Showing papers on "Multi-objective optimization published in 2014"

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints

Abstract: Having developed multiobjective optimization algorithms using evolutionary optimization methods and demonstrated their niche on various practical problems involving mostly two and three objectives, there is now a growing need for developing evolutionary multiobjective optimization (EMO) algorithms for handling many-objective (having four or more objectives) optimization problems. In this paper, we recognize a few recent efforts and discuss a number of viable directions for developing a potential EMO algorithm for solving many-objective optimization problems. Thereafter, we suggest a reference-point-based many-objective evolutionary algorithm following NSGA-II framework (we call it NSGA-III) that emphasizes population members that are nondominated, yet close to a set of supplied reference points. The proposed NSGA-III is applied to a number of many-objective test problems with three to 15 objectives and compared with two versions of a recently suggested EMO algorithm (MOEA/D). While each of the two MOEA/D methods works well on different classes of problems, the proposed NSGA-III is found to produce satisfactory results on all problems considered in this paper. This paper presents results on unconstrained problems, and the sequel paper considers constrained and other specialties in handling many-objective optimization problems.

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach

Abstract: In the precursor paper, a many-objective optimization method (NSGA-III), based on the NSGA-II framework, was suggested and applied to a number of unconstrained test and practical problems with box constraints alone. In this paper, we extend NSGA-III to solve generic constrained many-objective optimization problems. In the process, we also suggest three types of constrained test problems that are scalable to any number of objectives and provide different types of challenges to a many-objective optimizer. A previously suggested MOEA/D algorithm is also extended to solve constrained problems. Results using constrained NSGA-III and constrained MOEA/D show an edge of the former, particularly in solving problems with a large number of objectives. Furthermore, the NSGA-III algorithm is made adaptive in updating and including new reference points on the fly. The resulting adaptive NSGA-III is shown to provide a denser representation of the Pareto-optimal front, compared to the original NSGA-III with an identical computational effort. This, and the original NSGA-III paper, together suggest and amply test a viable evolutionary many-objective optimization algorithm for handling constrained and unconstrained problems. These studies should encourage researchers to use and pay further attention in evolutionary many-objective optimization.

Multi-objective Optimization

Abstract: Multi-objective optimization is an integral part of optimization activities and has a tremendous practical importance, since almost all real-world optimization problems are ideally suited to be modeled using multiple conflicting objectives. The classical means of solving such problems were primarily focused on scalarizing multiple objectives into a single objective, whereas the evolutionary means have been to solve a multi-objective optimization problem as it is. In this chapter, we discuss the fundamental principles of multi-objective optimization, the differences between multi-objective optimization and single-objective optimization, and describe a few well-known classical and evolutionary algorithms for multi-objective optimization. Two application case studies reveal the importance of multi-objective optimization in practice. A number of research challenges are then highlighted. The chapter concludes by suggesting a few tricks of the trade and mentioning some key resources to the field of multi-objective optimization.

Decomposition of a Multiobjective Optimization Problem Into a Number of Simple Multiobjective Subproblems

Abstract: This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better.

Moea/d with adaptive weight adjustment

Abstract: Recently, MOEA/D multi-objective evolutionary algorithm based on decomposition has achieved great success in the field of evolutionary multi-objective optimization and has attracted a lot of attention. It decomposes a multi-objective optimization problem MOP into a set of scalar subproblems using uniformly distributed aggregation weight vectors and provides an excellent general algorithmic framework of evolutionary multi-objective optimization. Generally, the uniformity of weight vectors in MOEA/D can ensure the diversity of the Pareto optimal solutions, however, it cannot work as well when the target MOP has a complex Pareto front PF; i.e., discontinuous PF or PF with sharp peak or low tail. To remedy this, we propose an improved MOEA/D with adaptive weight vector adjustment MOEA/D-AWA. According to the analysis of the geometric relationship between the weight vectors and the optimal solutions under the Chebyshev decomposition scheme, a new weight vector initialization method and an adaptive weight vector adjustment strategy are introduced in MOEA/D-AWA. The weights are adjusted periodically so that the weights of subproblems can be redistributed adaptively to obtain better uniformity of solutions. Meanwhile, computing efforts devoted to subproblems with duplicate optimal solution can be saved. Moreover, an external elite population is introduced to help adding new subproblems into real sparse regions rather than pseudo sparse regions of the complex PF, that is, discontinuous regions of the PF. MOEA/D-AWA has been compared with four state of the art MOEAs, namely the original MOEA/D, Adaptive-MOEA/D, -MOEA/D, and NSGA-II on 10 widely used test problems, two newly constructed complex problems, and two many-objective problems. Experimental results indicate that MOEA/D-AWA outperforms the benchmark algorithms in terms of the IGD metric, particularly when the PF of the MOP is complex.

Shift-Based Density Estimation for Pareto-Based Algorithms in Many-Objective Optimization

Abstract: It is commonly accepted that Pareto-based evolutionary multiobjective optimization (EMO) algorithms encounter difficulties in dealing with many-objective problems. In these algorithms, the ineffectiveness of the Pareto dominance relation for a high-dimensional space leads diversity maintenance mechanisms to play the leading role during the evolutionary process, while the preference of diversity maintenance mechanisms for individuals in sparse regions results in the final solutions distributed widely over the objective space but distant from the desired Pareto front. Intuitively, there are two ways to address this problem: 1) modifying the Pareto dominance relation and 2) modifying the diversity maintenance mechanism in the algorithm. In this paper, we focus on the latter and propose a shift-based density estimation (SDE) strategy. The aim of our study is to develop a general modification of density estimation in order to make Pareto-based algorithms suitable for many-objective optimization. In contrast to traditional density estimation that only involves the distribution of individuals in the population, SDE covers both the distribution and convergence information of individuals. The application of SDE in three popular Pareto-based algorithms demonstrates its usefulness in handling many-objective problems. Moreover, an extensive comparison with five state-of-the-art EMO algorithms reveals its competitiveness in balancing convergence and diversity of solutions. These findings not only show that SDE is a good alternative to tackle many-objective problems, but also present a general extension of Pareto-based algorithms in many-objective optimization.

Flower pollination algorithm: A novel approach for multiobjective optimization

Abstract: Multiobjective design optimization problems require multiobjective optimization techniques to solve, and it is often very challenging to obtain high-quality Pareto fronts accurately. In this article, the recently developed flower pollination algorithm (FPA) is extended to solve multiobjective optimization problems. The proposed method is used to solve a set of multiobjective test functions and two bi-objective design benchmarks, and a comparison of the proposed algorithm with other algorithms has been made, which shows that the FPA is efficient with a good convergence rate. Finally, the importance for further parametric studies and theoretical analysis is highlighted and discussed.

Flower Pollination Algorithm: A Novel Approach for Multiobjective Optimization

Abstract: Multiobjective design optimization problems require multiobjective optimization techniques to solve, and it is often very challenging to obtain high-quality Pareto fronts accurately. In this paper, the recently developed flower pollination algorithm (FPA) is extended to solve multiobjective optimization problems. The proposed method is used to solve a set of multobjective test functions and two bi-objective design benchmarks, and a comparison of the proposed algorithm with other algorithms has been made, which shows that FPA is efficient with a good convergence rate. Finally, the importance for further parametric studies and theoretical analysis are highlighted and discussed.

A Survey of Multiobjective Evolutionary Algorithms for Data Mining: Part I

Abstract: The aim of any data mining technique is to build an efficient predictive or descriptive model of a large amount of data. Applications of evolutionary algorithms have been found to be particularly useful for automatic processing of large quantities of raw noisy data for optimal parameter setting and to discover significant and meaningful information. Many real-life data mining problems involve multiple conflicting measures of performance, or objectives, which need to be optimized simultaneously. Under this context, multiobjective evolutionary algorithms are gradually finding more and more applications in the domain of data mining since the beginning of the last decade. In this two-part paper, we have made a comprehensive survey on the recent developments of multiobjective evolutionary algorithms for data mining problems. In this paper, Part I, some basic concepts related to multiobjective optimization and data mining are provided. Subsequently, various multiobjective evolutionary approaches for two major data mining tasks, namely feature selection and classification, are surveyed. In Part II of this paper, we have surveyed different multiobjective evolutionary algorithms for clustering, association rule mining, and several other data mining tasks, and provided a general discussion on the scopes for future research in this domain.

The Markowitz Optimization Enigma: Is 'Optimized' Optimal?

Abstract: The indifference of many investment practitioners to mean-variance optimization technology, despite its theoretical appeal, is understandable in many cases. The major problem with MV optimization is its tendency to maximize the effects of errors in the input assumptions. Unconstrained MV optimization can yield results that are inferior to those of simple equal-weighting schemes. Nevertheless, MV optimization is superior to many ad hoc techniques in terms of integration of portfolio objectives with client constraints and efficient use of information. Its practical value may be enhanced by the sophisticated adjustment of inputs and the imposition of constraints based on fundamental investment considerations and the importance of priors. The operating principle should be that, to the extent that reliable information is available, it should be included as part of the definition of the optimization procedure.

Adaptive Operator Selection With Bandits for a Multiobjective Evolutionary Algorithm Based on Decomposition

Abstract: Adaptive operator selection (AOS) is used to determine the application rates of different operators in an online manner based on their recent performances within an optimization process. This paper proposes a bandit-based AOS method, fitness-rate-rank-based multiarmed bandit (FRRMAB). In order to track the dynamics of the search process, it uses a sliding window to record the recent fitness improvement rates achieved by the operators, while employing a decaying mechanism to increase the selection probability of the best operator. Not much work has been done on AOS in multiobjective evolutionary computation since it is very difficult to measure the fitness improvements quantitatively in most Pareto-dominance-based multiobjective evolutionary algorithms. Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Thus, it is natural and feasible to use AOS in MOEA/D. We investigate several important issues in using FRRMAB in MOEA/D. Our experimental results demonstrate that FRRMAB is robust and its operator selection is reasonable. Comparison experiments also indicate that FRRMAB can significantly improve the performance of MOEA/D.

Bayesian Optimization with Inequality Constraints

Abstract: Bayesian optimization is a powerful framework for minimizing expensive objective functions while using very few function evaluations. It has been successfully applied to a variety of problems, including hyperparameter tuning and experimental design. However, this framework has not been extended to the inequality-constrained optimization setting, particularly the setting in which evaluating feasibility is just as expensive as evaluating the objective. Here we present constrained Bayesian optimization, which places a prior distribution on both the objective and the constraint functions. We evaluate our method on simulated and real data, demonstrating that constrained Bayesian optimization can quickly find optimal and feasible points, even when small feasible regions cause standard methods to fail.

A Population Prediction Strategy for Evolutionary Dynamic Multiobjective Optimization

Abstract: This paper investigates how to use prediction strategies to improve the performance of multiobjective evolutionary optimization algorithms in dealing with dynamic environments. Prediction-based methods have been applied to predict some isolated points in both dynamic single objective optimization and dynamic multiobjective optimization. We extend this idea to predict a whole population by considering the properties of continuous dynamic multiobjective optimization problems. In our approach, called population prediction strategy (PPS), a Pareto set is divided into two parts: a center point and a manifold. A sequence of center points is maintained to predict the next center, and the previous manifolds are used to estimate the next manifold. Thus, PPS could initialize a whole population by combining the predicted center and estimated manifold when a change is detected. We systematically compare PPS with a random initialization strategy and a hybrid initialization strategy on a variety of test instances with linear or nonlinear correlation between design variables. The statistical results show that PPS is promising for dealing with dynamic environments.

Stable Matching-Based Selection in Evolutionary Multiobjective Optimization

Abstract: Multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem into a set of scalar optimization subproblems and optimizes them in a collaborative manner. Subproblems and solutions are two sets of agents that naturally exist in MOEA/D. The selection of promising solutions for subproblems can be regarded as a matching between subproblems and solutions. Stable matching, proposed in economics, can effectively resolve conflicts of interests among selfish agents in the market. In this paper, we advocate the use of a simple and effective stable matching (STM) model to coordinate the selection process in MOEA/D. In this model, subproblem agents can express their preferences over the solution agents, and vice versa. The stable outcome produced by the STM model matches each subproblem with one single solution, and it tradeoffs convergence and diversity of the evolutionary search. Comprehensive experiments have shown the effectiveness and competitiveness of our MOEA/D algorithm with the STM model. We have also demonstrated that user-preference information can be readily used in our proposed algorithm to find a region that decision makers are interested in.

From Bandits to Monte-Carlo Tree Search: The Optimistic Principle Applied to Optimization and Planning

Abstract: This work covers several aspects of the optimism in the face of uncertainty principle applied to large scale optimization problems under finite numerical budget. The initial motivation for the research reported here originated from the empirical success of the so-called Monte-Carlo Tree Search method popularized in computer-go and further extended to many other games as well as optimization and planning problems. Our objective is to contribute to the development of theoretical foundations of the field by characterizing the complexity of the underlying optimization problems and designing efficient algorithms with performance guarantees. The main idea presented here is that it is possible to decompose a complex decision making problem (such as an optimization problem in a large search space) into a sequence of elementary decisions, where each decision of the sequence is solved using a (stochastic) multi-armed bandit (simple mathematical model for decision making in stochastic environments). This so-called hierarchical bandit approach (where the reward observed by a bandit in the hierarchy is itself the return of another bandit at a deeper level) possesses the nice feature of starting the exploration by a quasi-uniform sampling of the space and then focusing progressively on the most promising area, at different scales, according to the evaluations observed so far, and eventually performing a local search around the global optima of the function. The performance of the method is assessed in terms of the optimality of the returned solution as a function of the number of function evaluations. Our main contribution to the field of function optimization is a class of hierarchical optimistic algorithms designed for general search spaces (such as metric spaces, trees, graphs, Euclidean spaces, ...) with different algorithmic instantiations depending on whether the evaluations are noisy or noiseless and whether some measure of the ''smoothness'' of the function is known or unknown. The performance of the algorithms depend on the local behavior of the function around its global optima expressed in terms of the quantity of near-optimal states measured with some metric. If this local smoothness of the function is known then one can design very efficient optimization algorithms (with convergence rate independent of the space dimension), and when it is not known, we can build adaptive techniques that can, in some cases, perform almost as well as when it is known.

On the Road Between Robust Optimization and the Scenario Approach for Chance Constrained Optimization Problems

Abstract: We propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario-based methods. Our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods, nor is it based entirely on randomization as in the scenario approach. It instead involves solving a robust optimization problem with bounded uncertainty, where the uncertainty bounds are randomized and are computed using the scenario approach. To guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems. Our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost.

A Multiobjective Particle Swarm Optimization for Sizing and Placement of DGs from DG Owner's and Distribution Company's Viewpoints

Abstract: Distributed generations (DGs) have significant benefits in the electric power industry, such as a reduction in CO2 and NOX emissions in electricity generation, improvement of voltage profile in distribution feeders, amending voltage stability in heavy load levels, enhancement of reliability and power quality, as well as securing the power market. Despite the numerous advantages of DG technologies, weak capability in dispatching and management of DGs is a major challenge for distribution system operators. Hence, during recent years, several studies about various aspects of control, operation, placement, and sizing of DGs have been conducted. This paper presents a novel application of multiobjective particle swarm optimization with the aim of determining the optimal DGs places, sizes, and their generated power contract price. In the proposed multiobjective optimization, not only are the operational aspects, such as improving voltage profile and stability, power-loss reduction, and reliability enhancement taken into account, but also an economic analysis is performed based on the distribution company's and DG owner's viewpoints. The simulation study is performed on the IEEE 33-bus distribution test system and the consequent discussions prove the effectiveness of the proposed approach.

Multi-Objective Optimization and Design of Photovoltaic-Wind Hybrid System for Community Smart DC Microgrid

Abstract: Renewable energy sources continues to gain popularity. However, two major limitations exist that prevent widespread adoption: availability of the electricity generated and the cost of the equipment. Distributed generation, (DG) grid-tied photovoltaic-wind hybrid systems with centralized battery back-up, can help mitigate the variability of the renewable energy resource. The downside, however, is the cost of the equipment needed to create such a system. Thus, optimization of generation and storage in light of capital cost and variability mitigation is imperative to the financial feasibility of DC microgrid systems. PV and wind generation are both time dependent and variable but are highly correlated, which make them ideal for a dual-sourced hybrid system. This paper presents an optimization technique base on a Multi-Objective Genetic Algorithm (MOGA) which uses high temporal resolution insolation data taken at 10 seconds data rate instead of more commonly used hourly data rate. The proposed methodology employs a techno-economic approach to determine the system design optimized by considering multiple criteria including size, cost, and availability. The result is the baseline system cost necessary to meet the load requirements and which can also be used to monetize ancillary services that the smart DC microgrid can provide to the utility at the point of common coupling (PCC) such as voltage regulation. The hybrid smart DC microgrid community system optimized using high-temporal resolution data is compared to a system optimized using lower-rate temporal data to examine the effect of the temporal sampling of the renewable energy resource.

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

Abstract: 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.