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亲爱的您，
这里是LearningYard学苑！
今天小编给大家带来期刊论文精读，
欢迎您的用心访问！
本期推文阅读时长大约6分钟，请您耐心阅读。
Share interest, spread happiness, increase knowledge, and leave beautiful.
Dear you,
this is the LearningYard Academy!
Today, the editor brings you intensive reading of journal papers,
welcome your visit!
This tweet usually takes about 6 minutes to read. Please be patient and read.
今天小编将从思维导图、精读内容、知识补充三个板块为大家带来论文《属性关联的双极容度多属性决策 VIKOR方法》步骤四中相关内容，接下来我们开始今天的学习吧！
Today, the editor will introduce to you the relevant content in step four of the paper "VIKOR Method for Bipolar Capacity Multi attribute Decision Making with Attribute Correlation" from three sections: mind mapping, intensive reading content, and knowledge supplementation. Let's start today's learning!
思维导图
本节内容思维导图如下所示：
A mind map of the contents of this section is shown below.
精读内容
在上期推文中，我们搞清楚了最大熵模型的含义，接着咱们一起来最大熵模型中的原始问题。
In the previous tweet, we clarified the meaning of the maximum entropy model, and then let's discuss the original problem in the maximum entropy model together.
首先，让我们还是简单地回顾一下最大熵模型的含义吧。
Firstly, let's briefly review the meaning of the maximum entropy model.
由前面的学习可以知道，这其实是个优化问题，目的就是要找到使条件熵最大的那个概率分布，而这个分布是要从待选模型集合中选出。因为它的特点是概率分布，找到的所有集合就必须满足：
From the previous learning, it can be seen that this is actually an optimization problem, with the aim of finding the probability distribution that maximizes the conditional entropy, which needs to be selected from the set of models to be selected. Because its characteristic is probability distribution, all sets found must satisfy:
在解决最大熵模型问题之前，我们可以先学习一下一般的带有约束条件的优化问题，只要一般的问题会了，那么这个最大熵模型的知识掌握会相对容易理解一些。直接地，我们先从原始问题来看。
Before solving the maximum entropy model problem, we can first learn about general optimization problems with constraints. As long as the general problem is solved, the knowledge of this maximum entropy model will be relatively easy to understand. Directly, let's start with the original problem.
从上图中我们可以看到，原始问题的目的是找到可以使f(x)最小的向量X。观察约束条件公式，可以发现问题中的约束条件个数为（k+l）个。一个目标函数携带着n个未知数，以及（k+l）个约束条件，拉格朗日提出了拉格朗日乘子法，可以将约束的自由化问题写成广义的拉格朗日函数形式。如下所示。
From the above figure, we can see that the purpose of the original problem is to find the vector X that can minimize f (x). Observing the constraint formula, it can be observed that the number of constraint conditions in the problem is (k+l). An objective function carries n unknowns and (k+l) constraints. Lagrange proposed the Lagrange multiplier method, which can write the constraint liberalization problem in the form of a generalized Lagrange function. As shown below.
注意：这里的原始问题用来求最小值，那么我们可以将最大熵模型求最大值问题取反后求最小值，也是同样的含义。
Note: The original problem here is used to find the minimum value, so we can reverse the maximum entropy model problem and find the minimum value, which also has the same meaning.
对于每个约束条件前面都来加一个拉格朗日乘子，我们考虑关于x的函数，这就意味着代入已知的若干组 α和β、 找到可以使L(x， α，β)最大的值，接着把这个函数记为：
For each constraint condition, a Lagrange multiplier is added in front of it. We consider the function of x, which means that we can substitute several known groups α and β、 Find a way to make L (x α，β) The maximum value, then record this function as:
知识补充
上文中，我们学习了拉格朗日函数，接下来，和小编一起了解一下拉格朗日吧！
In the previous text, we learned about Lagrange functions. Next, let's learn about Lagrange with the editor!
拉格朗日是18世纪最伟大的数学家之一，拿破仑曾称赞拉格朗日是“一座高耸在数学界的金字塔”。他最突出的贡献是在把数学分析的基础脱离几何与力学方面起了决定性的作用。使数学的独立性更为清楚，而不仅是其他学科的工具。同时在使天文学力学化、力学分析化上也起了历史性作用，促使力学和天体力学更深入发展。
Lagrange was one of the greatest mathematicians of the 18th century, and Napoleon praised him as "a towering pyramid in the mathematical world". His most outstanding contribution is that he played a decisive role in separating the basis of mathematical analysis from geometry and mechanics. Make the independence of mathematics clearer, rather than just a tool for other disciplines. At the same time, it has also played a historic role in making astronomy more scientific and mechanical analysis, promoting the further development of mechanics and celestial mechanics.
约瑟夫·拉格朗日（1736~1813），法国著名数学家、物理学家。1736年1月25日生于意大利都灵，1813年4月10日卒于巴黎。他在数学、力学和天文学三个学科领域中都有历史性的贡献，其中尤以数学方面的成就最为突出。
Joseph Lagrange (1736-1813) was a famous French mathematician and physicist. Born on January 25, 1736 in Turin, Italy, and died on April 10, 1813 in Paris. He has made historic contributions in the fields of mathematics, mechanics, and astronomy, with his achievements in mathematics being the most prominent.
拉格朗日是数学分析的开拓者。牛顿和莱布尼兹以后的欧洲数学分裂为两派。英国仍坚持牛顿在《自然哲学中的数学原理》中的几何方法，进展缓慢；欧洲大陆则按莱布尼兹创立的分析方法（当时包括代数方法），进展很快，当时叫分析学。拉格朗日是仅次于欧拉的最大开拓者，在18世纪创立的主要分支中都有开拓性贡献。
Lagrange is a pioneer of mathematical analysis. After Newton and Leibniz, European mathematics split into two schools. The UK still adheres to Newton's geometric methods in "Mathematical Principles in Natural Philosophy", but progress is slow; The European continent followed the analytical method established by Leibniz (which at the time included algebraic methods) and made rapid progress, which was called analysis at that time. Lagrange was the largest pioneer, second only to Euler, and made pioneering contributions to the main branches established in the 18th century.
今天的分享就到这里了。
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让我们相约明天，
祝您今天过得开心快乐！
That's it for today's sharing.
If you have a unique idea about today’s article,
welcome to leave us a message,
let us meet tomorrow,
I wish you a nice day today!
参考资料：DeepL翻译、百度百科
参考文献：林萍萍, 李登峰, 江彬倩, 余高锋, 韦安鹏. 属性关联的双极容度多属性决策VIKOR方法 [J]. 系统工程理论与实践, 2021, 41(8): 2147-2156.
文案 |Yuan
排版 |Yuan
审核 |Wang