用户偏好的进化算法解决混合区间多目标优化问题

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本文主要探讨了"Evolutionary algorithms with user's preferences for solving hybrid interval multi-objective optimization problems"这一主题，发表在ApplIntell期刊上，DOI为10.1007/s10489-015-0658-x。该研究由Dunwei Gong、Yiping Liu、Xinfang Ji和Jing Sun四位作者共同完成，且收录于Springer Science+Business Media New York于2015年出版的一期。在现实世界的应用中，混合区间多目标优化问题（Hybrid Interval Multi-objective Optimization Problems, HIMOPs）非常常见，这类问题既包含明确的（explicit）目标，也包含隐含的（implicit）目标，且各目标值都是区间型的。以往的方法往往难以有效处理这类复杂问题。本文提出了一种创新的进化算法，它具有大规模种群以及用户偏好的集成，目的是为了更有效地解决此类问题。算法的核心在于用户参与：首先，通过一种基于相似性的策略，当用户未评价某个个体的隐含目标时，算法能够估算这些目标的区间值，从而减轻用户的评估负担。这降低了用户的认知负荷，提高了问题求解的用户体验。其次，用户对不同目标的偏好被精确地表示为区间，通过解决一个辅助优化问题来实现，使得算法能够更好地适应用户的需求和期望。排序操作在算法中也扮演着关键角色，通过对种群中的个体按照用户偏好进行排序，优化过程能够更加聚焦于那些满足用户优先级的目标。这种方法不仅考虑了问题的全局最优，还充分考虑了个体的局部满意，使得解决方案更具实用性。这篇研究论文提供了一个新颖的进化算法框架，它巧妙地融合了用户偏好和区间优化技术，对于实际工程中的混合区间多目标优化问题具有显著的优势。这种算法的引入有望推动多目标优化领域的发展，使得问题解决更为个性化和高效。

D. W. Gong et al.

obtained until the most satisfactory solution is found [14].

In the method proposed by Deb et al. a strictly increasing

function is constructed to reflect the user’s preferences after

the population has evolved for a number of generations set

in advance. Then the function is used to compare differ-

ent solutions to guide the population evolving towards the

most preferred region [15]. In the study of Sinha et al. a

preference-based methodology is proposed, where the pref-

erences provided by the user in the intermediate runs of an

evolutionary multi-objective optimization algorithm is used

to construct a polyhedral cone, which is used to eliminate

a part of search space and conduct a more focused search.

The dominance principle is modified to search for better

solutions lying in the preferred region [16].

Expression of the user’s preferences is also important

besides the means of embedding them. There are various

expression forms of the user’s preferences, e.g., reference

point [17], reference direction [18], solution ranking [19],

and so on. In the work of Bently et al. the importance of

objectives is defined, allowing evolutionary algorithms to

converge on a small subset of acceptable non-dominated

solutions [20]. It is convenient and common to adopt

weights to represent importance values of different objec-

tives. However, it is difficult and less reasonable to specify

a precise weight for the user aprioribecause of the uncer-

tainty of the user’s cognition. Therefore, the user’s prefer-

ences to different objectives were expressed with intervals

in this study to alleviate the user’s burden resulting from

assigning precise weights. Note that both the optimization

problem and the user’s preference have uncertainties, these

make the problem considered here more complex. There-

fore, appropriate methods are required to make the user’s

preferences precise. In the work of Wang et al., a method

of calculating the user’s precise preferences is obtained by

deduction, and different schemes are compared based on

the corresponding intervals accumulated from all attributes

[21]. In this study, the user’s interval preferences were made

precise by solving an auxiliary optimization problem in

view of the idea of [21].

3 Proposed method

3.1 Overview

In the framework of NSGA-II, a novel evolutionary opti-

mization algorithm with a user’s preferences to solve the

problem described with formula (1) is presented in this

section. In the algorithm, the user is requested to input

his/her interval preferences to different objectives before the

evolution; a large population is adopted and theK-means

clustering [25] is utilized to reasonably cluster it; then the

computer is employed to calculate the values of explicit

objectives of all evolutionary individuals, and the user eval-

uates only the values of implicit objectives of the centers

of all clusters; for evolutionary individuals not evaluated

by the user, a similarity-based estimation strategy is used

to estimate the values of implicit objectives described in

subsection 3.3. The user’s interval preferences to differ-

ent objectives are quantified by solving a single-objective

optimization problem with constraints, elaborated in sub-

section 3.4. In addition, a sorting scheme based on the user’s

preferences is proposed to steer the search towards the user’s

preferred region, given in subsection 3.5. The user can mod-

ify his/her preferences during the evolution. The steps of the

proposed method are as follows:

1. Set the values of evolutionary control parameters in the

method, and initialize a population x(t), t = 0.

2. Input the user’s interval preferences to each objective.

3. Divide x(t) into N

clusters using the method in [25].

4. Decode each evolutionary individual, evaluate all

implicit objectives of the centers of all clusters by the

user, estimate the values of all implicit objectives of the

rest evolutionary individuals in all clusters according to

(4), and calculate the values of all explicit objectives of

all evolutionary individuals by the computer.

5. Make the user’s interval preferences precise by solving

an auxiliary optimization problem.

6. Employ the tournament selection with size 2, and

perform crossover and mutation operations to create

offspring populations.

7. Combine parent and offspring populations.

8. Rank the combined population according to the sort-

ing scheme based on the user’s preferences, and select

superior individuals to form a new population, x(t + 1).

9. Judge whether termination criteria have been met or

not, if yes, stop the evolution and output the optimal

solutions; otherwise, let t = t + 1. If the user’s prefer-

ences are changed, go to Step 2; otherwise, go to Step

The flow chart of the proposed method is depicted as

Fig. 1 according to the above steps, and the shaded parts will

be introduced in the following subsection.

3.2 Relation to our previous work

As stated in Section 1, based on the characteristics of uncer-

tainties contained in the two types of objectives, HIMOP

is first transformed into one without uncertainties by using

interval analysis, and then an evolutionary optimization

algorithm is employed to effectively solve the transformed

problem by combining a traditional GA [4]. To be solved,

however, is the optimization problem without uncertain-

ties, transformed from the original one. In our recently

proposed method for interval MOPs, a novel interval

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