Benchmarking is essential to the effective conduct of algorithm development and selection. However, current benchmark suites in constrained multiobjective optimisation, consisting of constrained multiobjective optimisation problems (CMOPs), are lacking in many areas, particularly regarding their diversity and ability to differentiate among algorithms. This limits the ability to adequately conduct benchmarking in this domain. Despite the inadequacies of existing benchmark suites, there do exist CMOPs which are satisfactory according to certain criteria. This paper proposes a benchmark suite called the Selected Test Problems (STEP) suite. It is derived from a methodology for selecting the satisfactory CMOPs from the set of all CMOPs provided by the existing benchmark suites. The STEP suite is analysed in both the feature and performance spaces and is found to be diverse and able to distinguish among algorithms, in addition to other desirable characteristics.

Constrained multiobjective optimization has gained much interest in the past few years. However, constrained multiobjective optimization problems (CMOPs) are still unsatisfactorily understood. Consequently, the choice of adequate CMOPs for benchmarking is difficult and lacks a formal background. This paper takes a step towards addressing this issue by exploring CMOPs from a performance space perspective. First, it presents a novel performance assessment approach designed explicitly for constrained multiobjective optimization. This methodology offers a first attempt at simultaneously measuring the performance in approximating the Pareto front and constraint satisfaction. Secondly, it proposes an approach to measure the capability of the given optimization problem to differentiate among algorithm performances. Finally, this approach is used to compare eight frequently used artificial test suites of CMOPs. The experimental results reveal which suites are more efficient in discerning between four well-known multiobjective optimization algorithms.

A research problem studied extensively in recent years is the prediction of optimization algorithm performance. A common approach is using the landscape features of optimization problems to train machine learning models. These models are then used to predict algorithm performance. Due to the small number of constrained multiobjective optimization problems (CMOPs) available for benchmarking, training a machine learning model to predict algorithm performance is a hard task. To address this issue, this study uses the functions from the bbob and bbob-constrained benchmark problems to generate new CMOPs. These are then used as additional training examples for the machine learning models. Given the large number of generated CMOPs, the experiments in this study are limited to those with two objectives and two variables. The obtained results are promising. Using additional problems in the training phase improves the predictions in half of the defined classification tasks.

The benchmark problem of lunar lander landing site selection is a nonlinear constrained optimization problem introduced as a challenge for the 2018 Evolutionary Computation Symposium competition. In the single-objective variant, the task is to find the latitude and the longitude of the landing site maximizing the total communication time with Earth, and satisfying constraints for consecutive shade days and the landing site inclination angle. The requirements of the multiobjective variant are to maximize the total communication time, and minimize the consecutive shade days and the landing site inclination angle, while satisfying the constraints for the latter two objectives. We reexamine the problem by first characterizing it by means of problem landscape analysis, then performing a comparative study of algorithm performance, and finally relating the results of the two phases. The characterization relies on both the features known from the literature and the recently constructed ones for analyzing constrained and multiobjective problem landscapes, while the algorithmic study includes several single- and multiobjective evolutionary algorithms. The findings are insightful and allow for a better understanding of both the problem and the optimization algorithm behavior. They help identify appropriate algorithms for landing site selection and are advantageous for solving comparable problem instances.

Despite the increasing interest in constrained multiobjective optimization in recent years, constrained multiobjective optimization problems (CMOPs) are still insufficiently understood and characterized. For this reason, the selection of appropriate CMOPs for benchmarking is difficult and lacks a formal background. We address this issue by extending landscape analysis to constrained multiobjective optimization. By employing four exploratory landscape analysis techniques, we propose 29 landscape features (of which 19 are novel) to characterize CMOPs. These landscape features are then used to compare eight frequently used artificial test suites against a recently proposed suite consisting of real-world problems based on physical models. The experimental results reveal that the artificial test problems fail to adequately represent some realistic characteristics, such as strong negative correlation between the objectives and the overall constraint violation. Moreover, our findings show that all the studied artificial test suites have advantages and limitations, and that no “perfect” suite exists. Additionally, the effectiveness of the proposed features at predicting algorithm performance is demonstrated for two multiobjective optimization algorithms. Benchmark designers can use the obtained results to select or generate appropriate CMOP instances based on the characteristics they want to explore.

Cyclone dust separators (cyclones for short) are devices used to remove dispersed particles from a flue gas. They are widely applied because of their simple structure, low cost, ease of operation, and ability to operate in harsh environments. In contrast to previous work where the cyclone dust separator design was mainly regarded as an optimization problem with box constraints, we approach it as an optimization task involving the maximization of the collection efficiency and the minimization of the pressure drop, while regarding both box constraints and geometric constraints that limit the designs. We study three high-efficiency cyclone configurations, define four test instances for each configuration, and analyze the feasibility ratios resulting from the constraints. In addition, we investigate the performance of three multiobjective optimization algorithms on this design problem. We analyze the results, both in the objective and the decision space, investigate the effect of the decision space size on the solution quality, and gain insights into problem properties relevant to cyclone designers. Finally, we illustrate the results with selected optimized cyclone designs.

In multiobjective optimization, many techniques are used to visualize the results, ranging from traditional general-purpose data visualization techniques to approaches tailored to the specificities of multiobjective optimization. The number of specialized approaches rapidly grows in the recent years. To assist both the users and developers in this field, we propose a taxonomy of methods for visualizing Pareto front approximations. It builds on the nature of the visualized data and the properties of visualization methods rather than on the employed visual representations. It covers the methods for visualizing individual approximation sets resulting from a single algorithm run as well as multiple approximation sets produced in repeated runs. The proposed taxonomy categories are characterized and illustrated with selected examples of visualization methods. We expect that proposed taxonomy will be insightful to the multiobjective optimization community, make the communication among the participants easier and help focus further development of visualization methods.

In evolutionary multiobjective optimization, it is very important to be able to visualize approximations of the Pareto front (called approximation sets) that are found by multiobjective evolutionary algorithms. While scatter plots can be used for visualizing 2-D and 3-D approximation sets, more advanced approaches are needed to handle four or more objectives. This paper presents a comprehensive review of the existing visualization methods used in evolutionary multiobjective optimization, showing their outcomes on two novel 4-D benchmark approximation sets. In addition, a visualization method that uses prosection (projection of a section) to visualize 4-D approximation sets is proposed. The method reproduces the shape, range, and distribution of vectors in the observed approximation sets well and can handle multiple large approximation sets while being robust and computationally inexpensive. Even more importantly, for some vectors, the visualization with prosections preserves the Pareto dominance relation and relative closeness to reference points. The method is analyzed theoretically and demonstrated on several approximation sets.

In this paper, we present AMS-DEMO, an asynchronous master-slave implementation of DEMO, an evolutionary algorithm for multi-objective optimization. AMS-DEMO was designed for solving time-intensive problems efficiently on both homogeneous and heterogeneous parallel computer architectures. The algorithm is used as a test case for the asynchronous master-slave parallelization of multi-objective optimization that has not yet been thoroughly investigated. Selection lag is identified as the key property of the parallelization method, which explains how its behavior depends on the type of computer architecture and the number of processors. It is arrived at analytically and from the empirical results. AMS-DEMO is tested on a benchmark problem and a time-intensive industrial optimization problem, on homogeneous and heterogeneous parallel setups, providing performance results for the algorithm and an insight into the parallelization method. A comparison is also performed between AMS-DEMO and generational master-slave DEMO to demonstrate how the asynchronous parallelization method enhances the algorithm and what benefits it brings compared to the synchronous method.

The purpose of alternative energy supply systems is to produce electrical energy at the location of its consumption, independently from the supply grid, and exploiting renewable energy sources, such as sunlight and wind. Rapidly changing conditions on energy markets and strengthening environmental requirements make alternative energy supply systems more and more favored. However, designing such systems is a hard optimization problem because of numerous decision variables, conflicting criteria, and complex evaluation of the candidate designs. There are reports on techno-economic optimization of alternative energy supply systems in the literature that search for optimal system configurations maximizing technical performance and minimizing the overall costs. The authors employ stochastic optimization methods integrated with numerical models of the energy supply systems, and handle multiobjective optimization problems in the singleobjective manner through the weighted sum approach or transformation of selected criteria into constraints. This paper describes a Pareto multiobjective approach to techno-economic optimization of an alternative energy supply system based on differential evolution for multiobjective optimization. The paper reviews the related work, introduces the applied methodology and the considered problem, describes the experimental setup and the performed numerical experiments, and reports on the optimization results.

To address the needs of rapidly changing energy markets, an energy data management system capable of supporting higher utilization of renewable energy sources is being developed. The system receives flexible offers from producers and consumers of energy, aggregates them on a regional level and schedules the aggregated flexible offers to balance forecast energy supply and demand. This paper focuses on formulating and solving the optimization problem of scheduling aggregated flexible offers within such a system. Three metaheuristic scheduling algorithms (a randomized greedy search, an evolutionary algorithm and a hybrid between the two) tailored to this problem are introduced and their performance is assessed on a benchmark test problem and two realistic problems. The best results are achieved by the evolutionary algorithms, which can efficiently handle thousands of aggregated flex-offers.

In ideal multiobjective optimization, the result produced by an optimizer is a set of nondominated solutions approximating the Pareto optimal front. Visualization of this approximation set can help assess its quality as well as present various features of the problem. Most often, scatter plots are used to visualize 2D and 3D approximation sets, while no scatter plot equivalent exists for visualization in higher dimensions. This paper presents a method for visualizing 4D approximation sets which performs dimension reduction using prosections (projections of a section). The method yields a prosection matrix---a matrix of intuitive 3D scatter plots that well reproduce the shape, range and distribution of vectors in the observed approximation set. The performance of visualization with prosections is analyzed theoretically and demonstrated on two examples with approximation sets of state-of-the-art test optimization problems.

Tuning parameters of an evolutionary algorithm is the essential phase of a problem solving process since the parameter values significantly influence the algorithm efficiency. A traditional parameter tuning approach finds a setting of parameter values that is then used for solving various problem instances. Clearly, such parameter values may not perform well on specific problem instances. This paper suggests finding several parameter settings which are suitable for specific problem instances. However, this is not aimed at the level of each individual instance, but rather for specific types of problem instances. A new problem instance can then be solved using the tuned parameter values for its type. We demonstrate the approach by tuning parameters of an evolutionary algorithm for commodity transportation optimization with very heterogeneous problem instances. Numerical experiments show that the procedure improves the algorithm performance. Moreover, the analysis of empirical results reveals that there exist relations between the tuned parameter values and that they vary over types of problem instances.

Optimizing combinations of placements of parts, known as markers, is an important preparatory step in order-based industrial production of clothes. Given a work order in the form of a matrix of pieces in size numbers and designs, the task is to find a list of combinations of size numbers to complete the work order. The outcome of this step influences the number of cut out pieces, the amount of material used in the production phase, and the speed of the work order processing. The optimization task is demanding since a number of factors affect production costs and several conflicting criteria can be involved in marker assessment. We consider minimum number of markers per work order as an optimization criterion and transform the problem into the knapsack problem which is then solved with several variants of an evolutionary algorithm. Numerical experiments are performed on real problem instances from industrial clothes production and the results compare favorably with those produced by the algorithm regularly used in practice.

The paper presents a new distributed metaheuristic algorithm in an optimal control problem related to the cooling process in the continuous casting of steel. The optimization task is to tune 18 coolant flows in the caster secondary cooling system to achieve the target surface temperatures along the slab. Sequential search algorithms are proved inefficient for this problem because they take too much time to compute an appropriate solution. For this reason a new distributed search algorithm based on stigmergy perceived in ant colony was developed. The algorithm was run on the grid that allows us to solve this optimization problem in much shorter time. As a matter of fact, the computation time can be decreased from half a day to a few hours without any decrease in the solution quality.

Spam filtering poses a special problem in text categorization, of which the defining characteristic is that filters face an active adversary, which constantly attempts to evade filtering. Since spam evolves continuously and most practical applications are based on online user feedback, the task calls for fast, incremental and robust learning algorithms. In this paper, we investigate a novel approach to spam filtering based on adaptive statistical data compression models. The nature of these models allows them to be employed as probabilistic text classifiers based on character-level or binary sequences. By modeling messages as sequences, tokenization and other error-prone preprocessing steps are omitted altogether, resulting in a method that is very robust. The models are also fast to construct and incrementally updateable. We evaluate the filtering performance of two different compression algorithms; dynamic Markov compression and prediction by partial matching. The results of our empirical evaluation indicate that compression models outperform currently established spam filters, as well as a number of methods proposed in previous studies.

In continuous casting of steel a number of parameters have to be set, such as the casting temperature, casting speed and coolant flows that critically affect the safety, quality and productivity of steel production. We have implemented an optimization tool consisting in an optimization algorithm and casting process simulator. The paper describes the process, the optimization task, and the proposed optimization approach, and shows illustrative results of its application on an industrial casting machine where spray coolant flows were optimized. In the comparative study, two variants of an evolutionary algorithm and the downhill simplex method were used, and they were all able to significantly improve the manual setting of coolant flows.

The popularity of search heuristics has lead to numerous new approaches in the last two decades. Since algorithm performance is problem dependent and parameter sensitive, it is difficult to consider any single approach as of greatest utility overall problems. In contrast, differential evolution (DE) is a numerical optimization approach that requires hardly any parameter tuning and is very efficient and reliable on both benchmark and real-world problems. However, the results presented in this paper demonstrate that standard methods of evolutionary optimization are able to outperform DE on noisy problems when the fitness of candidate solutions approaches the fitness variance caused by the noise.

The paper presents an evolutionary scheduling algorithm designed for and applied to a group of production lines in an automobile factory. The cost of schedules is determined by the quality of energy consumption management over peak demand periods, and the task is to minimize the energy costs by appropriate scheduling of process interruptions. To explore the capabilities of the evolutionary scheduling algorithm, its results are compared with optimal solutions produced for selected problem instances by a constraint logic programming (CLP) approach. The comparison shows the evolutionary algorithm is capable of finding near-optimal schedules while successfully handling a larger set of possible problem instances than CLP.

A three-phased framework for learning dynamic system control is presented. A genetic algorithm is employed to derive control rules encoded as decision tables. Next, the rules are automatically transformed into comprehensible form by means of inductive machine learning. Finally, a genetic algorithm is applied again to optimize the numerical parameters of the induced rules. The approach is experimentally verified on a benchmark problem of inverted pendulum control, with special emphasis on robustness and reliability. It is also shown that the proposed framework enables exploiting available domain knowledge. In this case, genetic algorithm makes qualitative control rules operational by providing interpretation of symbols in terms of numerical values.>