Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.

THE CHARACTERIZATION OF EFFICIENCY AND SADDLE POINT CRITERIA FOR MULTIOBJECTIVE OPTIMIZATION PROBLEM WITH VANISHING CONSTRAINTS

In this article, we focus to study about modified objective function approach for multiobjective optimization problem with vanishing constraints. An equivalent η-approximated multiobjective optimization problem is constructed by a modification of the objective function in the original considered optimization problem. Furthermore, we discuss saddle point criteria for the aforesaid problem. Moreover, we present some examples to verify the established results.

An Objective Penalty Functions Algorithm for Multiobjective Optimization Problem

By using the penalty function method with objective parameters, the paper presents an interactive algorithm to solve the inequality constrained multi-objective programming (MP). The MP is transformed into a single objective optimal problem (SOOP) with inequality constrains;and it is proved that, under some conditions, an optimal solution to SOOP is a Pareto efficient solution to MP. Then, an interactive algorithm of MP is designed accordingly. Numerical examples show that the algorithm can find a satisfactory solution to MP with objective weight value adjusted by decision maker.

A Survey of Evolutionary Algorithms for Multi-Objective Optimization Problems With Irregular Pareto Fronts

Evolutionary algorithms have been shown to be very successful in solving multi-objective optimization problems(MOPs).However,their performance often deteriorates when solving MOPs with irregular Pareto fronts.To remedy this issue,a large body of research has been performed in recent years and many new algorithms have been proposed.This paper provides a comprehensive survey of the research on MOPs with irregular Pareto fronts.We start with a brief introduction to the basic concepts,followed by a summary of the benchmark test problems with irregular problems,an analysis of the causes of the irregularity,and real-world optimization problems with irregular Pareto fronts.Then,a taxonomy of the existing methodologies for handling irregular problems is given and representative algorithms are reviewed with a discussion of their strengths and weaknesses.Finally,open challenges are pointed out and a few promising future directions are suggested.

Solving Multitrip Pickup and Delivery Problem With Time Windows and Manpower Planning Using Multiobjective Algorithms

The multitrip pickup and delivery problem with time windows and manpower planning(MTPDPTW-MP)determines a set of ambulance routes and finds staff assignment for a hospital. It involves different stakeholders with diverse interests and objectives. This study firstly introduces a multiobjective MTPDPTW-MP(MO-MTPDPTWMP) with three objectives to better describe the real-world scenario. A multiobjective iterated local search algorithm with adaptive neighborhood selection(MOILS-ANS) is proposed to solve the problem. MOILS-ANS can generate a diverse set of alternative solutions for decision makers to meet their requirements. To better explore the search space, problem-specific neighborhood structures and an adaptive neighborhood selection strategy are carefully designed in MOILS-ANS. Experimental results show that the proposed MOILS-ANS significantly outperforms the other two multiobjective algorithms. Besides, the nature of objective functions and the properties of the problem are analyzed. Finally, the proposed MOILS-ANS is compared with the previous single-objective algorithm and the benefits of multiobjective optimization are discussed.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

A Novel Multiobjective Fireworks Algorithm and Its Applications to Imbalanced Distance Minimization Problems

Recently,multimodal multiobjective optimization problems(MMOPs)have received increasing attention.Their goal is to find a Pareto front and as many equivalent Pareto optimal solutions as possible.Although some evolutionary algorithms for them have been proposed,they mainly focus on the convergence rate in the decision space while ignoring solutions diversity.In this paper,we propose a new multiobjective fireworks algorithm for them,which is able to balance exploitation and exploration in the decision space.We first extend a latest single-objective fireworks algorithm to handle MMOPs.Then we make improvements by incorporating an adaptive strategy and special archive guidance into it,where special archives are established for each firework,and two strategies(i.e.,explosion and random strategies)are adaptively selected to update the positions of sparks generated by fireworks with the guidance of special archives.Finally,we compare the proposed algorithm with eight state-of-the-art multimodal multiobjective algorithms on all 22 MMOPs from CEC2019 and several imbalanced distance minimization problems.Experimental results show that the proposed algorithm is superior to compared algorithms in solving them.Also,its runtime is less than its peers'.

Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order??type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.

Lagrangian Relaxation Method for Multiobjective Optimization Methods: Solution Approaches

This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation method that leads to finding the optimal solution to the problem. Our analysis aims to find a suitable technique to generate Lagrangian multipliers, and later these multipliers are used in the relaxation method to solve Multiobjective optimization problems. We propose a search-based technique to generate Lagrange multipliers. In our paper, we choose a suitable and well-known scalarization method that transforms the original multiobjective into a scalar objective optimization problem. Later, we solve this scalar objective problem using Lagrangian relaxation techniques. We use Brute force techniques to sort optimum solutions. Finally, we analyze the results, and efficient methods are recommended.

基于双分类器辅助进化的多目标优化算法

在昂贵多目标优化问题中,常采用代理辅助进化算法以减少真实目标函数的评估次数,但传统的代理辅助进化算法因代理模型计算复杂而运行时间较长.为缩短运行时间,提出基于双分类器辅助进化的多目标优化算法(DC-MOEA),利用两个随机森林分类器,分别预测解的多样性优劣和解的收敛性优劣,选出同时具备优秀收敛性和多样性的解进行真实评估和环境选择.DC-MOEA对决策变量分类并用分类后的数据训练分类器,降低整体复杂度,减少运行时间.通过仿真试验,利用基准测试函数比较在不同问题上的性能,验证本算法在多样性探索和收敛性增强方面的能力.

山河纵横交错的工业园区能源多目标优化模型

由于山河纵横交错的工业园区受复杂地形影响大,导致其能源配置优化难度高。现面向山河纵横交错工业园区,提出将以园区能源配置成本最低为目标函数构建多目标线性优化模型。该模型既考虑传统的多能源约束条件,如供电系统的线路负荷、供水系统的水管网压降等,还考虑了工业园区复杂地形带来的阻隔限制。实验以某工业园区水电能源数据为输入求解模型,结果表明,所提方法能高效获取山河纵横交错的工业园区能源配置最佳方案。

Scalarizations for Approximate Quasi Efficient Solutions in Multiobjective Optimization Problems

In this paper,by reviewing two standard scalarization techniques,a new necessary and sufficient condition for characterizing(ε,.ε)-quasi(weakly)efficient solutions of multiobjective optimization problems is presented.The proposed procedure for the computation of(ε,.ε)-quasi efficient solutions is given.Note that all of the provided results are established without any convexity assumptions on the problem under consideration.And our results extend several corresponding results in multiobjective optimization.

Multidirection Update-Based Multiobjective Particle Swarm Optimization for Mixed No-Idle Flow-Shop Scheduling Problem

The Mixed No-Idle Flow-shop Scheduling Problem(MNIFSP)is an extension of flow-shop scheduling,which has practical significance and application prospects in production scheduling.To improve the efficacy of solving the complicated multiobjective MNIFSP,a MultiDirection Update(MDU)based Multiobjective Particle Swarm Optimization(MDU-MoPSO)is proposed in this study.For the biobjective optimization problem of the MNIFSP with minimization of makespan and total processing time,the MDU strategy divides particles into three subgroups according to a hybrid selection mechanism.Each subgroup prefers one convergence direction.Two subgroups are individually close to the two edge areas of the Pareto Front(PF)and serve two objectives,whereas the other one approaches the central area of the PF,preferring the two objectives at the same time.The MDU-MoPSO adopts a job sequence representation method and an exchange sequence-based particle update operation,which can better reflect the characteristics of sequence differences among particles.The MDU-MoPSO updates the particle in multiple directions and interacts in each direction,which speeds up the convergence while maintaining a good distribution performance.The experimental results and comparison of six classical evolutionary algorithms for various benchmark problems demonstrate the effectiveness of the proposed algorithm.

基于蚁群优化解决传感器网络中的能量洞问题

基于多跳的无线传感器网络,越靠近sink的传感器节点因需要转发更多的数据,其能量消耗就越快,从而在sink周围形成了一种称为"能量洞"的现象."能量洞"问题会导致整个网络由于内部节点能量过早耗尽而结束寿命,同时,网络中离sink较远的节点仍有大量能量剩余.研究"能量洞"现象,基于改进的分级环模型,总结出调节各环内节点的数据传输距离是实现网络节能的有效方法.证明搜索各区域最优的传输距离是一个多目标优化问题,即是NP难问题.从而提出一种基于蚁群优化的分布式算法,各区域根据其节点分布情况自适应地探索近似最优的传输距离,延长网络寿命.模拟实验结果表明,该算法在较短的时间内能够收敛到合理的解,并且得到的网络寿命接近于理想情况下的最优时间,与现有的类似算法相比,该算法提供了更长的网络寿命,并能适用于非均匀节点分布情况.

基于Pareto熵的多目标粒子群优化算法

粒子群优化算法因形式简洁、收敛快速和参数调节机制灵活等优点,同时一次运行可得到多个解,且能逼近非凸或不连续的Pareto最优前端,因而被认为是求解多目标优化问题最具潜力的方法之一.但当粒子群优化算法从单目标问题扩展到多目标问题时,Pareto最优解集的存储与维护、全局和个体最优解的选择以及开发与开采的平衡等问题亦随之出现.通过目标空间变换方法,采用Pareto前端在被称为平行格坐标系统的新目标空间中的分布熵及差熵评估种群的多样性及进化状态,并以此为反馈信息来设计进化策略,使得算法能够兼顾近似Pareto前端的收敛性和多样性.同时,引入格占优和格距离密度的概念来评估Pareto最优解的个体环境适应度,以此建立外部档案更新方法和全局最优解选择机制,最终形成了基于Pareto熵的多目标粒子群优化算法.实验结果表明:在IGD性能指标上,与另外8种对等算法相比,该算法在由ZDT和DTLZ系列组成的12个多目标测试问题集中表现出了显著的性能优势.

基于极大极小距离密度的多目标微分进化算法

微分进化(differential evolution)是一种新的简单而有效的直接全局优化算法,并在许多领域得到了成功应用.提出了基于极大极小距离密度的多目标微分进化算法.新算法定义了极大极小距离密度,给出了基于极大极小距离密度的Pareto候选解集的维护方法,保证了非劣解集的多样性.并根据个体间的Pareto支配关系和极大极小距离密度改进了微分进化的选择操作,保证了算法的收敛性,实现了利用微分进化算法求解多目标优化问题.通过对5个ZDT测试函数、两个高维测试函数的实验及与其他多目标进化算法的对比和分析,验证了新算法的可行性和有效性.

混合并行机调度问题的多目标优化模型及算法

针对生产工序的合并造成一种串并联共存的生产布局,研究了一种特殊的混合并行机调度问题,并考虑以最小化总流水时间和最小化总延迟工件数量为目标的多目标调度问题,建立了混合整数规划模型.针对模型特点,设计了一种改进的非支配排序遗传算法进行求解,采用基于启发式方法的初始种群生成方式以提高种群的质量和多样性,并引入一种局域搜索策略以改善求解算法所获得的非支配解的质量及分布性.通过对大量数值算例进行仿真实验,并与典型的多目标优化算法进行比较,结果表明所提出的模型和算法在收敛性、分布性及极端点质量方面均具有优势,能够较好的解决多目标混合并行机调度问题.

Pareto最大最小蚂蚁算法及其在热轧批量计划优化中的应用

针对双目标旅行商问题提出了基于Pareto概念的最大最小蚂蚁算法(P--MMAS).通过重新设计状态转移策略、信息素更新策略及局部搜索策略,同时引入基于自适应网格的多样性保持策略与信息素平滑机制,使算法能够快速搜索到在目标空间上均匀分布的近似Pareto前端.通过在6个标准测试函数上的实验及在热轧批量计划优化中的应用,表明P--MMAS具有良好的优化性能及实用性.

基于多目标遗传算法求解时间窗车辆路径问题

有别于传统的单目标方法,将带时间窗约束的车辆路径问题描述成为一个多目标最优化问题,并为之提出了一种多目标遗传算法。在算法中设计了擂台法则作为构造非支配集的方法,提出了可变爬山率的局部爬山法,并通过将组合种群分成多层非支配集来实现精英保留策略。实验结果表明,该算法能有效地求解车辆路径问题并且为决策者提供了强有力的决策支持。

基于自适应分解的多任务协作型昂贵多目标优化算法

现实世界的工程优化问题通常需要同时优化多个冲突的目标,且这些目标函数的评估由于依赖仿真、物理实验而十分昂贵,这类问题被称为昂贵多目标优化问题.使用机器学习方法建立代理模型用于估计候选解的目标函数值是求解此类问题的一种有效手段.高斯代理模型适用于训练样本数较少的中小规模问题,且能提供评估的不确定性,因此常作为代理模型被应用于昂贵优化.分解是处理多目标优化问题的一种有效手段.一个多目标优化问题可被分解为多个单目标优化子问题,且多个子问题可被进一步划分为代理模型学习的一个目标任务.现有基于分解的昂贵多目标优化算法大多将固定数量的子问题静态地划分到同一任务,从而构造多个固定任务并对其建立多任务高斯代理模型进行求解.这未能充分利用数据的相关信息动态反映出任务间的相关性,限制了多任务高斯过程模型的预测精度以及优化算法的最终性能.为此,本文提出了一种自适应多任务多种群协作搜索算法(AMMCS).AMMCS使用相似性指标实时度量已评估的解集,获得子问题间的相关性,从而自适应地划分任务,提升多任务模型的预测质量.此外,AMMCS使用一个解集(种群)优化一个任务,并通过多种群的协作搜索实现多任务高斯模型的批量优化,提高了采样效率,提升了算法的收敛效率.通过AMMCS与六个代理辅助进化算法进行多组实验对比和分析,显示了AMMCS具有良好的性能.我们同时也设计实验验证了算法中自适应分解以及多种群协作搜索的有效性.