解这一问题,这也是本算法有别于其他优化算法的独特优势,为某些最优解所在区间难以估计的无约束非线性规划问题的优化提供了一条有效可行的途径.系统阐述了区间自适应遗传算法的原理,给出了算法优化无约束非线性规划问题的步骤,以MatlabR2016b仿真方式对算法进行了实例测试,结果表明,方法是一种计算稳定、正确、有效、可靠实用的无约束非线性规划优化方法.     

一种求解半向量二层规划问题的基于KKT背离度量方程的粒子群优化算法

下层多目标规划问题的Pareto最优解的精确性对于成功求解半向量二层规划问题具有决定性作用.本文基于多目标规划问题的KKT背离度量方程,设计了具有确定性终止准则的半向量二层规划问题的粒子群算法.最后,利用线性半向量二层规划算例和非线性半向量二层规划算例进行数值仿真,仿真结果表明,算法中的KKT背离度量方程能有效控制下层问题Pareto最优解的精度,从而确保问题最优解的真实有效性.     

灾情巡视问题的蚁群两目标优化算法研究

分析目前灾情巡视问题求解方法存在的缺陷,归纳出灾情巡视问题两目标优化模型.针对灾情巡视问题模型特点,引入蚁群算法和多目标优化理论,提出两个灾情巡视问题的蚁群两目标优化算法:算法1将灾情巡视问题的道路网络转化为完全图,增加m-1个(m为巡视组数)虚拟巡视起点,将灾情巡视两目标优化问题转化为单旅行商两目标优化问题,然后使用蚁群算法和多目标优化理论进行迭代求解.算法2使用一只蚂蚁寻找一个子回路,m个子回路构成一个灾情巡视可行方案,采用罚函数法和多目标优化理论构建增广两目标优化评价函数,使用g组,共g×m只蚂蚁共同协作来发现灾情巡视问题的最优解.算法特点:①算法1将灾情巡视两目标优化问题转化为单旅行商两目标优化问题,可以充分利用已有蚁群算法求解单旅行商问题的研究成果;②两个算法引入蚁群算法,提高了算法效率;③两个算法克服目前灾情巡视问题的求解方法不严密性缺陷;④两目标优化算法可以为用户提供多个满足约束条件的Pareto组合解,扩大了用户选择范围,增强了算法的适用性.算法测试表明:灾情巡视问题的蚁群两目标优化算法是完全可行和有效的.     

求解双层规划的多目标布谷鸟算法

双层规划是一类具有主从递阶结构的优化问题,属于NP-hard范畴。本文利用KKT条件将双层规划问题转化为等价的单层约束规划问题,通过约束处理技术进一步转化为带偏好双目标无约束优化问题,提出多目标布谷鸟算法求解策略。该算法采用Pareto支配和ε-个体比较准则,充分利用种群中优秀不可行解的信息指导搜索过程;设置外部档案集存储迭代过程中的优秀个体并通过高斯扰动改善外部档案集的质量,周期性替换群体中的劣势个体,引导种群不断向可行域或最优解逼近。数值实验及其参数分析验证了算法的有效性。     

改进的NSGA-Ⅱ算法研究风力机叶片多目标优化

将一种采用精英控制策略和动态拥挤方法用于快速非支配排序遗传算法（NSGA-Ⅱ）,并应用到风力机叶片的优化研究中,获得了一种新颖的风力机叶片多目标优化设计方法．作为应用算例,以设计风速下的功率系数最大和叶片质量最小为优化目标,用该方法设计了5 MW大型风力机叶片．优化结果表明,此算法在处理风力机多目标优化问题取得了良好的效果,给出的是一个Pareto最优解集,而不是传统优化方法追求的单个最优解,为风力机多目标优化设计提供新的思路和通用的算法．     

区间数线性规划问题的解法

基于区间数与实数之间的关系,提出了区间数线性规划的激进最优解,保守最优解的定义.利用约束集之间以及目标函数值之间的关系,在原有区间数线性规划的基础之上,给出了两个求解激进最优解、保守最优解的方法.数值例子验证了该方法的有效性和可行性.     

考虑帕累托最优解的多目标优化进化算法

大多数现有的进化算法在处理多目标优化问题(multi-objective optimization problem,MOP)时会遇到Pareto最优解稀疏的困难,特别是当决策变量的数目很大时,如旨在从大量候选特征中找出小部分特征的特征选择.为此,提出了一种求解大规模稀疏MOP的进化算法.算法考虑Pareto最优解的稀疏性,提出了一种新的种群初始化策略和遗传算子,以保证解的稀疏性.此外,还设计了一个测试套件来评估该算法在大规模稀疏MOP中的性能,实验结果和应用实例证明了该算法在处理大规模稀疏MOP问题上的优越性.     

多目标条件风险值的一种近似求解方法

本文研究了一种求解多目标条件风险值问题的近似方法,首先引入了多个损失函数在对应的置信水平下关于一个证券组合的α-VaR损失值,以及α-CVaR损失值概念．α-CVaR损失值表明了在给定的证券组合于置信水平对应的最小信用风险值的条件期望损失值,那么求出这样的最小条件期望损失值的模型构成了一个求解α-CVaR损失值的多目标问题,它的解就是最小条件期望损失值的有效证券组合,即Pareto弱有效解．为了求解它的Pareto弱有效解,我们引进了损失函数对应的优化问题(SCVaR),可以通过求解非线性规划问题(SCVaR)的最优解近似地刻画α—CVaR损失值,这样使得求解α-CVaR损失值变得容易．     

区间数线性规划及其区间解的研究

区间数线性规划可用于处理含有离散区间数的不确定性优化问题。针对已有算法所求区间解可能包含非可行解的缺陷,基于可能度概念提出了区间数线性规划的有效解、弱有效解、最优解及其解域的定义,给出了改进解法,所得区间解为以上解域的子集。以一个数值模型为例求解,将运算结果与已有算法所得区间解作了对比,说明了改进解法的有效性。     

随机混流U型拆卸线平衡排序问题多目标进化算法优化

针对混流U型拆卸线平衡排序问题,考虑拆卸时间不确定,建立了该问题最小拆卸线平均闲置率、尽早拆卸危害和高需求零部件、最小化平均方向改变次数的多目标优化模型,并提出一种基于分解和动态邻域搜索的混合多目标进化算法(Hybrid Multi-objective Evolutionary Algorithm Based on Decomposition, HMOEA/D)。该算法通过采用弹性任务分配策略、动态邻域结构和动态调整权重以保证解的可行性并搜索得到分布较好的非劣解集。最后,仿真求解实验设计技术(DOE)生成的测试算例,结果表明HMOEA/D较其它算法能得到更接近Pareto最优、分布更好的近似解集。     

Characterizations of solution sets for parametric multiobjective optimization problems

In this article, we investigate the nonemptiness and compactness of the weak Pareto optimal solution set of a multiobjective optimization problem with functional constraints via asymptotic analysis. We then employ the obtained results to derive the necessary and sufficient conditions of the weak Pareto optimal solution set of a parametric multiobjective optimization problem. Our results improve and generalize some known results.     

UNIFORM APPROXIMATION OF A MULTIOBJECTIVE OPTIMIZATION SEQUENCE

In this paper the Pareto efficiency of a uniformly convergent multiobjective optimization sequence is studied. We obtain some relation between the Pareto efficient solutions of a given multiobjective optimization problem and those of its uniformly convergent optimization sequence and also some relation between the weak Pareto efficient solutions of the same optimization problem and those of its uniformly convergent optimization sequence. Besides, under a compact convex assumption for constraints set and a certain convex assumption for both objective and constraint functions, we also get some sufficient and necessary conditions that the limit of solutions of a uniformly convergent multiobjective optimization sequence is the solution of a given multiobjective optimization problem.     

多目标最优化G-恰当有效解集的存在性和连通性

本文证明了非空紧凸集上拟凸多目标最优化问题的G-恰当有效解的存在性．在此基础上，得到了向量目标函数既是似凸又是拟凸的多目标最优化问题的G-恰当有效解集是连通的结论．同时，还给出一个关于Pareto有效解集连通性的新结果．     

Convergence analysis of Tikhonov-type regularization algorithms for multiobjective optimization problems

In this paper, we introduce a vector-valued Tikhonov-type regularization algorithm for an extended-valued multiobjective optimization problem. Under some mild conditions, we prove that any sequence generated by this algorithm converges to a weak Pareto optimal solution of the multiobjective optimization problem. Our results improve and generalize some known results.     

On the calculation of a membership function for the solution of a fuzzy linear optimization problem

In the present paper the fuzzy linear optimization problem (with fuzzy coefficients in the objective function) is considered. Recent concepts of fuzzy solution to the fuzzy optimization problem based on the level-cut and the set of Pareto optimal solutions of a multiobjective optimization problem are applied. Chanas and Kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem, which is based on calculating the sum of lengths of certain intervals. The purpose of this paper is to determine a method for realizing this idea. We derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered.     

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.     

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

G恰当有效点集和G恰当有效解集的连通性

与多目标规划问题的G恰当有效解相应，引进了集合的G恰当有效点的概念，并互研究了G恰当有效点集和G恰当有效解集的连通性．利用所得的结果，还获得多目标规划问题的Pareto有效解集是连通的一个新的结论。     

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.