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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?
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  • $\begingroup$ 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. $\endgroup$
    – xd y
    Commented Dec 13, 2023 at 9:48
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    $\begingroup$ 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。 $\endgroup$
    – xd y
    Commented Dec 13, 2023 at 9:50
  • $\begingroup$ 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。 $\endgroup$ Commented Jan 25 at 2:29

1 Answer 1

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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 enter image description here

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

The dual problem is enter image description here

Is anything wrong?

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