本文是“支持向量机系列”的第四篇,参见本系列的其他文章。
在最开始讨论支持向量机的时候,我们就假定,数据是线性可分的,亦即我们可以找到一个可行的超平面将数据完全分开。后来为了处理非线性数据,使用 Kernel 方法对原来的线性 SVM 进行了推广,使得非线性的的情况也能处理。虽然通过映射
用黑圈圈起来的那个蓝点是一个 outlier ,它偏离了自己原本所应该在的那个半空间,如果直接忽略掉它的话,原来的分隔超平面还是挺好的,但是由于这个 outlier 的出现,导致分隔超平面不得不被挤歪了,变成途中黑色虚线所示(这只是一个示意图,并没有严格计算精确坐标),同时 margin 也相应变小了。当然,更严重的情况是,如果这个 outlier 再往右上移动一些距离的话,我们将无法构造出能将数据分开的超平面来。
为了处理这种情况,SVM 允许数据点在一定程度上偏离一下超平面。例如上图中,黑色实线所对应的距离,就是该 outlier 偏离的距离,如果把它移动回来,就刚好落在原来的超平面上,而不会使得超平面发生变形了。具体来说,原来的约束条件
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0054.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2265.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
现在变成
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0054.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2265.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2212.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
其中
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0043.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
其中
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0073.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0074.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0043.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0054.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2265.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2212.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2265.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
用之前的方法将限制加入到目标函数中,得到如下问题:
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/004C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0072.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0043.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2013.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0054.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2212.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/0029.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2013.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0072.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
分析方法和前面一样,转换为另一个问题之后,我们先让
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/004C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/004C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/004C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2202.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BE.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/21D2.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0077.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/21D2.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/21D2.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0043.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2212.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2212.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0072.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
将
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0061.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2013.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/27E8.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/27E9.png)
不过,由于我们得到
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/006D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0061.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0073.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0074.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2013.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/27E8.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0078.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006A.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/27E9.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2264.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2264.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0043.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2026.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Size2/Regular/120/2211.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/085/0031.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/006E.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B1.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/0079.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/085/0069.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png)
![](http://blog.pluskid.org/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png)
和之前的结果对比一下,可以看到唯一的区别就是现在 dual variable