NSGA3算法详解：改进版NSGA2处理约束与自适应优化

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NSGA3论文深入探讨了多目标优化领域的最新进展，它是NSGA2算法的一个重要改进版本。NSGA2（非支配排序遗传算法II）以其在解决无约束多目标优化问题中的卓越性能而闻名，但面对实际问题中的约束条件，其局限性逐渐显现。NSGA3论文旨在解决这一挑战，通过结合参考点基非支配排序方法，将NSGA2框架扩展到处理广泛的多目标优化问题，并包括约束条件。首先，论文介绍了NSGA3的核心求解流程，它在原有的NSGA2基础上，引入了新的策略来处理约束问题。这可能包括对个体适应度函数的修改，使其不仅考虑目标函数的优化，还要满足约束条件。作者提出了三种类型的可扩展约束测试问题，这些问题能够适应任意数量的目标，同时对多目标优化算法构成了不同类型的挑战。通过这些测试问题，研究者可以评估算法在处理约束情况下的稳健性和效率。此外，论文还涉及对先前提出的MOEA/D（多目标演化算法/分解）算法的扩展，使其也能适应有约束的优化环境。MOEA/D是一种广泛应用的多目标优化算法，但通过与NSGA3的对比，研究者发现NSGA3在处理具有大量目标的约束问题时表现更为出色。这表明NSGA3在解决复杂优化问题时具有更高的灵活性和有效性。最后，NSGA3算法被设计成一种自适应策略，这意味着它能够在运行过程中动态调整自身的参数和行为，以更好地适应不断变化的问题特性。这种适应性使得NSGA3能够更高效地在实际应用中找到平衡点，即在满足约束的同时优化多个目标。这篇NSGA3论文提供了重要的理论支持和实践指导，对于多目标优化算法的发展以及在工程、科学等领域的应用具有重要意义。通过理解和分析这篇论文，研究人员和工程师可以了解到如何有效应对多目标优化问题中的约束条件，提高算法的实用性和性能。

606 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 4, AUGUST 2014

TABLE I

Number of Reference Points/Directions and Corresponding

Population Sizes Used in Constrained NSGA-III

and C-MOEA/D Algorithms

TABLE II

Parameter Values Used in Constrained NSGA-III and

C- MOEA/D. n Is the Number of Variables

do not use the DE operator, instead a real-coded GA with SBX

and polynomial mutation operators are used for creating the

offspring population. We also propose the use of PBI metric

(instead of the Tchebycheff metric), as PBI metric was found

to work better in the original study [1]. We name this version

of MOEA/D as constrained MOEA/D or simply C-MOEA/D.

V. Results

In this section, we present simulation results of the proposed

constrained NSGA-III and C-MOEA/D approaches. For this

purpose, we use a number of constrained test problems with

three to 15 objectives, designed to introduce different types of

difﬁculties to an algorithm. The problems are scalable both in

the number of objectives and in the number of variables.

For each problem, 20 different runs with different initial

populations are carried out, and the best, median, and worst

IGD performance values (which can only be computed for a

test problem with a known Pareto-optimal front) are reported.

To compute IGD values, ﬁrst, we compute the targeted points

eff

) on the known Pareto-optimal front from the supplied

reference points or directions in the normalized objective

space. Then, for an algorithm, we obtain the ﬁnal nondom-

inated points (set A) in the objective space. Now, we compute

the IGD metric value as the average Euclidean distance of

points in set Z

eff

with their nearest members of all points in

set A

IGD(A, Z

eff



i=1

|A|

min

j=1

d(z

, a

) (3)

where d(z

, a

)=z

− a



. For both algorithms, the

population members from the ﬁnal generation are presented

and used for computing the above IGD metric. The number

of reference points, population size, and other parameters

are kept in agreement with the original study [1] and are

tabulated in Tables I and II. In the case of C-MOEA/D,

two parameters δ (probability with which the parent solutions

are selected from the neighborhood) and n

(maximal number

of solutions replaced by an offspring solution) are set as 0.9

and 2, respectively, as suggested by the developers in [15].

In contrast, the proposed constrained-handling NSGA-III does

not require to set any new parameter.

A. Constrained Problems of Type-1

In Type-1 constrained problems, the original Pareto-optimal

front is still optimal, but there is an infeasible barrier in ap-

proaching the Pareto-optimal front. This is achieved by adding

a constraint to the original problem. The barrier provides

infeasible regions in the objective space that an algorithm must

learn to overcome, thereby providing a difﬁculty in converging

to the true Pareto-optimal front. DTLZ1 and DTLZ3 problems

[24] are modiﬁed according to this principle in this paper.

For the type-1 constrained DTLZ1 (or C1-DTLZ1), only a

part of objective space that is close to Pareto-optimal front is

made feasible, as shown in Fig. 1. The objective functions are

kept the same as they were in the original DTLZ1 problem,

while the following constraint is now added:

c(x)=1−

(x)

0.6

−

M−1



i=1

(x)

0.5

≥ 0. (4)

The feasible region and the Pareto-optimal front are shown

for a two-objective C1-DTLZ1 problem in Fig. 1. In all

simulations, we use k = 5 variables for the original g-function

[24], thereby making a total of (M + 4) variables to the

M-objective C1-DTLZ1 problem.

In the case of the C1-DTLZ3 problem, a band of infeasible

space is introduced adjacent to the Pareto-optimal front, as

shown in Fig. 2. Again, the objective functions are kept

the same as in the original DTLZ3 problem [24], while the

following constraint is added:

c(x)=





i=1

(x)

− 16





i=1

(x)

− r



≥ 0 (5)

where r = {9, 12.5, 12.5, 15, 15} is the radius of the hyper-

sphere for M = {3, 5, 8, 10, 15}. For C1-DTLZ3, we use k =

10 so that total number of variables is (M+9)inanM-objective

problem.

Both algorithms (NSGA-III and C-MOEA/D) are tested on

three- to 15-objective versions of the above two problems.

Fig. 3 shows that in the case of three-objective C1-DTLZ1

problem, NSGA-III is able to reach the feasible region and

ﬁnd a well distributed set of points on the entire Pareto-optimal

front. C-MOEA/D is also able to ﬁnd a nice distribution of

points (see Fig. 4). However, as is evident from Table III,

in most cases for the C1-DTLZ1 problem, NSGA-III per-

forms better than C-MOEA/D in terms of the IGD metric.

Interestingly, the best performance of C-MOEA/D is in most

cases better than that of NSGA-III. However, as the number

of objectives increases (10- and 15-objective problems), the

performance of NSGA-III is clearly better.

In addition, we compute the GD metric value for NSGA-III

solutions and tabulate the best, median, and worst values

in Table III. Small GD values indicate that NSGA-III so-

lutions are close to the true Pareto-optimal fronts in each

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