> For the complete documentation index, see [llms.txt](https://windmising.gitbook.io/lihang-tongjixuexifangfa/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://windmising.gitbook.io/lihang-tongjixuexifangfa/2/5.md).

# 凸二次规划问题求解

要求解的带约束最优化问题为：

$$
\begin{aligned}
min\_{w,b} \quad \frac {1}{2}||w||^2 \\
s.t.  \quad y\_i(w \cdot x\_i + b)-1 \ge 0, i = 1,2,\cdots,N   && {1}
\end{aligned}
$$

将原始问题转换为对偶最优化问题（[推导过程](https://windmising.gitbook.io/lihang-tongjixuexifangfa/svm/6)）：

$$
\begin{aligned}
\min\_a \quad \frac{1}{2}\sum\_{i=1}^N\sum\_{j=1}^Na\_ia\_jy\_iy\_j(x\_i\cdot x\_j) - \sum\_{i=1}^na\_i   \\
s.t. \quad \sum\_{i=1}^Na\_iy\_i = 0 \\
\quad a\_i \ge 0, i = 1,2,\cdots, N && {2}
\end{aligned}
$$

通过公式（2）求解$$a\*$$（[推导过程](/lihang-tongjixuexifangfa/2/5.md)）

通过$$a\*$$求出$$w\*$$和$$b\*$$：

$$
\begin{aligned}
w\* = \sum\_{i=1}^Na\_i^*y\_ix\_i  \\
b* = y\_j - \sum\_{i=1}^Na\_i^\*y\_i(x\_i \cdot x\_j)  && {3}
\end{aligned}
$$

通过公式（3）得到：\
分离超平面：

$$
\sum\_{i=1}^Na\_i^*y\_i(x\cdot x\_i) + b^* = 0
$$

分类决策函数（[推导过程](/lihang-tongjixuexifangfa/2/5.md)）：

$$
f(x) = sign(\sum\_{i=1}^Na\_i^*y\_i(x\cdot x\_i) + b^*)
$$


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