A continuation approach to mode-finding of multivariate Gaussian mixtures and kernel density estimates

This paper presents a novel method of multi-objective optimization by learning automata (MOLA) to solve complex multi-objective optimization problems. MOLA consists of multiple automata which perform sequential search in the solution domain. Each automaton undertakes dimensional search in the selected dimension of the solution domain, and each dimension is divided into a certain number of cells. Each automaton performs a continuous search action, instead of discrete actions, within cells. The merits of MOLA have been demonstrated, in comparison with a multi-objective evolutionary algorithm based on decomposition (MOEA/D) and non-dominated sorting genetic algorithm II (NSGA-II), on eleven multi-objective benchmark functions and an optimal problem in the midwestern American electric power system which is integrated with wind power, respectively. The simulation results have shown that MOLA can obtain more accurate and evenly distributed Pareto fronts, in comparison with MOEA/D and NSGA-II.     

A new orthogonal evolutionary algorithm based on decomposition for multi-objective optimization

The diversity of solutions is very important for multi-objective evolutionary algorithms to deal with multi-objective optimization problems (MOPs). In order to achieve the goal, a new orthogonal evolutionary algorithm based on objective space decomposition (OEA/D) is proposed in this paper. To be specific, the objective space of an MOP is firstly decomposed into a set of sub-regions via a set of direction vectors, and OEA/D maintains the diversity of solutions by making each sub-region have a solution to the maximum extent. Also, the quantization orthogonal crossover (QOX) is used to enhance the search ability of OEA/D. Experimental studies have been conducted to compare this proposed algorithm with classic MOEA/D, NSGAII, NICA and D2MOPSO. Simulation results on six multi-objective benchmark functions show that the proposed algorithm is able to obtain better diversity and more evenly distributed Pareto fronts than other four algorithms.     

Computing multiobjective Markov chains handled by the extraproximal method

This paper suggests a new method for generating the Pareto front in multi-objective Markov chains, which overcomes some existing drawbacks in multi-objective methods: a fundamental issue is to find strong Pareto policies which are policies whose cost-function value is the closest in Euclidean norm to the utopian point. Each strong Pareto policy is reached when each cost-function, constrained by the strategy of others, cannot improve further its own criterion. Constraints associated to the objective function are implemented formulating the problem as a bi-level optimization approach. We convert the problem into a single level optimization approach by introducing a generalized Lagrangian function to represent the original multi-objective problem in terms of a related nonlinear programming problem. Then, we apply the Tikhonov regularization method to the objective function. The regularization method ensures that all the possible Pareto policies to be generated along the Pareto front are strong Pareto policies. For solving the problem we employ the extra-proximal method. The method effectively approximates to every optimal Pareto point, which in this case is a strong Pareto point, in the Pareto front. The experimental result, applied to the route selection for counter-kidnapping problem, validates the effectiveness and usefulness of the method.     

Evolutionary multiobjective optimization in noisy problem environments

This paper presents a multiobjective evolutionary algorithm (MOEA) capable of handling stochastic objective functions. We extend a previously developed approach to solve multiple objective optimization problems in deterministic environments by incorporating a stochastic nondomination-based solution ranking procedure. In this study, concepts of stochastic dominance and significant dominance are introduced in order to better discriminate among competing solutions. The MOEA is applied to a number of published test problems to assess its robustness and to evaluate its performance relative to NSGA-II. Moreover, a new stopping criterion is proposed, which is based on the convergence velocity of any MOEA to the true Pareto optimal front, even if the exact location of the true front is unknown. This stopping criterion is especially useful in real-world problems, where finding an appropriate point to terminate the search is crucial.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem

Evolutionary algorithms are applied to problems that are not well understood as well as to problems in combinatorial optimization. The analysis of these search heuristics has been started for some well-known polynomial solvable problems. Such analyses are starting points for the analysis of evolutionary algorithms on difficult problems. We present the first runtime analysis of a multi-objective evolutionary algorithm on a NP-hard problem. The subject of our analysis is the multi-objective minimum spanning tree problem for which we give upper bounds on the expected time until a simple evolutionary algorithm has produced a population including for each extremal point of the Pareto front a corresponding spanning tree. These points are of particular interest as they give a 2-approximation of the Pareto front. We show that in expected pseudopolynomial time a population is produced that includes for each extremal point a corresponding spanning tree.     

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。     

ParEGO extensions for multi-objective optimization of expensive evaluation functions

This paper deals with multi-objective optimization in the case of expensive objective functions. Such a problem arises frequently in engineering applications where the main purpose is to find a set of optimal solutions in a limited global processing time. Several algorithms use linearly combined criteria to use directly mono-objective algorithms. Nevertheless, other algorithms, such as multi-objective evolutionary algorithm (MOEA) and model-based algorithms, propose a strategy based on Pareto dominance to optimize efficiently all criteria. A widely used model-based algorithm for multi-objective optimization is Pareto efficient global optimization (ParEGO). It combines linearly the objective functions with several random weights and maximizes the expected improvement (EI) criterion. However, this algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy. Such an innovation allows to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Set solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations. This new algorithm is compared against ParEGO and several MOEA algorithms on a standard benchmark problems. Finally, an automotive engineering problem allowing to illustrate the applicability of the proposed approach is given as an example of a real application: the parameter setting of an indirect tire pressure monitoring system.     

Box-constrained multi-objective optimization: A gradient-like method without “a priori” scalarization

The aim of this paper is the development of an algorithm to find the critical points of a box-constrained multi-objective optimization problem. The proposed algorithm is an interior point method based on suitable directions that play the role of gradient-like directions for the vector objective function. The method does not rely on an “a priori” scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box. The key tool to define the mentioned vector field is the notion of vector pseudogradient. We prove that the limit points of the solutions of the system satisfy the Karush–Kuhn–Tucker (KKT) first order necessary condition for the box-constrained multi-objective optimization problem. These results allow us to develop an algorithm to solve box-constrained multi-objective optimization problems. Finally, we consider some test problems where we apply the proposed computational method. The numerical experience shows that the algorithm generates an approximation of the local optimal Pareto front representative of all parts of optimal front.     

Robustness in multi-objective optimization using evolutionary algorithms

This work discusses robustness assessment during multi-objective optimization with a Multi-Objective Evolutionary Algorithm (MOEA) using a combination of two types of robustness measures. Expectation quantifies simultaneously fitness and robustness, while variance assesses the deviation of the original fitness in the neighborhood of the solution. Possible equations for each type are assessed via application to several benchmark problems and the selection of the most adequate is carried out. Diverse combinations of expectation and variance measures are then linked to a specific MOEA proposed by the authors, their selection being done on the basis of the results produced for various multi-objective benchmark problems. Finally, the combination preferred plus the same MOEA are used successfully to obtain the fittest and most robust Pareto optimal frontiers for a few more complex multi-criteria optimization problems.     

A fast steady-state <Emphasis Type="Italic">ε</Emphasis>-dominance multi-objective evolutionary algorithm

Multi-objective evolutionary algorithms (MOEAs) have become an increasingly popular tool for design and optimization tasks in real-world applications. Most of the popular baseline algorithms are pivoted on the use of Pareto-ranking (that is empirically inefficient) to improve the convergence to the Pareto front of a multi-objective optimization problem. This paper proposes a new ε-dominance MOEA (EDMOEA) which adopts pair-comparison selection and steady-state replacement instead of the Pareto-ranking. The proposed algorithm is an elitist algorithm with a new preservation technique of population diversity based on the ε-dominance relation. It is demonstrated that superior results could be obtained by the EDMOEA compared with other algorithms: NSGA-II, SPEA2, IBEA, ε-MOEA, PESA and PESA-II on test problems. The EDMOEA is able to converge to the Pareto optimal set much faster especially on the ZDT test functions with a large number of decision variables.     

Solving multiobjective,multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive ε-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive ε -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).     

An experimental analysis of evolutionary heuristics for the biobjective traveling purchaser problem

Given a set of markets and a set of products to be purchased on those markets, the Biobjective Traveling Purchaser Problem (2TPP) consists in determining a route through a subset of markets to collect all products, minimizing the travel distance and the purchasing cost simultaneously. As its single objective version, the 2TPP is an NP-hard Combinatorial Optimization problem. Only one exact algorithm exists that can solve instances up to 100 markets and 200 products and one heuristic approach that can solve instances up to 500 markets and 200 products. Since the Transgenetic Algorithms (TAs) approach has shown to be very effective for the single objective version of the investigated problem, this paper examines the application of these algorithms to the 2TPP. TAs are evolutionary algorithms based on the endosymbiotic evolution and other interactions of the intracellular flow interactions. This paper has three main purposes: the first is the investigation of the viability of Multiobjective TAs to deal with the 2TPP, the second is to determine which characteristics are important for the hybridization between TAs and multiobjective evolutionary frameworks and the last is to compare the ability of multiobjective algorithms based only on Pareto dominance with those based on both decomposition and Pareto dominance to deal with the 2TPP. Two novel Transgenetic Multiobjective Algorithms are proposed. One is derived from the NSGA-II framework, named NSTA, and the other is derived from the MOEA/D framework, named MOTA/D. To analyze the performance of the proposed algorithms, they are compared with their classical counterparts. It is also the first time that NSGA-II and MOEA/D are applied to solve the 2TPP. The methods are validated in 365 uncapacitated instances of the TPPLib benchmark. The results demonstrate the superiority of MOTA/D and encourage further researches in the hybridization of Transgenetic Algorithms and Multiobjective Evolutionary Algorithms specially the ones based on decomposition.     

A parallel multiple reference point approach for multi-objective optimization

This paper presents a multiple reference point approach for multi-objective optimization problems of discrete and combinatorial nature. When approximating the Pareto Frontier, multiple reference points can be used instead of traditional techniques. These multiple reference points can easily be implemented in a parallel algorithmic framework. The reference points can be uniformly distributed within a region that covers the Pareto Frontier. An evolutionary algorithm is based on an achievement scalarizing function that does not impose any restrictions with respect to the location of the reference points in the objective space. Computational experiments are performed on a bi-objective flow-shop scheduling problem. Results, quality measures as well as a statistical analysis are reported in the paper.     

Hybrid immune algorithm with Lamarckian local search for multi-objective optimization

Lamarckian learning has been introduced into evolutionary computation as local search mechanism. The relevant research topic, memetic computation, has received significant amount of interests. In this study, a novel Lamarckian learning strategy is designed for improving the Nondominated Neighbor Immune Algorithm, a novel hybrid multi-objective optimization algorithm, Multi-objective Lamarckian Immune Algorithm (MLIA), is proposed. The Lamarckian learning performs a greedy search which proceeds towards the goal along the direction obtained by Tchebycheff approach and generates the improved progenies or improved decision vectors, so single individual will be optimized locally and the newcomers yield an enhanced exploitation around the nondominated individuals in less-crowded regions of the current trade-off front. Simulation results based on twelve benchmark problems show that MLIA outperforms the original immune algorithm and NSGA-II in approximating Pareto-optimal front in most of the test problems. When compared with the state of the art algorithm MOEA/D, MLIA shows better performance in terms of the coverage of two sets metric, although it is laggard in the hypervolume metric.     

Multi-objective and prioritized berth allocation in container ports

This paper considers a berth allocation problem (BAP) which requires the determination of exact berthing times and positions of incoming ships in a container port. The problem is solved by optimizing the berth schedule so as to minimize concurrently the three objectives of makespan, waiting time, and degree of deviation from a predetermined priority schedule. These objectives represent the interests of both port and ship operators. Unlike most existing approaches in the literature which are single-objective-based, a multi-objective evolutionary algorithm (MOEA) that incorporates the concept of Pareto optimality is proposed for solving the multi-objective BAP. The MOEA is equipped with three primary features which are specifically designed to target the optimization of the three objectives. The features include a local search heuristic, a hybrid solution decoding scheme, and an optimal berth insertion procedure. The effects that each of these features has on the quality of berth schedules are studied.     

Hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear systems with fuzzy parameters

This paper proposes a decomposition method for hierarchical generation of α-Pareto optimal solutions in large-scale multi-objective non-linear programming (MONLP) problems with fuzzy parameters in the objective functions and in the constraints (FMONLP). These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept of α-Pareto optimality introduced by extending the ordinary Pareto optimality based on the α-level sets of fuzzy numbers. The decomposition method is based on the principle of decompose the original problem into interdependent sub-problems. In this method, the global multi-objective non-linear problem is decomposed into smaller multi-objective sub-problems. The smaller sub-problems, which obtained solved separately by using the weighting method and through an operative procedure. All these solution are coordinates in such a way that an optimal solution for the global problem achieved. In addition, an interactive fuzzy decision-making algorithm for hierarchical generation of α-Pareto optimal solution through the decomposition method is developed. Finally, two numerical examples given to illustrate the results developed in this paper.     

Annealing-tabu PAES: a multi-objective hybrid meta-heuristic

Most real-life optimization problems require taking into account not one, but multiple objectives simultaneously. In most cases these objectives are in conflict, i.e. the improvement of some objectives implies the deterioration of others. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined, but rather a set of solutions. In the last decade most papers dealing with multi-objective optimization use the concept of Pareto-optimality. The goal of Pareto-based multi-objective strategies is to generate a front (set) of non-dominated solutions as an approximation to the true Pareto-optimal front. However, this front is unknown for problems with large and highly complex search spaces, which is why meta-heuristic methods have become important tools for solving this kind of problem. Hybridization in the multi-objective context is nowadays an open research area. This article presents a novel extension of the well-known Pareto archived evolution strategy (PAES) which combines simulated annealing and tabu search. Experiments on several mathematical problems show that this hybridization allows an improvement in the quality of the non-dominated solutions in comparison with PAES, and also with its extension M-PAES.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

The hypervolume based directed search method for multi-objective optimization problems

We present a new hybrid evolutionary algorithm for the effective hypervolume approximation of the Pareto front of a given differentiable multi-objective optimization problem. Starting point for the local search (LS) mechanism is a new division of the decision space as we will argue that in each of these regions a different LS strategy seems to be most promising. For the LS in two out of the three regions we will utilize and adapt the Directed Search method which is capable of steering the search into any direction given in objective space and which is thus well suited for the problem at hand. We further on integrate the resulting LS mechanism into SMS-EMOA, a state-of-the-art evolutionary algorithm for hypervolume approximations. Finally, we will present some numerical results on several benchmark problems with two and three objectives indicating the strength and competitiveness of the novel hybrid.