Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization

The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods.     

A New Hybrid Evolutionary Multiobjective Algorithm Guided by Descent Directions

Hybridization of local search based algorithms with evolutionary algorithms is still an under-explored research area in multiobjective optimization. In this paper, we propose a new multiobjective algorithm based on a local search method. The main idea is to generate new non-dominated solutions by adding a linear combination of descent directions of the objective functions to a parent solution. Additionally, a strategy based on subpopulations is implemented to avoid the direct computation of descent directions for the entire population. The evaluation of the proposed algorithm is performed on a set of benchmark test problems allowing a comparison with the most representative state-of-the-art multiobjective algorithms. The results show that the proposed approach is highly competitive in terms of the quality of non-dominated solutions and robustness.     

A new multiobjective procedure for solving nonconvex environmental/economic power dispatch

In this article, an improved multiobjective chaotic interactive honey bee mating optimization (CIHBMO) is proposed to find the feasible optimal solution of the environmental/economic power dispatch problem with considering operational constraints of the generators. The three conflicting and noncommensurable: fuel cost, pollutant emissions, and system loss, should be minimized simultaneously while satisfying certain system constraints. To achieve a good design with different solutions in a multiobjective optimization problem, Pareto dominance concept is used to generate and sort the dominated and nondominated solutions. Also, fuzzy set theory is used to extract the best compromise solution. The propose method has been individually examined and applied to the standard Institute of Electrical and Electronics Engineers (IEEE) 30‐bus six generator, IEEE 180‐bus 14 generator and 40 generating unit (with valve point effect) test systems. The computational results reveal that the multiobjective CIHBMO algorithm has excellent convergence characteristics and is superior to other multiobjective optimization algorithms. Also, the result shows its great potential in handling the multiobjective problems in power systems. © 2014 Wiley Periodicals, Inc. Complexity 20: 47–62, 2014     

A fast Pareto genetic algorithm approach for solving expensive multiobjective optimization problems

We present a new multiobjective evolutionary algorithm (MOEA), called fast Pareto genetic algorithm (FastPGA), for the simultaneous optimization of multiple objectives where each solution evaluation is computationally- and/or financially-expensive. This is often the case when there are time or resource constraints involved in finding a solution. FastPGA utilizes a new ranking strategy that utilizes more information about Pareto dominance among solutions and niching relations. New genetic operators are employed to enhance the proposed algorithm’s performance in terms of convergence behavior and computational effort as rapid convergence is of utmost concern and highly desired when solving expensive multiobjective optimization problems (MOPs). Computational results for a number of test problems indicate that FastPGA is a promising approach. FastPGA yields similar performance to that of the improved nondominated sorting genetic algorithm (NSGA-II), a widely-accepted benchmark in the MOEA research community. However, FastPGA outperforms NSGA-II when only a small number of solution evaluations are permitted, as would be the case when solving expensive MOPs.     

The influence of the fitness evaluation method on the performance of multiobjective search algorithms

In this paper we are concerned with finding the Pareto optimal front or a good approximation to it. Since non-dominated solutions represent the goal in multiobjective optimisation, the dominance relation is frequently used to establish preference between solutions during the search. Recently, relaxed forms of the dominance relation have been proposed in the literature for improving the performance of multiobjective search methods. This paper investigates the influence of different fitness evaluation methods on the performance of two multiobjective methodologies when applied to a highly constrained two-objective optimisation problem. The two algorithms are: the Pareto archive evolutionary strategy and a population-based annealing algorithm. We demonstrate here, on a highly constrained problem, that the method used to evaluate the fitness of candidate solutions during the search affects the performance of both algorithms and it appears that the dominance relation is not always the best method to use.     

Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning

This paper examines the interpretability-accuracy tradeoff in fuzzy rule-based classifiers using a multiobjective fuzzy genetics-based machine learning (GBML) algorithm. Our GBML algorithm is a hybrid version of Michigan and Pittsburgh approaches, which is implemented in the framework of evolutionary multiobjective optimization (EMO). Each fuzzy rule is represented by its antecedent fuzzy sets as an integer string of fixed length. Each fuzzy rule-based classifier, which is a set of fuzzy rules, is represented as a concatenated integer string of variable length. Our GBML algorithm simultaneously maximizes the accuracy of rule sets and minimizes their complexity. The accuracy is measured by the number of correctly classified training patterns while the complexity is measured by the number of fuzzy rules and/or the total number of antecedent conditions of fuzzy rules. We examine the interpretability-accuracy tradeoff for training patterns through computational experiments on some benchmark data sets. A clear tradeoff structure is visualized for each data set. We also examine the interpretability-accuracy tradeoff for test patterns. Due to the overfitting to training patterns, a clear tradeoff structure is not always obtained in computational experiments for test patterns.     

Bird mating optimizer: An optimization algorithm inspired by bird mating strategies

Thanks to their simplicity and flexibility, evolutionary algorithms (EAs) have attracted significant attention to tackle complex optimization problems. The underlying idea behind all EAs is the same and they differ only in technical details. In this paper, we propose a novel version of EAs, bird mating optimizer (BMO), for continuous optimization problems which is inspired by mating strategies of bird species during mating season. BMO imitates the behavior of bird species metaphorically to breed broods with superior genes for designing optimum searching techniques. On a large set of unimodal and multimodal benchmark functions, BMO represents a competitive performance to other EAs.     

Multiobjective optimization using differential evolution for real-world portfolio optimization

Portfolio optimization is an important aspect of decision-support in investment management. Realistic portfolio optimization, in contrast to simplistic mean-variance optimization, is a challenging problem, because it requires to determine a set of optimal solutions with respect to multiple objectives, where the objective functions are often multimodal and non-smooth. Moreover, the objectives are subject to various constraints of which many are typically non-linear and discontinuous. Conventional optimization methods, such as quadratic programming, cannot cope with these realistic problem properties. A valuable alternative are stochastic search heuristics, such as simulated annealing or evolutionary algorithms. We propose a new multiobjective evolutionary algorithm for portfolio optimization, which we call DEMPO??Differential Evolution for Multiobjective Portfolio Optimization. In our experimentation, we compare DEMPO with quadratic programming and another well-known evolutionary algorithm for multiobjective optimization called NSGA-II. The main advantage of DEMPO is its ability to tackle a portfolio optimization task without simplifications, while obtaining very satisfying results in reasonable runtime.     

Evaluation of the multiobjective ant colony algorithm performances on biobjective quadratic assignment problems

The difficulty of resolving the multiobjective combinatorial optimization problems with traditional methods has directed researchers to investigate new approaches which perform better. In recent years some algorithms based on ant colony optimization (ACO) metaheuristic have been suggested to solve these multiobjective problems. In this study these algorithms have been reported and programmed both to solve the biobjective quadratic assignment problem (BiQAP) instances and to evaluate the performances of these algorithms. The robust parameter sets for each 12 multiobjective ant colony optimization (MOACO) algorithms have been calculated and BiQAP instances in the literature have been solved within these parameter sets. The performances of the algorithms have been evaluated by comparing the Pareto fronts obtained from these algorithms. In the evaluation step, a multi significance test is used in a non hierarchical structure, and a performance metric (P metric) essential for this test is introduced. Through this study, decision makers will be able to put in the biobjective algorithms in an order according to the priority values calculated from the algorithms’ Pareto fronts. Moreover, this is the first time that MOACO algorithms have been compared by solving BiQAPs.     

On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods

In solving multi-objective optimization problems, evolutionary algorithms have been adequately applied to demonstrate that multiple and well-spread Pareto-optimal solutions can be found in a single simulation run. In this paper, we discuss and put together various different classical generating methods which are either quite well-known or are in oblivion due to publication in less accessible journals and some of which were even suggested before the inception of evolutionary methodologies. These generating methods specialize either in finding multiple Pareto-optimal solutions in a single simulation run or specialize in maintaining a good diversity by systematically solving a number of scalarizing problems. Most classical generating methodologies are classified into four groups mainly based on their working principles and one representative method from each group is chosen in the present study for a detailed discussion and for its performance comparison with a state-of-the-art evolutionary method. On visual comparisons of the efficient frontiers obtained for a number of two and three-objective test problems, the results bring out interesting insights about the strengths and weaknesses of these approaches. The results should motivate researchers to design hybrid multi-objective optimization algorithms which may be better than each of the individual methods.     

Two-Sided Pareto Front Approximations

A new approach to derive Pareto front approximations with evolutionary computations is proposed here. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. However, accuracy of such approximations is known only if the Pareto front is known, which makes their usefulness questionable. Here we propose to exploit also elements outside feasible sets to derive pairs of such Pareto front approximations that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between elements of a pair. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.     

Global Multiobjective Optimization Using Evolutionary Algorithms

Since the 60s, several approaches (genetic algorithms, evolution strategies etc.) have been developed which apply evolutionary concepts for simulation and optimization purposes. Also in the area of multiobjective programming, such approaches (mainly genetic algorithms) have already been used (Evolutionary Computation 3(1), 1–16).In our presentation, we consider a generalization of common approaches like evolution strategies: a multiobjective evolutionary algorithm (MOEA) for analyzing decision problems with alternatives taken from a real-valued vector space and evaluated according to several objective functions. The algorithm is implemented within the Learning Object-Oriented Problem Solver (LOOPS) framework developed by the author. Various test problems are analyzed using the MOEA: (multiobjective) linear programming, convex programming, and global programming. Especially for hard problems with disconnected or local efficient regions, the algorithms seems to be a useful tool.     

Robustness to uncertain optimization using scalarization techniques and relations to multiobjective optimization

In this paper, we propose two kinds of robustness concepts by virtue of the scalarization techniques (Benson’s method and elastic constraint method) in multiobjective optimization, which can be characterized as special cases of a general non-linear scalarizing approach. Moreover, we introduce both constrained and unconstrained multiobjective optimization problems and discuss their relations to scalar robust optimization problems. Particularly, optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem, and these solutions are efficient under uniqueness assumptions. Two examples are employed to illustrate those results. Finally, the connections between robustness concepts and risk measures in investment decision problems are also revealed.     

A novel evolution strategy for multiobjective optimization problem

Recent literatures have suggested that multiobjective evolutionary algorithms (MOEAs) can serve as a more exploratory and effective tool in solving multiobjective optimization problems (MOPs) than traditional optimizers. In order to contain a good approximation of Pareto optimal set with wide diversity associated with the inherent characters and variability of MOPs, this paper proposes a new evolutionary approach—(μ, λ) multiobjective evolution strategy ((μ, λ)-MOES). Following the highlight of how to balance proximity and diversity of individuals in exploration and exploitation stages respectively, some cooperative techniques are devised. Firstly, a novel combinatorial exploration operator that develops strong points from Gaussian mutation of proximity exploration and from Cauchy mutation of diversity preservation is elaborately designed. Additionally, we employ a complete nondominance selection so as to ensure maximal pressure for proximity exploitation while a fitness assignment determined by dominance and population diversity information is simultaneous used to ensure maximal diversity preservation. Moreover, a dynamic external archive is introduced to store elitist individuals as well as relatively better individuals and exchange information with the current population when performing archive increase scheme and archive decrease scheme. By graphical presentation and examination of selected performance metrics on three prominent benchmark test functions, (μ, λ)-MOES is found to outperform SPEA-II to some extent in terms of finding a near-optimal, well-extended and uniformly diversified Pareto optimal front.     

A software framework based on a conceptual unified model for evolutionary multiobjective optimization: ParadisEO-MOEO

This paper presents a general-purpose software framework dedicated to the design and the implementation of evolutionary multiobjective optimization techniques: ParadisEO-MOEO. A concise overview of evolutionary algorithms for multiobjective optimization is given. A substantial number of methods has been proposed so far, and an attempt of conceptually unifying existing approaches is presented here. Based on a fine-grained decomposition and following the main issues of fitness assignment, diversity preservation and elitism, a conceptual model is proposed and is validated by regarding a number of state-of-the-art algorithms as simple variants of the same structure. This model is then incorporated into the ParadisEO-MOEO software framework. This framework has proven its validity and high flexibility by enabling the resolution of many academic, real-world and hard multiobjective optimization problems.     

A predictive gradient strategy for multiobjective evolutionary algorithms in a fast changing environment

An essential feature of a dynamic multiobjective evolutionary algorithm (MOEA) is to converge quickly to the Pareto-optimal Set before it changes. In cases where the behavior of the dynamic problem follows a certain trend, convergence can be accelerated by anticipating the characteristics of future changes in the problem. A prediction model is usually used to exploit past information and estimate the location of the new Pareto-optimal Set. In this work, we propose the novel approach of tracking and predicting the changes in the location of the Pareto Set in order to minimize the effects of a landscape change. The predicted direction and magnitude of the next change, known as the predictive gradient, is estimated based on the history of previously discovered solutions using a weighted average approach. Solutions updated with the predictive gradient will remain in the vicinity of the new Pareto-optimal Set and help the rest of the population to converge. The prediction strategy is simple to implement, making it suitable for fast-changing problems. In addition, a new memory technique is introduced to exploit any periodicity in the dynamic problem. The memory technique selects only the more promising stored solutions for retrieval in order to reduce the number of evaluations used. Both techniques are incorporated into a variant of the multi-objective evolutionary gradient search (MO-EGS) and two other MOEAs for dynamic optimization and results indicate that they are effective at improving performance on several dynamic multiobjective test problems.     

A novel elitist multiobjective optimization algorithm: Multiobjective extremal optimization

Recently, a general-purpose local-search heuristic method called extremal optimization (EO) has been successfully applied to some NP-hard combinatorial optimization problems. This paper presents an investigation on EO with its application in numerical multiobjective optimization and proposes a new novel elitist (1 + λ) multiobjective algorithm, called multiobjective extremal optimization (MOEO). In order to extend EO to solve the multiobjective optimization problems, the Pareto dominance strategy is introduced to the fitness assignment of the proposed approach. We also present a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. The proposed approach is validated using five popular benchmark functions. The simulation results indicate that the proposed approach is highly competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered a good alternative to solve numerical multiobjective optimization problems.     

Comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d-dimensional knapsack problems

This study analyzes multiobjective d-dimensional knapsack problems (MOd-KP) within a comparative analysis of three multiobjective evolutionary algorithms (MOEAs): the ε-nondominated sorted genetic algorithm II (ε-NSGAII), the strength Pareto evolutionary algorithm 2 (SPEA2) and the ε-nondominated hierarchical Bayesian optimization algorithm (ε-hBOA). This study contributes new insights into the challenges posed by correlated instances of the MOd-KP that better capture the decision interdependencies often present in real world applications. A statistical performance analysis of the algorithms uses the unary ε-indicator, the hypervolume indicator and success rate plots to demonstrate their relative effectiveness, efficiency, and reliability for the MOd-KP instances analyzed. Our results indicate that the ε-hBOA achieves superior performance relative to ε-NSGAII and SPEA2 with increasing number of objectives, number of decisions, and correlative linkages between the two. Performance of the ε-hBOA suggests that probabilistic model building evolutionary algorithms have significant promise for expanding the size and scope of challenging multiobjective problems that can be explored.     

Scale factor local search in differential evolution

This paper proposes the scale factor local search differential evolution (SFLSDE). The SFLSDE is a differential evolution (DE) based memetic algorithm which employs, within a self-adaptive scheme, two local search algorithms. These local search algorithms aim at detecting a value of the scale factor corresponding to an offspring with a high performance, while the generation is executed. The local search algorithms thus assist in the global search and generate offspring with high performance which are subsequently supposed to promote the generation of enhanced solutions within the evolutionary framework. Despite its simplicity, the proposed algorithm seems to have very good performance on various test problems. Numerical results are shown in order to justify the use of a double local search instead of a single search. In addition, the SFLSDE has been compared with a standard DE and three other modern DE based metaheuristic for a large and varied set of test problems. Numerical results are given for relatively low and high dimensional cases. A statistical analysis of the optimization results has been included in order to compare the results in terms of final solution detected and convergence speed. The efficiency of the proposed algorithm seems to be very high especially for large scale problems and complex fitness landscapes.     

A survey of simulated annealing as a tool for single and multiobjective optimization

This paper presents a comprehensive review of simulated annealing (SA)-based optimization algorithms. SA-based algorithms solve single and multiobjective optimization problems, where a desired global minimum/maximum is hidden among many local minima/maxima. Three single objective optimization algorithms (SA, SA with tabu search and CSA) and five multiobjective optimization algorithms (SMOSA, UMOSA, PSA, WDMOSA and PDMOSA) based on SA have been presented. The algorithms are briefly discussed and are compared. The key step of SA is probability calculation, which involves building the annealing schedule. Annealing schedule is discussed briefly. Computational results and suggestions to improve the performance of SA-based multiobjective algorithms are presented. Finally, future research in the area of SA is suggested.