Given a matrix $\Gamma_1 \in \mathbb R^{n \times n}$, find a matrix $\Gamma \in \mathbb R^{n \times n}$ that minimizes the Kullback-Leibler divergence $D_{\text{KL}}$

$$\begin{array}{ll} \min_\Gamma & D_{\text{KL}}( \Gamma \| \Gamma_1 )\quad \text{subject to} & \Gamma 1_n = 1_n\\ & \Gamma^{\top} 1_n = 1_n\end{array}$$

How can I solve it using MATLAB? I find someone uses Iterative Bregman Projections. Or can this be solved by using proximal methods?

What about adding another constraint $C_{n \times n} \Gamma =1_n 1_n^{T}$. Is there any methods to solve it. As it's very important for my current research, I hope someone could help me.

Given matrices $\Gamma_1, C \in \mathbb R^{n \times n}$, find a matrix $\Gamma \in \mathbb R^{n \times n}$ that minimizes the matrix norm of $\Gamma - \Gamma_1$ subject to constraints

$$\begin{array}{ll} \text{minimize} & \| \Gamma - \Gamma_1 \|\\ \text{subject to} & \Gamma 1_n = 1_n\\ & \Gamma^{\top} 1_n = 1_n\\ & C \Gamma = 1_n 1_n^{\top}\end{array}$$

How can I solve it using MATLAB? I find someone uses Iterative Bregman Projections. Or can this be solved by using proximal methods?

Are there any methods to solve it. It's very important for my current research. I hope someone could help me.

Given a matrix $\Gamma_1 \in \mathbb R^{n \times n}$, find a matrix $\Gamma \in \mathbb R^{n \times n}$ that minimizes the Kullback-Leibler divergence $D_{\text{KL}}$

$$\begin{array}{ll} \min_\Gamma & D_{\text{KL}}( \Gamma \| \Gamma_1 )\quad \text{subject to} & \Gamma 1_n = 1_n\\ & \Gamma^{\top} 1_n = 1_n\end{array}$$

How can I solve it using MATLAB? I find someone uses Iterative Bregman Projections. Or can this be solved by using proximal methods?

What about adding another constraint $C_{n \times n} \Gamma =1_n ^{T} 1_n$. Is there any methods to solve it. As it's very important for my current research, I hope someone could help me.

Given a matrix $\Gamma_1 \in \mathbb R^{n \times n}$, find a matrix $\Gamma \in \mathbb R^{n \times n}$ that minimizes the Kullback-Leibler divergence $D_{\text{KL}}$

$$\begin{array}{ll} \min_\Gamma & D_{\text{KL}}( \Gamma \| \Gamma_1 )\quad \text{subject to} & \Gamma 1_n = 1_n\\ & \Gamma^{\top} 1_n = 1_n\end{array}$$

How can I solve it using MATLAB? I find someone uses Iterative Bregman Projections. Or can this be solved by using proximal methods?

What about adding another constraint $C_{n \times n} \Gamma =1_n 1_n^{T}$. Is there any methods to solve it. As it's very important for my current research, I hope someone could help me.