I Introduction
In multiobjective optimization, researchers generally assume that the objectives are conflicting to each other. In practical, however, this may not always be true. For example, when dealing with a dynamic optimization scenario, an engineer may not look carefully into the connection of the existing objectives, but rather add new objectives to accommodate new requirements. This may lead to some added objectives harmonious with the existing objectives (or their combinations). Such problems are called degenerate problems [1]. The degenerate problems widely exist in real world, such as multispeed gearbox design [2], stormdrainage system planning [3], car structure design [4, 5], and optimal product selection in software engineering [6].
Degenerate problems appear relatively rare in the evolutionary multiobjective optimization research. And more importantly, they are designed to serve some particular purpose or in accordance with specific landscape patterns, thus failing to represent the variety of realworld scenarios. For example, DebThieleLaumannsZitzler (DTLZ) 5 [7], DTLZ6 [7], and Walking fish group (WFG) 3 [8, 9] are three degenerate test problems whose PFs lie on 1D curves independent of the number of objectives, and all of the objectives except the last one objective are multiples of the first objective on their PFs. The DTLZ5 problem is further extended into DTLZ5(,) [10], an objective problem with specifiable essential objectives. To help the researchers to easily view and understand the search behavior of multiobjective optimizers, the multipoint distance minimization problem (MPDMP) [11, 12, 13] and the multiline distance minimization problem (MLDMP) [14, 15] are developed. MPDMP is to simultaneously minimize the distances of a point to a prespecified set of target points, and MLDMP aims to simultaneously minimize the distances of a point to a set of target lines. Since the Paretooptimal regions of these two problems in the decision space typically lie on a D manifold (regardless of the number of the objectives and the decision variables) [15], their PFs are also on a D manifold, and this makes them be degenerate problems. In [16], a set of problems whose PFs lie on D or D manifold are presented to stress the complexity of the Pareto optimal solutions in the decision space. In contrast, in [17] to emphasize the effectiveness of the Pareto optimal solutions in the objective space, another set of problems are proposed, where the redundant objectives are all equal to zero and the degenerate PF is determined only by the first part of the objectives.
On the other hand, objective reduction techniques have been receiving increasing attention in the evolutionary multiobjective optimization area [18, 19, 20, 21, 22, 13, 23]. However, the lack of a set of comprehensive degenerate problems may limit systematic investigations of their performance. Algorithms which perform well on existing degenerate test problems with particular properties (e.g., DTLZ5(, )) may not be able to work in reallife degenerate cases, where the correlation between objectives can be of high complexity.
In this paper, we propose a set of degenerate test problems aiming to reflect the generality of degenerate problems. The main contribution of this paper is twofold: 1) by analyzing the relation among the objectives of problems, we capture three characteristics of degenerate problems; 2) based on a uniform formulation and the captured characteristics, five test problems have been proposed. These problems contain a variety of representative characteristics and features, which enable researchers to investigate working mechanisms of different MOEAs on degenerate problems, particularly the objective reductionbased algorithms.
The remainder of this paper is organized as follows. Section II and Section III are devoted to the design principles and the description of three characteristics of test problems with degenerate PFs, respectively. Based on the principles and the analysis in the previous two sections, five test problems are proposed in Section IV. Section V Section V presents empirical results of 10 stateoftheart MOEAs on the proposed test problems and also the impact of the presented three characteristics for existing objectivereduction techniques. Finally, Section VI concludes the paper.
Ii Design Principles
In order to extend and generalize the test problems easily, we follow four basic principles to design the test problems as suggested in [7], [9] and [24].

The test problems can be constructed with a uniform formulation.

The test problems should be scalable to the number of the decision variables.

The test problems should be scalable to the number of the objectives.

The resulting PF of the problem should be exactly known, and the corresponding decision variable values should also be easy to find.
In this paper, we use the following uniform formulation for all the proposed test problems:
(1) 
with
(2) 
where
is the first part of decision vector and
is the other part, the functions are used to define the shape of Pareto front, known as the shape functions, define the fitness landscape, known as the landscape functions [24], are essential objective functions, are problem objectives, and are transforming functions which define the relation between the problem objectives and the essential objectives.Denoting , , , , and , we can rewrite the formulation as
(3) 
with
(4) 
where the symbol denotes a entrywise product operation, and is a vector of ones. Please note that (4) defines essential objective functions, and (3) transforms these essential objectives into another highdimensional space and obtains the final objective functions for the test problems.
For a given test problem formulated by (4) and (3), the goal of a multiobjective optimizer is to find the Pareto decision vectors such that and . Following this manner, we can use the designed to evaluate the ability of an algorithm to converge to the PF, and use the designed to test an algorithm’s ability to obtain diverse solutions.
Since this paper aims to present some important characteristics of the problems with degenerate PFs, we focus on the design of the transformations in (3), which controls the relationships between the final objectives and the essential objectives . For the definition of the essential objectives in (4), we simply select/design based on the existing definitions in [7] and [9]. In the next section, we will present the detailed characteristics of the proposed test problems with respect to the design of in (3).
Iii Problem Characteristics
As we mentioned before, the correlation between the objectives is not systematically considered in the existing test problems. In this section, we present the following three characteristics of the degenerate problems.
Iiia Explicitly Redundant Objectives
In manyobjective optimization, there exist some problems that explicitly have redundant objectives, i.e., the existence of these objectives do not have any impact on the solutions for the optimization problem. With regard to this case, we have the following theorem (for ease of explanation, assume that there are two essential objectives):
Theorem 1.
Supposing that there is an optimization problem with two conflicting objectives and , we further add another objective , where is a nondecreasing function with respective to and , then the new problem with the three objectives has the same Pareto solution set as the original twoobjective problem.
Proof:
See Appendix A. ∎
Based on Theorem 1, we can define test problems with arbitrary number of redundant objectives. If an algorithm can find the essential objectives of a problem, it can ignore all redundant objectives and potentially obtain the a good set of solutions for the original problem. Here, we give a general formulation of this type of problems. Let be conflicting objectives, then we add redundant objectives to the problem as:
(5) 
where are nondecreasing functions with respect to their corresponding inputs. From the definition of DTLZ5(, ) in [10], we can see that it is a special case of this test problem.
IiiB Implicitly Redundant Objectives
In the realworld applications, the number of essential objectives may be smaller than the number of the objectives of the underlying problem, while the essential objective set is not a subset of the objective set of the problem. For this case, let us first consider a objective minimization problem:
(6) 
with
(7) 
where is the number of decision variables, and . Fig. 1(a) shows the PF of this problem, from which we can see that the PF lies in a D manifold since the problem has only two essential objectives and . Generally, this problem cannot be well solved with the objective selectionbased reduction methods. The projections of the original objectives on subspaces , , and are plotted in Fig. 1(b)(d), respectively. From the plots, we can see that only part of the Pareto optimal solution set (PS) of the original objective problem and the PSs of the reduced objective MOPs are overlapped. Nevertheless, the PS of this problem can be obtained by optimizing the problem with the objective two essential objectives and .
Following the manner in the above example, we assume the essential (conflicting) objectives are , and then we construct a minimization problem with objectives:
(8) 
where are increasing functions, and . With regard to this case, we have the following theorem.
Theorem 2.
Suppose there exists an optimization problem with conflicting objectives , we construct another problem via (8), then the new problem with the objectives has the same PS as the problem with the objectives .
Proof:
See Appendix B. ∎
Since the essential objectives are not all included in the objective set of the problem, this type of problems cannot be well solved with objective selectionbased MOEAs. This type of problems is proposed to test the ability of algorithms to extract essential objectives.
IiiC Partially Redundant Objectives
The correlation between objectives may differ in different regions of the objective space. Two objectives may be harmonious on some parts of the PF, while they are unrelated/conflicting on some other parts of the PF. Let us consider a objective minimization problem:
(9) 
with
(10) 
where is the number of decision variables, and . Fig. 2 shows the PF of this problem, from which we can see that and are conflicting at , while they are equal at , and it makes the problem has a degenerate PF at this region, i.e., it leads to a partially degenerate test problem.
This type of test problems is proposed to test the ability of algorithms to discover degenerate segments of PF in the objective space.
Iv Problem Instances
Based on the basic principles and the above three characteristics, we present here a representative set of test problems with degenerate PFs, called as DPF^{1}^{1}1The MATLAB code of the proposed test problems is available at http://machineilab.org/users/zhenliangli/code/dpf.zip.. The essential objectives are carefully selected/designed with diverse properties which cover a good representation of various realworld scenarios, such as being multimodal, disconnected, partially separable, biased, and having different shapes of PFs. The characteristics and features of these five test problems are summarized in Tab. I. More test problems can also be constructed by designing different essential objective functions and transforming functions.
Problem  Redundant  Correlation  PF shape  Other features 

DPF1  Explicitly  Linear  Linear  Multimodal 
DPF2  Explicitly  Nonlinear  Mixed  Disconnected, 
DPF3  Implicitly  Linear  Concave  Bias 
DPF4  Implicitly  Nonlinear  Concave  Multimodal 
DPF5  Partially  Linear  Convex  Partially separable 
Iva Test Problem DPF1
In the first test problem, we construct an objective minimization problem with essential objectives. The explicitly redundant objectives are linearly correlated with the essential objectives. The objective functions of DPF1 are defined as
(11) 
with
(12) 
where is the number of decision variables, (typically set as ) denotes the number of elements in , and are column vectors with nonnegative elements.
The essential objectives simply refer to that in DTLZ1 [7]. The Pareto optimal solution corresponds to and the objective vectors lie on the linear hyperplane: . The difficulty of this problem is to select the essential objectives and converge to the hyperplane.
IvB Test Problem DPF2
The second constructed objective minimization problem has essential conflicting objectives as well. While the explicitly redundant objectives are nonlinearly correlated with the essential objectives. The problem is to minimize the following objective functions:
(13) 
with
(14) 
where is the number of decision variables, (typically set as ) denotes the number of elements in , are column vectors with nonnegative elements, and are nonlinearly and nondecreasingly mapping functions.
IvC Test Problem DPF3
The third problem is a objective minimization problem with essential objectives. The essential objectives do not explicitly exist in the problem, but implicitly exist as follows
(15) 
with
(16) 
where is the number of decision variables, (typically set as ) denotes the number of elements in , and . Please note that the obtained objective functions are partially linear with the essential objective functions in DPF3. From the definition of the essential objective functions, we can see that the search space has a variable density of solutions due to the bias transforming on the decision variables. The essential objective is locally linear correlated with the last objectives of the defined problem.
IvD Test Problem DPF4
In the fourth test problem, we construct an objective minimization problem with essential objectives. Different from DPF3, the objectives of this problem are nonlinearly correlated with the essential objectives as
(17) 
with
(18) 
where is the number of decision variables, (typically set as ) denotes the number of elements in , , and are nonlinear and nondecreasing mapping functions. This problem is multimodal as has local minima. Extracting essential objectives in this problem is more difficult than that in DPF3 because of the nonlinear mapping between the problem objective set and the essential objective set.
IvE Test Problem DPF5
In the fifth test problem, we construct a minimization problem whose correlations between the objectives differ in different PF segments as
(19) 
with
(20) 
where is the number of decision variables, (typically set as ) denotes the number of elements in . From the definition of DPF5, we can see that the objective vectors of the PF satisfy that , and the decision variables are partially separable. The PF of this problem contains a degenerate Paretooptimal segment of dimensions and a nondegenerate Paretooptimal segment of dimensions.
IvF Setting of Parameters and Mapping Functions in DPF
We assign randomly generated numbers to the elements of in DPF1DPF2 and the parameters in DPF3DPF4. Despite that the test instances are easy to implement, they may differ in different runs due to the randomly generated parameters, which makes the comparisons of the statistical results hard. To guarantee that these parameters are unchanged in different independent runs, we use a chaosbased pseudo random number generator by following [24]. The generated numbers are
(21) 
where and are parameters for the logistic map in (21), and typically set as and , respectively. The generated numbers are sequently assigned to the elements of in DPF1DPF2, and the parameters in DPF3DPF4 are set as the increasedly sorted results of the generated numbers.
To preserve the dominance relation between the decision vectors, the nonlinearly mapping functions in DPF2 and DPF4 have to be nondecreasingly functions and increasingly functions, respectively. The choice of the mapping functions is critical for the test problems. Many widelyused increasing functions can be selected to construct the instances of our proposed test problems. However, it is notable that different mapping functions induce different levels of difficulties to the test problems.
Fig. 3 shows two solution sets obtained by NSGAII on DPF2 with the mappings of the quadratic function
From Fig. 3(a), we can see that NSGAII can obtain a solution set that has a good convergence and diversity to the PF of the problem. While from Fig. 3(b), it is clear that most of the solutions are far from the PF. It illustrates that DPF2 with the sigmoid function is more difficult to be optimized than DPF2 with the quadratic function. The potential reason may be that the input values of the sigmoid function are mapped to values that are approximately equal to one, which decreases the distinction between two different inputs. In this paper, we adopt the quadratic function as the nonlinearly mapping function for DPF2 and DPF4.
Fig. 4 shows the scatter plots of the PFs of DPF1 to DPF5 with and . From the results, we can see that the PFs of DP1DPF4 lie in a D manifold, and the PF of DPF5 contains a curve and a part of a D spherical surface. Furthermore, this test suite has a variety of features, i.e., the Pareto optimal geometry, modality, PF shape, etc, and a set of recommendations, i.e., scalable number of objectives and variables, Pareto optima known, dissimilar tradeoff ranges, etc.
V Computational evaluations
This section is devoted to the experimental investigation of the proposed test problem, with the focus on its difference from existing degenerate problems. To do so, we first examine the performance of several stateoftheart MOEAs, most of which have been found to be promising in the existing degenerate problems. Then, we look into the impact of the proposed three characters and compare the proposed problems with a dominantlyused degenerate problem by demonstrating the performance difference of five representative objective reduction methods on them.
Va Tested MOEAs
In the experiments,
MOEAs are tested on the proposed test problems, including classical MOEAs, the algorithms designed specially for MaOPs, and MOEAs that are based on objective reduction. These ten MOEAs are the nondominated sorting genetic algorithm II (NSGAII)
[25], the multiobjective evolutionary algorithm based on decomposition (MOEA/D) [26], the indicatorbased evolutionary algorithm (IBEA) [27], the reference vector guided evolutionary algorithm (RVEA) [28], the strength Pareto evolutionary algorithm 2 [29]with the shiftbased density estimation (SDE) strategy
[30] (SPEA2+SDE), the algorithm for minimum objective subset problem (MOSS) [31], the algorithm for finding a minimum objective subset of size k with minimum error (kEMOSS) [31], the objective space participation based evolutionary algorithm (OSP) [21], the objective reduction algorithm based on nonlinear correlation information entropy (NCIE) [22], and the objective reduction algorithm with multiobjective search (ORMOS) [23]. Please note that the last five methods are objective reduction methods, and first four of them are incorporated into NSGAII and the ORMOS method is incorporated into SPEA2+SDE, to obtain the PS of MaOPs in our experiments.VB Parameter Settings for Tested MOEAs
A simulated binary crossover (SBX) with the probability
and a polynomial mutation with the probability (where denotes the number of decision variables) are used for all MOEAs, and their distribution indexes are both set as as recommended in [32]. The parameters in the MOEAs are set by following the suggestion in their original papers. MOEA/D has two commonlyused achievement scalarizing functions, Tchebycheff and penaltybased boundary intersection (PBI). In this study, we use the later one, and set the neighborhood size as and the penalty parameter as . For IBEA, we set the parameter as . For RVEA, the adaptive frequency and the parameter are set as and , respectively. The parameter is set as in MOSS. The parameter is set as the same value as in the test instance for kEMOSS. For OSP, the number of subsets of the partition is set as , and we set the threshold value of as for ORMOS. Since the algorithms MOSS, kEMOSS, NCIE, and OSP are incorporated into NSGAII, and ORMOS is incorporated into SPEA2+SDE, we execute the objective reduction for every generations in the NSGAII and SPEA2+SDE.VC Performance Metrics
To compare the performance of the MOEAs on the proposed test problems, the inverted generational distance (IGD) [33, 34] is adopted in the experiments.
IGD measures the average distance from the points in the PF to their closest solution in the obtained solution set. It can provide a combined information about convergence and diversity of a solution set [34]. Mathematically, let be a reference set representing the PF, and be a set of solutions obtained by an MOEA. The IGD value between and is defined as
(24) 
where denotes the minimal Euclidean distance from to the elements in . A small IGD value indicates that the obtained solution set is close to the PF and has a good distribution as well. To calculate the value of IGD, we have to provide a reference set representing the PF. In our experiments, points are uniformly sampled from the true PFs to construct the set of .
VD Results of MOEAs
In the proposed test problems, three parameters should be provided, i.e., the number of objectives , the number of essential objectives , and the number of decision variables . In the first experiment, we test the instances with and , respectively. The number of decision variables is set as the recommending value as stated in the corresponding problems. The maximum number of generations is taken as the termination condition, which is set to and for the instances with , and objectives, respectively. For MOEA/D and RVEA, the population size is determined by simplexlattice design factor and the number of objectives. We follow the setting in [28] where the population size is specified to and for the problems with , and objectives, respectively. In this experiment, times of Monte Carlo simulations are conducted on each instance for each algorithm, and the statistical results of the MOEAs are reported in Tab. II.
For objective test instances, the Paretobased algorithms NSGAII and SPEA2+SDE achieve competitive performance results. The objective reductionbased algorithms, MOSS, kEMOSS, and NCIE, improve the performance of NSGAII slightly on DPF1 and DPF2, while they are inferior to NSGAII on DPF3 and DPF4. This is due to that these four algorithms select a subset of the original objective set as the criterions to optimize the problem, they perform well on the problems with explicitly redundant objectives, e.g., DPF1 and DPF2, but might not work on the problems with implicitly redundant objectives, e.g., DPF3 and DPF4. ORMOS performs worse than SPEA2+SDE on all of the test problems and fails to converge to the PF on DPF4. The decompositionbased methods, MOEA/D and RVEA perform not well on these degenerate problems as only a small proportion of the weight vectors are close to the PF. The kEMOSS algorithm performs worst on DPF5 since kEMOSS only selects two of the objectives to optimize at one time but DPF5 has a nondegenerate PF region.
For objective test instances, SDE and ORMOS obtain the best IGD values on DPF1 and DPF2, respectively. IBEA achieves the best results on DPF3 and DPF5, and NSGAII outperforms the others on DPF4. In addition, the objective selectionbased methods, MOSS, kEMOSS, NCIE, OSP, and ORMOS can obtain competitive performance compared with the best performance algorithms on DPF1 and DPF2, where the essential objectives are explicitly included in the objective set of the problem. While they are much inferior to the best performance algorithms on DPF3 and DPF4, where the essential objectives are not explicitly included. The objective reduction methods have to extract the essential objectives instead of selecting them from the objective set of the problem in DPF3 and DPF4.
In terms of objective test instances, SPEA2+SDE achieves the best IGD values on DPF1 and DPF5, IBEA on DPF2 and DPF3, and NSGAII on DPF4. Some of the objective reductionbased algorithms perform better than its integrated method, and some of them perform worse than its integrated method. However, the objective reductionbased algorithms potentially have a much lower computational time cost than the integrated methods since much fewer objectives are considered in the selection stage of the evolution. We can also see that the performance of all tested MOEAs drop dramatically on DPF2, DPF3, and DPF4 when compared with their performance on objective test instances. To better analyze the results on different test problems, we show the parallel coordinates plot of the results obtained by the algorithms that achieved the best IGD values in the runs in Fig. 5. From the results, we can see that even SPEA2+SDE obtains the best IGD value on DPF1 and DPF5, it fails to find the solutions on the boundary of the PF, and only few of its solutions cover the degenerate part of the PF since the solutions on the degenerate PF should have the same value on the first objectives in Fig. 5(e). From Fig. 5(d), we can see that the objective value range of the solutions obtained by NSGAII is approximately from to , which is far from the objective value range of the PF from to .
From the above, it can be found that
1) Paretobased algorithms (or their variants) are good options for lowdimensional degenerate problems, such as NSGAII and SPEA2+SDE;
2) Decompositionbased algorithms fail to obtain good results on degenerate problems since a large proportion of the weight vectors may be far from the PF;
3) None of all the tested algorithms can obtain good performance on the highdimensional degenerate instances, which has shown the difficulty of the proposed problem suite.
Test instance  Method  DPF1  DPF2  DPF3  DPF4  DPF5 

m = 3, d = 2  NSGAII  3.80E03 (1.22E03)  2.09E02 (7.12E04)  5.17E03 (2.86E04)  6.59E03 (3.55E04)  4.06E02 (2.02E03) 
MOEA/D  8.26E03 (5.44E03)  1.85E+00 (8.76E02)  6.31E02 (1.38E01)  1.85E01 (4.28E03)  4.60E02 (3.07E05)  
IBEA  8.28E02 (1.06E02)  2.32E02 (6.79E04)  3.93E02 (1.43E01)  7.42E01 (2.41E02)  5.27E02 (2.37E03)  
RVEA  8.93E02 (1.32E01)  4.11E01 (8.69E02)  7.36E02 (1.35E01)  6.00E01 (4.42E01)  4.62E02 (3.77E05)  
SPEA2+SDE  2.46E03 (2.16E04)  2.05E02 (8.16E04)  9.88E03 (1.78E03)  9.16E02 (2.76E02)  4.84E02 (2.83E03)  
MOSS  3.52E03 (9.31E04)  2.10E02 (4.16E03)  3.74E02 (1.47E01)  2.74E02 (6.60E02)  4.14E02 (2.31E03)  
kEMOSS  3.38E03 (9.21E04)  2.00E02 (6.61E04)  2.71E01 (9.01E02)  2.26E01 (4.14E04)  6.27E01 (1.30E01)  
NCIE  3.49E03 (1.21E03)  2.01E02 (6.07E04)  9.27E02 (9.53E03)  2.48E+03 (1.47E+03)  1.23E01 (2.05E01)  
OSP  2.91E02 (5.24E02)  5.14E02 (2.95E02)  4.85E02 (1.78E02)  2.35E01 (1.76E01)  2.90E01 (8.31E02)  
ORMOS  1.63E02 (6.06E02)  2.06E02 (6.51E04)  3.81E02 (1.32E01)  5.27E+04 (9.95E+04)  4.84E02 (1.90E03)  
m = 6, d = 3  NSGAII  2.72E02 (1.36E03)  3.40E01 (1.37E01)  6.17E02 (2.34E03)  8.77E02 (4.72E03)  3.20E01 (1.94E02) 
MOEA/D  4.49E02 (1.82E04)  6.82E+00 (1.49E+00)  1.84E01 (1.31E01)  3.30E01 (4.72E03)  2.66E01 (1.49E05)  
IBEA  1.75E01 (2.25E02)  4.34E01 (3.65E01)  6.16E02 (3.43E03)  9.50E01 (1.93E02)  2.44E01 (4.47E03)  
RVEA  1.06E01 (3.43E02)  3.90E+00 (2.37E+00)  1.82E01 (5.46E02)  3.00E01 (4.68E02)  2.75E01 (1.37E04)  
SPEA2+SDE  2.05E02 (2.44E04)  2.76E01 (1.77E02)  6.40E02 (3.23E03)  1.95E01 (1.80E02)  2.54E01 (9.90E03)  
MOSS  5.33E02 (2.20E02)  7.02E01 (1.20E+00)  6.66E02 (3.30E03)  1.33E01 (3.70E02)  3.15E01 (1.42E02)  
kEMOSS  2.70E02 (1.15E03)  4.09E01 (1.46E01)  2.34E01 (7.53E02)  3.58E01 (2.07E02)  7.96E01 (5.23E02)  
NCIE  2.73E02 (1.17E03)  2.88E+00 (1.47E+00)  3.21E01 (2.48E01)  5.47E01 (1.19E01)  9.78E01 (1.78E01)  
OSP  3.58E02 (2.25E03)  1.32E+00 (8.98E01)  2.01E01 (6.58E02)  2.48E01 (7.38E02)  7.29E01 (2.96E02)  
ORMOS  1.84E01 (1.04E02)  2.50E01 (1.48E02)  1.08E+00 (2.13E01)  3.15E+05 (1.46E+05)  2.55E01 (1.10E02)  
m = 10, d = 5  NSGAII  9.28E02 (2.32E03)  2.12E+00 (5.86E02)  1.85E01 (3.08E03)  2.56E01 (5.67E03)  2.45E+00 (2.55E01) 
MOEA/D  9.27E02 (8.07E04)  2.19E+01 (5.97E+00)  3.94E01 (1.01E01)  5.48E01 (7.69E03)  4.22E01 (1.44E04)  
IBEA  2.15E01 (2.04E02)  1.90E+00 (9.37E02)  1.75E01 (2.86E03)  1.17E+00 (9.51E03)  4.11E01 (3.38E03)  
RVEA  1.65E01 (1.74E02)  2.01E+01 (4.61E+00)  3.89E01 (2.87E02)  4.78E01 (2.95E02)  4.23E01 (4.38E04)  
SPEA2+SDE  5.42E02 (2.10E04)  2.13E+00 (1.37E01)  1.80E01 (3.84E03)  3.75E01 (1.23E02)  4.00E01 (4.26E03)  
MOSS  9.35E02 (1.07E02)  2.11E+00 (7.44E02)  1.94E01 (5.37E03)  3.48E01 (1.46E01)  2.40E+00 (2.36E01)  
kEMOSS  8.37E02 (1.18E02)  6.79E+00 (2.37E+00)  2.92E01 (6.05E03)  4.09E01 (7.45E03)  7.45E01 (8.71E02)  
NCIE  1.91E01 (9.07E02)  6.13E+00 (2.22E+00)  4.67E01 (9.09E02)  9.06E01 (1.17E01)  1.10E+00 (6.66E02)  
OSP  9.09E02 (5.11E03)  1.93E+00 (2.21E01)  2.19E01 (1.76E02)  2.87E01 (1.59E02)  6.56E01 (4.27E02)  
ORMOS  1.74E01 (3.26E02)  2.50E+00 (9.32E02)  1.41E+00 (2.09E01)  3.25E+05 (1.41E+05)  7.97E01 (2.07E01) 
The statistical results (mean and standard deviation) of the IGD values on the proposed test problems. The best result regarding the mean for each problem instance is highlighted in boldface.
VE Impact of the Proposed Three Characteristics
In the last section, we have presented the performance of the current state of the arts on the proposed functions and have found that these functions provide big challenges to the algorithms. However, we may not be able to conclude that this underperformance of the considered algorithms is caused by the proposed three characteristics since the tested functions are of different features with respect to their essential objectives (e.g., multimodal, bias and disconnected). To investigate the impact of the proposed three characteristics, in this section we modify the original DPF by making them have the same essential objectives as DTLZ5(, ), called DPF1A–DPF5A (see Appendix C). Therefore, the performance difference of algorithms between these functions can fully boil down to the proposed characteristics.
In this experiment, we compare the performance of five objective reductionbased MOEAs on DPF1ADPF4A, DPF5 and DTLZ5(, ) [10]. We test the instances with for DTLZ5(, ), and for the proposed problems. The number of decision variables of DTLZ5(, ) is set as by following the recommendation in [10], and the same setting for DPF1A–DPF5A and DPF5. We run the tested MOEAs for times with the population size of and the maximum number of executing generations .
The results of the run with the best IGD value (measured in the essential objective space) on DTLZ5(, ) and on the proposed problems are shown in Fig. 6 and Fig. 7, respectively. From the results, we have the following observations:
1) The five algorithms, MOSS, kEMOSS, NCIE, OSP and ORMOS, all can obtain a solution set that has a good convergence and diversity to the PF of DTLZ5(, ). Since the first objectives of DTLZ5(, ) are linearly dependent with each other, it makes easy for the objective reduction algorithms to discover the essential objectives of the problem.
2) The solution sets of OSP and ORMOS have a poor diversity on DPF1A compared with the results on DTLZ5(, ), and the other three algorithms can obtain fairly well results. On DPF2A, the performance of kEMOSS and NCIE decrease slightly compared with the results on DPF1A, and OSP and ORMOS still cannot obtain diverse solution sets. In addition, the results obtained by these algorithms on DPF1A and DPF2A are not as diverse as the results on DTLZ5(, ) since the relation between the redundant objectives and the essential objectives is mutually linearly correlated in DTLZ5(, ), linearly correlated (but not mutually linearly correlated) in DPF1A, and nonlinearly correlated in DPF2A.
3) ORMOS fails to converge to the PF on DPF3A and DPF4A. The other four methods can converge to the PF, but they fail to maintain the diversity of the solution set. These two test problems are harder than DPF1A and DPF2A since their redundant objectives exist implicitly whereas the redundant objectives of DPF1A and DPF2A exist explicitly.
4) None of these algorithms can obtain good results on the degenerate part of the PF on DPF5A. We also show the parallel coordinate plot of the whole solution sets of these five algorithms in Fig. 8, from which we can see that a large number of the solutions obtained by MOSS do not converge to the PF. The objective value range of MOSS’s solutions is approximately from to , which is far from that of the PF (from to ). The value of the last objective for the degenerate part of DPF5’s PF is larger than . OSP has only one solution lie on the degenerate part of the PF as shown in Fig. 7(x). Even though kEMOSS, NCIE and ORMOS can obtain some solutions that lie on the degenerate part of the PF on DPF5A, they fail to obtain diverse solutions in the nondegenerate part of the PF. It can be seen from Fig. 8(b), Fig. 8(c) and Fig. 8(e) that a large part of the PF is not overlaid with the solutions of kEMOSS, NCIE and ORMOS.
From the above observations, we can see that the proposed three characteristics have a significant impact on the performance of the existing algorithms. They bring different types of difficulties for objective reduction techniques. The implicit redundancy among objectives posts a big challenge to the objective reduction methods based on objective selection. The partial redundancy makes all the methods struggle to find/maintain welldistributed solutions on both degenerate and nondegenerate segments of the PF.
Vi Conclusion
This paper discusses three characteristics that lead to MOPs degenerate, i.e., explicitly redundant objectives, implicitly redundant objectives, and partially redundant objectives. The first two characteristics make the problem with a complete degenerate PF, while the third one results in a partially degenerate PF for the problem.
Five test problems are instantiated based on these three characteristics with a uniform formulation. Among them, DPF1 and DPF2 have explicitly redundant objectives, DFP3 and DPF4 have implicitly redundant objectives, and DPF5 has partially redundant objectives. DPF1 and DPF2 are designed to test the algorithm’s ability of objective selection, DPF3 and DPF4 to test the algorithm’s ability of objective extraction, and DPF5 to test the algorithm’s ability to maintain different subpopulations on the degenerate and the nondegenerate segments of the PF.
Ten representative MOEAs have been tested on the proposed problems. In contrast to existing degenerate problems, our problems have introduced new features (with varying difficulty) that can challenge various objective reduction methods. This has been evidenced in our experimental studies where none of tested MOEAs is able to well solve all the proposed problems. This, therefore, suggests a need of developing new methods to solve MOPs with degenerate PFs.
Appendix A Proof of Theorem 1
Proof of Sufficiency: For any , if in the original objective space, we have that
(25) 
and satisfies
(26) 
Considering a new objective , we obtain that
(27) 
Combining (27) and (25), and based on the fact that is a nondecreasing function corresponding to , it flows that
(28) 
such that we can draw the conclusion that the in the new objective space.
Proof of Necessity: For any , if in the new objective space. Since the set of the original objectives is a subset of the set of the new objectives, there are two cases:
a) in first objective space, which directly completes the proof.
b) the objective values of and are equal on the original objectives, i.e.,
(29) 
and satisfies
(30) 
From the definition of the objective , we have
(31) 
Appendix B Proof of Theorem 2
Let us first consider the situation of the problem with three objectives, and there are only two essential objectives.
Proof of Sufficiency: For any , if in the original objective space, we have that
(32) 
and satisfies
(33) 
Supposing that
(34) 
we obtain that
(35) 
and the following three possibilities:
Case 1):
(36) 
Case 2):
(37) 
Case 3):
(38) 
Proof of Necessity: For any , if in the new objective space, we have that
(39) 
and satisfies
(40) 
If , it is clear that
(41) 
If , we have that
(42) 
and the following two possibilities:
Case 1):
(43) 
Case 2):
(44) 
Combining the results in (41) and (42)(44), we find that holds in the original objective space as well.
It is clear that the above analysis is also true to the situation with any number of objectives. This completes the proof.
Appendix C Essential Objectives of DPF1ADPF5A
The definition of the essential objectives of DPF1ADPF4A and the degenerate part of DPF5A is the same as that of DTLZ5(, ) [10]:
(45) 
where is the number of decision variables, and .
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