An important issue in hypervolume-based evolutionary multi-objective optimization (EMO) algorithms is the specification of a reference point for hypervolume calculation. However, its appropriate specification has not been carefully studied in the literature. Some recent studies have pointed out the importance and difficulty of the reference point specification. Its appropriate specification depends on problem characteristics such as the Pareto front shape and the number of objectives. In this paper, the difficulty of the reference point specification in hypervolume-based EMO algorithms is circumvented by using two reference points. Instead of using only a single reference point, we propose a new hypervolume-based EMO algorithm that can effectively utilize two reference points cooperatively. Experimental results show that the proposed algorithm has good and robust performance on a wide range of test problems. In comparison to hypervolume-based EMO algorithms with only a single reference point, the proposed algorithm can find a wider and more uniformly distributed solution set. On a recently proposed real-world problem suite, the proposed algorithm shows competitive performance in comparison to state-of-the-art algorithms.

In real-world multi-objective problems, each objective has a totally different scale. However, some frequently-used multi-objective evolutionary algorithms (MOEAs) have no objective space normalization mechanisms. The effect of objective space normalization on the performance of decomposition-based MOEAs (e.g., MOEA/D and NSGA-III) has already been examined for artificial test problems (e.g., DTLZ and WFG) in the literature. In this paper, we examine its practical usefulness for real-world multi-objective problems using various MOEAs. Our experimental results clearly show that objective space normalization is needed not only in decomposition-based MOEAs but also in hypervolume-based MOEAs. We also explain why objective space normalization is needed in these two types of MOEAs.

Real-world multiobjective optimization problems (MOPs) usually have conflicting and differently-scaled objectives. To deal with such problems, objective space normalization is widely used in multiobjective evolutionary algorithm (MOEA) design, especially, in the design of decomposition-based MOEAs. It has been demonstrated that uniformly-distributed solutions can be obtained for badly-scaled MOPs by decomposition-based MOEAs with objective space normalization. Recently, weight vector scaling has also been used for badly-scaled MOPs. In some studies, it was argued that weight vector scaling and objective space normalization are essentially the same when applied to decomposition-based MOEAs. In this paper, we theoretically and empirically show the relation between objective space normalization and weight vector scaling. Our results demonstrate that similarities and differences between the two methods depend on the choice of a scalarizing function. How the choice between normalization and weight vector scaling affects decomposition-based MOEAs with solution assignment mechanisms is also analyzed.

Hypervolume subset selection (HSS) has received considerable attention in the field of evolutionary multiobjective optimization (EMO). It aims to select a representative subset from a candidate solution set so that the hypervolume (HV) of the selected subset is maximized. A number of HSS methods have been proposed in the literature, attempting to either reduce the computation time of subset selection or improve the subset quality (i.e., the HV of the selected subset). However, when selecting from a large candidate set (e.g., from hundreds of thousands of candidate solutions), most HSS methods fail to strike a balance between the computation time and the subset quality. In this article, we propose a new local search HSS method and its extended version. Three strategies are proposed. The first two strategies are applied to the proposed method to obtain a good subset within a small computation time, and the third one is applied to the extended version to further improve the obtained subset. The experimental results on various candidate sets demonstrate that the proposed method and its extended version are much more efficient and effective than the existing HSS methods.

Multiobjective evolutionary algorithms (MOEAs) have been widely used to solve multiobjective optimization problems (MOPs). Conventional MOEAs usually require a large number of function evaluations (FEs) for evaluating the quality of solutions. However, only a limited number of FEs are affordable in many real-world optimization problems, where the FEs are computationally or economically expensive. The use of a large number of FEs decreases the effectiveness of MOEAs in solving these problems. This article proposes a dual-fuzzy-classifier-based surrogate model (DFC) and a DFC-based MOEA (DFC-MOEA) framework for expensive optimization problems. The DFC model is used for offspring preselection to choose high-quality offspring solutions and for categorization to divide unevaluated solutions into three categories. In the proposed framework, two fuzzy classifiers are built to predict the quality of each unevaluated solution. To strike a balance between convergence and diversity, one classifier is designed for predicting the dominance relation of unevaluated solutions (i.e., convergence), and another classifier is designed for predicting the crowdedness of unevaluated solutions (i.e., diversity). The integration of the proposed framework in three different types of MOEAs (i.e., a dominance-based MOEA, an indicator-based MOEA, and a decomposition-based MOEA) demonstrates its usefulness in handling expensive optimization problems. Comprehensive experiments on four well-known test suites and several real-world optimization problems demonstrate that the proposed framework is able to improve the performance of MOEAs under a limited number of FEs.

Hypervolume-based multi-objective evolutionary algorithms (HV-MOEAs) are one of the popular algorithm classes in the evolutionary multi-objective optimization (EMO) community. HV-MOEAs, which can directly optimize the HV of a solution set, are useful in various applications. However, the computation time of HV-MOEAs is very long for many-objective problems since the calculation of the hypervolume contribution (HVC) is computationally expensive. Therefore, a number of approximation methods for the HVC calculation were proposed to reduce its time cost. An R2-based hypervolume contribution approximation (R2-HVC) method was proposed for HVC approximation. However, for HV-MOEAs, the point is to find the worst solution, instead of accurately approximating the HVC of each solution. In this paper, a novel method (i.e., two-stage R2-HVC) is proposed for improving the ability of R2-HVC to correctly identify the worst solution (i.e., the solution with the smallest HVC value) in a solution set. In the proposed method, some candidate solutions are selected based on rough HVC approximation in the first stage, and they are carefully evaluated in the second stage. It is shown through computational experiments that the proposed method performs much better than the original R2-HVC method.

Normalization is an important algorithmic component for multiobjective evolutionary algorithms (MOEAs). Different normalization methods have been proposed in the literature. Recently, several studies have been conducted to examine the effects of normalization methods. However, the existing evaluation methods for investigating the effects of normalization are limited due to their drawbacks. In this paper, we discuss the limitations of the existing evaluation methods. A new metric has been proposed to facilitate the investigation of normalization methods. Our analysis clearly shows the superiority of the proposed metric over the existing methods. We also use the proposed metric to compare three popular normalization methods on problems with different Pareto front shapes.

The quality of solutions in multiobjective evolutionary algorithms (MOEAs) is usually evaluated by objective functions. However, function evaluations (FEs) are usually time-consuming in real-world problems. A large number of FEs limit the application of MOEAs. In this paper, we propose a fuzzy classifier-based selection strategy to reduce the number of FEs of MOEAs. First, all evaluated solutions in previous generations are used to build a fuzzy classifier. Second, the built fuzzy classifier is used to predict each unevaluated solution's label and its membership degree. The reproduction procedure is repeated to generate enough offspring solutions (classified as positive by the classifier). Next, unevaluated solutions are sorted based on their membership degrees in descending order. The same number of solutions as the population size are selected from the top of the sorted unevaluated solutions. Then, the best half of the chosen solutions are selected and stored in the new population without evaluations. The other half solutions are evaluated. Finally, the evaluated solutions are used together with evaluated current solutions for environmental selection to form another half of the new population. The proposed strategy is integrated into two MOEAs. Our experimental results demonstrate the effectiveness of the proposed strategy on reducing FEs.

Hypervolume subset selection is a hot topic in the field of evolutionary multi-objective optimization (EMO) due to the increasing needs of selecting a small set of representative solutions from a large set of non-dominated solutions (e.g., unbounded external archive). To maximize the hypervolume (HV) of the selected subset, a number of HV subset selction (HSS) methods have been proposed. Greedy forward selection (GFS) subset selection method is the most popular one, which has been actively investigated in the literature. However, few studies focus on local search (LS) HSS method, which is similar to the mechanism of SMS-EMOA. The time cost of the LS method is usually high, and the quality of the subset selected by this method is always poor. To address these two issues, in this paper, we first adopt an HV contribution update strategy to the original LS method to significantly reduce its time cost. In addition, two efficient strategies are proposed to improve the performance of the LS method to get a better subset. Finally, experiments are conducted to show the effectiveness of the improved LS method.

A real-world multiobjective optimization problem (MOP) usually has differently scaled objectives. Objective space normalization has been widely used in multiobjective optimization evolutionary algorithms (MOEAs). Without objective space normalization, most of the MOEAs may fail to obtain uniformly distributed and well-converged solutions on MOPs with differently scaled objectives. Objective space normalization requires information on the Pareto front (PF) range, which can be acquired from the ideal and nadir points. Since the ideal and nadir points of a real-world MOP are usually not knowna priori, many recently proposed MOEAs tend to estimate and update the two points adaptively during the evolutionary process. Different methods to estimate ideal and nadir points have been proposed in the literature. Due to inaccurate estimation of the two points (i.e., inaccurate estimation of the PF range), objective space normalization may deteriorate the performance of an MOEA. Different methods have also been proposed to alleviate the negative effects of inaccurate estimation. This article presents a comprehensive survey of objective space normalization methods, including ideal point estimation methods, nadir point estimation methods, and different methods based on the utilization of the estimated PF range.

Hypervolume is widely used as a performance indicator in the field of evolutionary multiobjective optimization (EMO). It is used not only for performance evaluation of EMO algorithms (EMOAs) but also in indicator-based EMOAs to guide the search. Since its initial proposal in the late 1990s, a wide variety of studies have been done on various topics, including hypervolume calculation, optimal μ-distribution, subset selection, hypervolume-based EMOAs, and extensions of the hypervolume indicator. However, currently there is no work to systematically survey the hypervolume indicator for these topics whereas it has been frequently used in the EMO field. This article aims to fill this gap and provide a comprehensive survey on the hypervolume indicator. We expect that this survey will help EMO researchers to understand the hypervolume indicator more deeply and thoroughly, and promote further utilization of the hypervolume indicator in the EMO field.

Objective space normalization is important since areal-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.

A set of uniformly sampled weight vectors from a unit simplex has been frequently used in decomposition-based multi-objective algorithms. The number of the generated weight vectors is controlled by a user-defined parameter H. In the literature, good results are often reported on test problems with triangular Pareto fronts since the shape of the Pareto fronts is consistent with the distribution of the weight vectors. However, when a problem has an inverted triangular Pareto front, well-distributed solutions over the entire Pareto front are not obtained due to the inconsistency between the Pareto front shape and the weight vector distribution. In this paper, we demonstrate that the specification of H has an unexpected large effect on the performance of decomposition-based multi-objective algorithms when the test problems have inverted triangular Pareto fronts. We clearly explain why their performance is sensitive to the specification of H in an unexpected manner (e.g., H = 3 is bad but H = 4 is good for three-objective problems whereas H = 3 is good but H = 4 is bad for four-objective problems). After these discussions, we suggest a simple weight vector specification method for inverted triangular Pareto fronts.

Performance of evolutionary multi-objective and many-objective optimization algorithms is usually evaluated by computational experiments on a number of test problems. Thus, performance comparison results depend on the choice of test problems. For fair comparison, it is needed to use a wide variety of test problems with various characteristics. However, most of well-known and frequently-used scalable test problems have the same type of Pareto fronts called "regular" Pareto fronts: Their shape is triangular. In this paper, we discuss the reality of this type of Pareto fronts. First, we show that a triangular Pareto front has some unrealistic properties as the Pareto front of a real-world multi-objective problem. Next, we examine the shape of the Pareto fronts of some other multi-objective test problems with independently generated objectives (i.e., with objectives that are not derived from a pre-specified shape of Pareto fronts). It is shown that the Pareto fronts of those test problems are inverted triangular (i.e., not regular). Then, we demonstrate that the shape of Pareto fronts (i.e., triangular or inverted triangular) has large effects on the performance of decomposition-based and hypervolume-based algorithms. Finally, we show difficulties of hypervolume-based performance evaluation for many-objective problems with inverted triangular Pareto fronts.

A low-power CMOS voltage reference with active-feedback frequency compensation is proposed for power-constrained IoT applications whereby power supply ripple rejection (PSRR) performance is critical for the survival of the devices. The proposed voltage reference consists of MOS transistors operating in the sub-threshold region to allow for low-voltage and low-power operations. An active-feedback frequency compensation technique has been used to make the loop stable and improve the PSRR with a very small compensation capacitor while allowing a relatively large output capacitor. The circuit is fabricated in a standard 0.18-μm CMOS process. The measured power consumption is 22nW at 0.7V power supply. The measured temperature coefficient (TC) is 38ppm/°C in a range from -40 to +110°C, and the line regulation is 200μV/V in a supply voltage range of 0.7~2V. The measured PSRR at 10 Hz, 1 kHz, and 100 kHz is -64dB, -56dB, and -52dB, respectively. The active chip area is 0.04mm2.