# Dual of second order cone programming with both equality and inequality constraints

How to derive the dual of a second order cone programming with both equality and inequality constraints? Here is the optimization problem I want to handle:

$$\begin{array}{rl}\min_{\mathbf{x},\mathbf{y}}&\frac{1}{2}\mathbf{x}^{{T}}\mathbf{P}_0\mathbf{x}+\mathbf{q}_0^{{T}}\mathbf{x}+\mathbf{r}_0\\ \text{s.t.}&\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{y}=\mathbf{f}\\ &\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{y}\le\mathbf{g}\\ &\frac{1}{2}\mathbf{x}^{{T}}{\mathbf{P}}_{i}\mathbf{x}+\mathbf{q}_{{i}}^{{T}}\mathbf{x}+\mathbf{r}_{{i}}\leq0,{i}=1,\ldots,k\\ & \left\|\mathbf{S}_{i}\mathbf{y}+\mathbf{t}_{i}\right\|_2\leq\mathbf{u}_{i}^{T}\mathbf{y}+{v}_{i},{i}=1,\ldots,l\end{array}$$ where $$\mathbf P_0,\ldots,\mathbf P_m$$ are $$n\times n$$ matrices and $$\mathbf x \in \mathbb R^n, \mathbf y \in \mathbb R^m$$ is the optimization variable. The optimization variables are separated into $$\mathbf x$$ and $$\mathbf y$$ as they follow different constraints.

In my setting, $$\mathbf P_0,\ldots,\mathbf P_m$$ are all positive semidefinite and thus the problem is convex. I wanna ask:

1. What is the dual of this problem?
2. How to ensure strong duality?
• Cone programming have same ways to derive dual problem based on dual cone. I recommend this book from Prof. Yinyu Ye. Your model is convex but not a standard SOCP. So you may need convert it to standard form first.
– xd y
Commented Dec 13, 2023 at 9:48
• Please allow me to attach a Chinese version here :) 所有的锥规划的对偶问题都可以用基于对偶锥的方式推导。推荐叶老师的这本书。你给出的模型是凸的，但是不是标准的二阶锥规划，所以你可能要先转换一下。
– xd y
Commented Dec 13, 2023 at 9:50
• Thank you. Actually what I really want to deal with is the following model. 谢谢，其实我是想推导下面这个问题的对偶。 Commented Jan 25 at 2:29

Variables: $$\mathbf r_{u}, \mathbf r_{d},\mathbf q_{g}, \Delta, \mathbf c_{iing}, \mathbf c_{ij}, \mathbf s_{ij}, \mathbf p_{f}, \mathbf q_{f}, \mathbf p_{t}, \mathbf q_{t}, \mathbf S_f, \mathbf S_t, \mathbf D_{1}, \mathbf D_{2}, \mathbf D_{3}, \mathbf D_{4}$$

Known values: $$\mathbf p_{g},\mathbf c_{iig}, \hat{\mathbf w}, \mathbf w$$

Fixed coefficients: $$\mathbf c_{ru}, \mathbf c_{rd}, \mathbf B_{g}, \mathbf{agc}, \mathbf C_{rp}, \dots$$

Now the model is as follows

Take all the inequalities in the form of $$g(\cdot)\leq0$$. The Lagrange function is

The dual problem is

Is anything wrong?