The support vector machine (SVM) is one of the successful approaches to the classification problem. Since the values of features are typically affected by uncertainty, it is important to incorporate uncertainty into t...
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The support vector machine (SVM) is one of the successful approaches to the classification problem. Since the values of features are typically affected by uncertainty, it is important to incorporate uncertainty into the SVM formulation. This paper focuses on developing a robust optimization (RO) model for SVM. A key distinction from existing literature lies in the timing of optimizing decision variables. To the best of our knowledge, in all existing RO models developed for SVM, a common assumption is that all decision variables are decided before the uncertainty realization, which leads to an overly conservative decision boundary. However, this paper adopts a different strategy by determining the variables that assess the misclassification error of data points or their fall within the margin post-realization, resulting in a less conservative model. The RO models where decisions are made in two stages (some before and the rest after the uncertainty resolution), are called adjustable RO models. This adjustment results in a three-level optimization model for which two decomposition-based algorithms are proposed. In these algorithms, after providing a bi-level reformulation, the model is divided into a masterproblem (MP) and a sub-problem the interaction of which yields the optimal solution. Acceleration of algorithms via incorporating valid inequalities into MP is another novelty of this paper. Computational results over simulated and real-world datasets confirm the efficiency of the proposed model and algorithms.
Objective space normalization is important since a real-world multiobjective problem usually has differently scaled objective functions. Recently, bad effects of the commonly used simple normalization method have been...
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ISBN:
(纸本)9781728169293
Objective space normalization is important since a real-world multiobjective problem usually has differently scaled objective functions. Recently, bad effects of the commonly used simple normalization method have been reported for the popular decomposition-based algorithm MOEA/D. However, the effects of recently proposed sophisticated normalization methods have not been investigated. In this paper, we examine the effectiveness of these normalization methods in MOEA/D. We find that these normalization methods can cause performance deterioration. We also find that the sophisticated normalization methods are not necessarily better than the simple one. Although the negative effects of inaccurate estimation of the nadir point are well recognized in the literature, no solution has been proposed. In order to address this issue, we propose two dynamic normalization strategies which dynamically adjust the extent of normalization during the evolutionary process. Experimental results clearly show the necessity of considering the extent of normalization.
Dominance resistant solutions (DRSs) in multi-objective problems have very good values for some objectives and very bad values for other objectives. Whereas DRSs are far away from the Pareto front, they are hardly dom...
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ISBN:
(纸本)9781450371285
Dominance resistant solutions (DRSs) in multi-objective problems have very good values for some objectives and very bad values for other objectives. Whereas DRSs are far away from the Pareto front, they are hardly dominated by other solutions due to some very good objective values. It is well known that the existence of DRSs severely degrades the search ability of Pareto dominance-basedalgorithms such as NSGA-II and SPEA2. In this paper, we examine the effect of DRSs on the search ability of NSGA-II on the DTLZ test problems with many objectives. We slightly change their problem formulation to increase the size of the DRS region. Through computational experiments, we show that DRSs have a strong negative effect on the search ability of NSGA-II whereas they have almost no effect on MOEA/D with the PBI function. We also show that a slightly modified NSGA-II for decreasing the negative effect of DRSs works well on many-objective DTLZ test problems (its performance is similar to NSGA-III and MOEA/D). These results suggest that DTLZ is not an appropriate test suite for evaluating many-objective evolutionary algorithms. This issue is further addressed through computational experiments on newly formulated test problems with no distance function.
We consider essential challenges related to the elicitation of indirect preference information in interactive evolutionary algorithms for multiple objective optimization. The methods in this stream use holistic judgme...
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ISBN:
(纸本)9781450371285
We consider essential challenges related to the elicitation of indirect preference information in interactive evolutionary algorithms for multiple objective optimization. The methods in this stream use holistic judgments provided by the Decision Maker (DM) to progressively bias the evolutionary search toward his/her most preferred region in the Pareto front. We enhance such an interactive process using three targeted developments and illustrate their efficiency in the context of a decomposition-based evolutionary framework. Firstly, we present some active learning strategies for selecting solutions from the current population that should be critically compared by the DM. These strategies implement the paradigm of maximizing the potential information gain derived from the DM's answer. Secondly, we discuss the procedures for deciding when the DM should be questioned for preference information. In this way, we refer to a more general problem of distributing the DM's interactions with the method in a way that ensures sufficient evolutionary pressure. Thirdly, we couple the evolutionary schemes with different types of indirect preferences, including pairwise comparisons, preference intensities, best-of-k. judgments, and complete orders of a small subset of solutions. A thorough experimental analysis indicates that the three introduced advancements have a positive impact on the DM-perceived quality of constructed solutions.
The frequently used basic version of MOEA/D (multi-objective evolutionary algorithm based on decomposition) has no normalization mechanism of the objective space, whereas the normalization was discussed in the origina...
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The frequently used basic version of MOEA/D (multi-objective evolutionary algorithm based on decomposition) has no normalization mechanism of the objective space, whereas the normalization was discussed in the original MOEA/D paper. As a result, MOEA/D shows difficulties in finding a set of uniformly distributed solutions over the entire Pareto front when each objective has a totally different range of objective values. Recent variants of MOEA/D have normalization mechanisms for handling such a scaling issue. In this paper, we examine the effect of the normalization of the objective space on the performance of MOEA/D through computational experiments. A simple normalization mechanism is used to examine the performance of MOEA/D with and without normalization. These two types of MOEA/D are also compared with recently proposed many-objective algorithms: NSGA-III, MOEA/DD, and 0-DEA. In addition to the frequently used many-objective test problems DTLZ and WFG, we use their minus versions. We also propose two variants of the DTLZ test problems for examining the effect of the normalization in MOEA/D. Test problems in one variant have objective functions with totally different ranges. The other variant has a kind of deceptive nature, where the range of each objective is the same on the Pareto front but totally different over the entire feasible region. Computational experiments on those test problems clearly show the necessity of the normalization. It is also shown that the normalization has both positive and negative effects on the performance of MOEA/D. These observations suggest that the influence of the normalization is strongly problem dependent.
This paper proposes a decomposition-based heuristic for a network delivery problem in which relief workers acquire valuable emergency supplies from relief warehouses, and transport them to meet the urgent needs of dis...
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This paper proposes a decomposition-based heuristic for a network delivery problem in which relief workers acquire valuable emergency supplies from relief warehouses, and transport them to meet the urgent needs of distressed population centres. The problem context dictates that the relief items reach these population centres before critical deadlines. However, co-ordination challenges and random disruptions introduce uncertainty in both network travel times and the destination deadlines. Hence, relief workers have to negotiate the tension between ensuring a high probability of punctual delivery and maximising the combined value of the relief supplies delivered. For an arbitrary routing scheme which guarantees punctual delivery in an uncertainty-free state of nature, the heuristic yields an upper bound on the probability that, under uncertainty, the routing scheme described will lead to tardy delivery. We demonstrate our solution approach on a small numerical example and glean insights from experiments on a realistically sized problem. Overall, our central model and proposed solution approach are useful to managers who need to evaluate routing options and devise effective operational delivery plans in humanitarian crisis situations. (C) 2016 Elsevier Ltd. All rights reserved.
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