Linear Programming (LP)
$$\begin{array}{lcl}
\text{minimize} & c^{T}x + d \\
\text{subject to} & Gx \le h \\
& Ax = b \\
\text{where }G \in \mathbb{R}^{m \times n} \text{ and } A \in \mathbb{R}^{p \times n}& \\
\end{array}$$
Quadratic Programming (QP)
$$\begin{array}{lcl}
\text{minimize} & \frac{1}{2}x^{T}Px + q^{T}x + r \\
\text{subject to} & Gx \le h \\
& Ax = b \\
\text{where }G \in \mathbb{R}^{m \times n} \text{ and } A \in \mathbb{R}^{p \times n}& \\
\end{array}$$
Quadratically Constrained Quadratic Programming (QCQP)
$$\begin{array}{lcl}
\text{minimize} & \frac{1}{2}x^{T}P_{0}x + q_{0}^{T}x + r_{0} \\
\text{subject to} & \frac{1}{2}x^{T}P_{i}x + q_{i}^{T}x + r_{i} \le 0, i=1, \cdots, m \\
& Ax = b \\
\text{where }P \in \mathbb{S}^{n}_{+} \text{for } i=1,\cdots,m \text{ and } A \in \mathbb{R}^{p \times n}& \\
\end{array}$$
Second-Order Cone Programming (SOCP)
$$\begin{array}{lcl}
\text{minimize} & f^{T}x \\
\text{subject to} & \parallel A_{i}x + b_{i} \parallel_{2} \leq c_{i}^{T}x + d_{i}, i=1,\cdots,m \\
& Fx = g \\
\text{where }x \in \mathbb{R}^{n} \text{ is the optimization variable, } &\\
A_{i} \in \mathbb{R}^{n_{i} \times n} \text{ and } F \in \mathbb{R}^{p \times n} &\\
\end{array}$$
Semidefinite Programming (SDP)
$$\begin{array}{lcl}
\text{minimize} & c^{T}x + d \\
\text{subject to} & x_{1}F_{1} + \cdots + x_{n}F_{n} + G \leq 0 \\
& Ax = b \\
\text{where }G, F_{1}, \cdots, F_{n} \in \mathbb{S}^{k} \text{ and } A \in \mathbb{R}^{p \times n}& \\
\end{array}$$
Conic Programming (CP)
$$\begin{array}{lcl}
\text{minimize} & c^{T}x + d \\
\text{subject to} & Fx + g \leq_{K} 0 \\
& Ax = b \\
\text{where }c, x \in \mathbb{R}^{n}, A \in \mathbb{R}^{p \times n} \text{ and } b \in \mathbb{R}^{p} & \\
\end{array}$$
Reference