目标（Objective）

Our objective is to transform non-convex functions to a convex functions, i.e.,
$$\begin{array}{ll}& \min f_0(x)\to \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\\ & s.t.f_i(x)\le 0,i=1,...,m\to \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\\ & s.t.a_i^{\mathrm{T}}x=b_i,(h_i(x)=0)i=1,...,p\to \mathrm{A}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\end{array}$$

等价变换（Equivalent Transformation for Conditions）

$$\begin{array}{ll}& \min f_0(x)=x_1^2+x_2^2\\ & s.t.f_1(x)=\frac{x_1}{1+x_2^2}(\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x})\\ & h_i(x)=x_1^2+x_2^2=0(\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e})\end{array}$$ The above equation can be rewrriten as
$$\begin{array}{ll}& \min f_0(x)=x_1^2+x_2^2\\ & s.t.f_1(x)=x_1\le 0\\ & h_i(x)=x_1+x_2=0\end{array}$$

降维（Dimension Reduction）

Generally, for reducing the complication of the problem, we need to reduce the dimensionality of functions. But, sometime, dimensionality reduction will lead the probloem to be difficult.
For example, a convex problem can be expressed as
$$\begin{array}{ll}P0:& \min f_0(x)\\ & s.t.f_i(x)\le 0,i=1,...,m\\ & s.t.a_i^{\mathrm{T}}x=b_i,i=1,...,p\end{array}$$ Letting $Fz=x_0$, the above equation can be rewritten as
$$\begin{array}{ll}& \min f_0(Fz=x_0)\\ & s.t.f_i(Fz=x_0)\le 0,i=1,...,m\end{array}$$

升维（Dimension Raising）

Here, we introduce slack variable $s_i$ to raise the dimensionality of the problem.
Problem $P0$ can be rewritten as
$$\begin{array}{ll}& \min f_0(x)\\ & s.t.s_i\le 0,i=1,...,m\\ & f_i(x)-s_i=0,i=1,...,m\\ & a_i^{\mathrm{T}}x=b_i,i=1,...,p\end{array}$$