Bayesian Preference Elicitation for Decision Support in Multi‐Objective Optimization

In multiobjective optimization, it is nontrivial for decision makers to articulate preferences without a priori knowledge, which is particularly true when the number of objectives becomes large. Depending on the shape of the Pareto front, optimal solutions such as knee points may be of interest. Although several multi- and many-objective optimization test suites have been proposed, little work has been reported focusing on designing multiobjective problems whose Pareto front contains complex knee regions. Likewise, few performance indicators dedicated to evaluate an algorithm's ability of accurately locating all knee points in high-dimensional objective space have been suggested. This paper proposes a set of multiobjective optimization test problems whose Pareto front consists of complex knee regions, aiming to assess the capability of evolutionary algorithms to accurately identify all knee points. Various features related to knee points have been taken into account in designing the test problems, including symmetry, differentiability, and degeneration. These features are also combined with other challenges in solving the optimization problems, such as multimodality, linkage between decision variables, nonuniformity, and scalability of the Pareto front. The proposed test problems are scalable to both decision and objective spaces. Accordingly, new performance indicators are suggested for evaluating the capability of optimization algorithms in locating the knee points. The proposed test problems, together with the performance indicators, offer a new means to develop and assess preference-based evolutionary algorithms for solving multi- and many-objective optimization problems.

Product improvement and cost reduction have always been important goals in the metal forming industry. The rise of finite element (FEM) simulations for processes has contributed to these goals in a major way. More recently, coupling FEM simulations to mathematical optimisation techniques has shown the potential to make a further giant contribution to product improvement and cost reduction. Much research on the optimisation of metal forming processes has been published during the last couple of years. Although the results are impressive, the optimisation techniques are generally only applicable to specific optimisation problems for specific products and specific metal forming processes. As a consequence, applying optimisation techniques to other metal forming problems requires a lot of optimisation expertise, which forms a barrier for more general industrial application of these techniques. In this paper, we overcome this barrier by proposing a generally applicable optimisation strategy that makes use of FEM simulations of metal forming processes. It consists of a structured methodology for modelling optimisation problems related to metal forming. Subsequently, screening is applied to reduce the size of the optimisation problem by selecting only the most important design variables. Finally, the reduced optimisation problem is solved by an efficient optimisation algorithm. The strategy is generally applicable in a sense that it is not constrained to a certain type of metal forming problem, product or process. Also, any FEM code may be included in the strategy. Furthermore, the structured approach for modelling and solving optimisation problems should enable non-optimisation specialists to apply optimisation techniques to improve their products and processes. The optimisation strategy has been successfully applied to a hydroforming process, which demonstrates the potential of the optimisation of metal forming processes in general and more specific the proposed optimisation strategy.

Structural design optimization in precise or imprecise environment plays an important role in civil engineering as well as in mathematics. Conventionally in most of the situation structural designs are treated as single objective optimization problem with cost as objective function. Practically there exists some real life problems where multiple and conflicting objects frequently exist.So a methodology known as Multi-objective Four Valued Refined Neutrosophic Optimization (MOFVRNO) technique has been introduced in this paper.This optimization technique is generalization of fuzzy set (FS)theory,Intuitionistic Fuzzy Set (IFO)Theory and above all Single Valued  Neutrosophic Set Optimization (SVNSO)theory.In this problem cost and deflection of the beam have been considered as objective functions. Also in this welded beam design problem maximum shear stress in the weld group, maximum bending stress in the beam, buckling load of the beam have been considered as constraints. The design variables are length ,depth ,height,width in this optimization problem.Now from the mathematical point of view indeterminacy is again refined as uncertain(U),Contradiction(Truth∧Falsity,T) in Four valued Refined Neutrosophic Optimization technique.The results obtained by different methods like nonlinear optimization technique,Fuzzy optimization technique ,Intuitionistic Fuzzy Optimization technique ,Single Valued Neutrosophic Optimization technique andFour Valued Refined Neutrosophic Optimization technique have been discussed in the perspective of welded beam design problem in this paper. It helps to make comparison among them to find most cost effective method.

This paper proposes a new multi-objective optimization algorithm - scored Pareto mind evolutionary computation (SP-MEC), which introduces the theory of Pareto into mind evolutionary computation (MEC) for the multi-objective optimization. In our SP-MEC, the selection of individuals is based on their scores that include the Pareto dominance and density information among the individuals. The SP-MEC is compared with the VEGA, NSGA, SPEA, and Pareto-MEC on the basis of four different test problems: convexity, non-convexity, discreteness, as well as non-uniformity. Especially, both the Pareto-MEC and SPEA have shown promising performances in solving various optimization problems. On the test problems, SP-MEC outperforms all the four reference algorithms concerning three measures: the distance from trade-off front to Pareto-optimal front, the uniformity of solutions, and the spread of solutions. Impersonal termination criterion is used in SP-MEC and Pareto-MEC instead of the preset number of generations in other algorithms. SP-MEC has a higher computational efficiency than the VEGA, NSGA, and SPEA. Compared with another our algorithm, Pareto-MEC, the computational efficiency of SP-MEC is a little lower. However, the solution quality of SP-MEC is higher than that of the Pareto-MEC. Therefore, it can be concluded the SP-MEC is a powerful algorithm for multi-objective optimization.

Pairwise comparison is a popular method for decision making. Analytic Hierarchy Process (AHP) is based on pairwise comparison (PC), which the rating results of PC forms pairwise reciprocal matrices (PRM) further to be derived as the priority values by the Eigen method. Prioritization method using Eigen method is still open to discuss, although many applications using this method in AHP have been made. This study proposes the Canonical Particle Swarm Optimized Direct Least Squares Prioritization (CPSO-DLSP) method for deriving the priority values from a PRM. The direct least squares function leads to the least total square variance between ground PRM and the derived priority values. The Canonical Particle Swarm Optimization (CPSO) algorithm is one of the promising methods to effectively solve the direct least squares optimization problem. An application of mutual fund selection using the conventional AHP method is revised and demonstrated by the proposed method.

Cuckoo search is a swarm-intelligence-based algorithm that is very effective for solving highly nonlinear optimization problems. In this paper, the multiobjective cuckoo search is extended so as to obtain high-quality Pareto fronts more accurately for multiobjective optimization problems with complex constraints. The proposed approach uses a combination of the cuckoo search with non-dominated sorting and archiving techniques. The performance of the proposed approach is validated by seven test problems. The convergence property and diversity as well as uniformity are compared with those of the NSGA-II. The results show that the proposed approach can find Pareto fronts with better uniformity and quicker convergence.

The selection mechanisms that are most commonly adopted by multi-objective evolutionary algorithms (MOEAs) are based on Pareto optimality. However, recent studies have provided theoretical and experimental evidence regarding the unsuitability of Pareto-based selection mechanisms when dealing with problems having four or more objectives. In this paper, we propose a novel MOEA designed for solving many-objective optimization problems. The selection mechanism of our approach is based on the transformation of a multi-objective optimization problem into a linear assignment problem, which is solved by the Kuhn–Munkres’ (Hungarian) algorithm. Our proposed approach is compared with respect to three state-of-the-art MOEAs, designed for solving many-objective optimization problems (i.e., problems having four or more objectives), adopting standard test problems and performance indicators taken from the specialized literature. Since one of our main aims was to analyze the scalability of our proposed approach, its validation was performed adopting test problems having from two to nine objective functions. Our preliminary experimental results indicate that our proposal is very competitive with respect to all the other MOEAs compared, obtaining the best results in several of the test problems adopted, but at a significantly lower computational cost.

A novel scalable test problem suite for multimodal multiobjective optimization

The experimental results reported in many papers suggest that making an appropriate a priori choice of an evolutionary method for a non-linear parameter optimisation problem remains an open question. It seems that the most promising approach at this stage of research is experimental, involving a design of a scalable test suite of constrained optimisation problems, in which many features could be tuned easily. It would then be possible to evaluate the merits and drawbacks of the available methods as well as test new methods efficiently. In this chapter, we discuss a new test-case generator for constrained parameter optimisation techniques, which deals with deficiencies of the generators proposed earlier. This generator, TCG-2, is capable of creating various test problems with different characteristics, including the dimensionality of the problem, number of local optima, number of active constraints at the optimum, topology of the feasible search space, etc. Such a test-case generator is very useful for analysing and comparing different constraint-handling techniques.KeywordsEvolutionary computationnon-linear programmingconstrained optimisationtest-case generator

When facing dynamic optimization problems the goal is no longer to find the extrema, but to track their progression through the space as closely as possible. Over these kind of over changing, complex and ubiquitous real-world problems, the explorative-exploitive subtle counterbalance character of our current state-of-the-art search algorithms should be biased towards an increased explorative behavior. While counterproductive in classic problems, the main and obvious reason of using it in severe dynamic problems is simple: while we engage ourselves in exploiting the extrema, the extrema moves elsewhere. In order to tackle this subtle compromise, we propose a novel algorithm for optimization in dynamic binary landscapes, stressing the role of negative feedback mechanisms. The Binary Ant Algorithm (BAA) mimics some aspects of social insects' behavior. Like Ant Colony Optimization (ACO), BAA acts by building pheromone maps over a graph of possible trails representing pseudo-solutions of increasing quality to a specific optimization problem. Main differences rely on the way this search space is represented and provided to the colony in order to explore/exploit it, while and more important, we enrol in providing strong evaporation to the problem-habitat. By a process of pheromone reinforcement and evaporation the artificial insect's trails over the graph converge to regions near the ideal solution of the optimization problem. Over each generation, positive feedbacks made available by pheromone reinforcement consolidate the best solutions found so far, while enhanced negative feedbacks given by the evaporation mechanism provided the system with population diversity and fast self-adaptive characteristics, allowing BAA to be particularly suitable for severe complex dynamic optimization problems. Experiments made with some well known test functions frequently used in the Evolutionary Algorithms' research field illustrate the efficiency of the proposed method. BAA was also compared with other algorithms, proving to be more able to track fast moving extrema on several test problems.

Improving NSGA-III algorithms with information feedback models for large-scale many-objective optimization

We consider optimization problems for minimizing conditional value-at-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers.

A novel modified bat algorithm hybridizing by differential evolution algorithm

Genetic algorithms (GAs), which are directed stochastic hill climbing algorithms, are a commonly used optimization technique and are generally applied to single criterion optimization problems with fairly complex solution landscapes. There has been some attempts to apply GA to multicriteria optimization problems. The GA selection mechanism is typically dependent on a single-valued objective function and so no general methods to solve multicriteria optimization problems have been developed so far. In this paper, a new method of transformation of the multiple criteria problem into a single-criterion problem is presented. The problem of transformation brings about the need for the introduction of thePareto set estimation method to perform the multicriteria optimization using GAs. From a given solution set, which is the population of a certain generation of the GA, the Pareto set is found. The fitness of population members in the next GA generation is calculated by a distance metric with a reference to the Pareto set of the previous generation. As we are unable to combine the objectives in some way, we resort to this distance metric in the positive Pareto space of the previous solutions, as the fitness of the current solutions. This new GA-based multicriteria optimization method is proposed here, and it is capable of handling any generally formulated multicriteria optimization problem. The main idea of the method is described in detail in this paper along with a detailed numerical example. Preliminary computer generated results show that our approach produces better, and far more Pareto solutions, than plain stochastic optimization methods.

Population-based (gradient-free) methods have become attractive for solving optimization problems since they have been known to locate "optimal" (best) positions within a search space. In this study, we investigate the use of three different gradient-free optimization methods namely: Iterative Latin Hypercube Sampling (ILHS), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA) for well placement and well controls optimization problem of the CO2 underground storage in a 2D saline aquifer with bound constraints on the design variables. We also consider sequential optimization of well locations and well controls using the gradient-free methods. To evaluate the performance of these algorithms, we test the methods on well-known test (or toy) problems such as the multi-dimensional (40 design variables) Rosenbrock function, as well as the Traveling Salesman Problem (TSP) with 20 cities (20 design variables). We further extend the study to a more complex problem which involves finding the optimal well types, well locations, or well controls for a CO2 injection problem in a synthetic saline aquifer compositional model built in a commercial simulator. The objective function formulated for the CO2 injection problem is the net present environmental value (NPEV) involving discrete design variables (well types and well locations) and real design variables (well controls). Results obtained from the performance comparison of these gradient-free methods on the toy problems demonstrate the superior performance of the ILHS over PSO and GA for the Rosenbrock test function, while GA outperforms ILHS and PSO in solving the TSP. We also observe that for the simple well locations optimization problems involving one or two wells, all three algorithms obtained comparable results, given the same number of samples and number of iterations. However, for more complex cases such as the sequential or simultaneous optimization problems, when multiple wells which may include injectors and producers are present, we observe significant differences in the selection of the optimal well types and well locations among the three optimizers. In the simultaneous optimization of well types, well locations, and well controls, when both injectors and producers are BHP controlled, the results obtained showed that it is more optimal for at least one producer to be present, and for the injectors to be operated at the upper bound of the injector BHP, while the producers operated at the lower bound of the producer BHP to maximize the NPEV.