I have a large scale convex optimization problem as the following shows.
$$\begin{array}{ll} \underset{\mathbf{x} \in \mathbb{R}^K}{\text{minimize}} & -\alpha^T[\mathrm{ln}(\mathbf{1}+\mathbf{x})-\mathbf{Ax}]\\ \text{subject to} & \mathbf{Bx}\le \mathbf{b}\\ & (\mathbf{1}+\mathbf{Px})\odot(\mathbf{1}+\mathbf{Px})-\mathbf{Qx}\le \mathbf{d}\end{array}$$
where $\alpha\in \mathbb{R}^K$, $\odot$ denotes the Hadamard product and $\mathbf{A}, \mathbf{P}, \mathbf{Q} \in {\mathbb{R}^{K\times K}}, \mathbf{B} \in \mathbb{R}^{M \times K}$.
I would like to solve it via using barrier function based interior point method and proximity operator.
What I have tried...
$$\text{minimize} -\alpha^T[\mathrm{ln}(\mathbf{1}+\mathbf{x})-\mathbf{Ax}] - \mu[\sum_\limits{i=1}^{M}\mathrm{ln}(\mathbf{b}_i - \mathbf{B}_{i,:}x) + \sum_\limits{i=1}^K\mathrm{ln}(\mathbf{d}_i-(1+\mathbf{P}_{i,:}\mathbf{x})^2-\mathbf{Q}_{i,:}\mathbf{x})] $$
If I set $$F(\mathbf{x})=-\alpha^T[\mathrm{ln}(\mathbf{1}+\mathbf{x})-\mathbf{Ax}]- \mu\sum_\limits{i=1}^K\mathrm{ln}(\mathbf{d}_i-(1+\mathbf{P}_{i,:}\mathbf{x})^2-\mathbf{Q}_{i,:}\mathbf{x})$$ and $$G(\mathbf{x})=-\mu\sum_\limits{i=1}^{M}\mathrm{ln}(\mathbf{b}_i - \mathbf{B}_{i,:}\mathbf{x}).$$
Firstly, do a forward step for $F(\mathbf{x})$, i.e., $$\mathbf{x}_{k+1} = \mathrm{prox}_{\mu_k\gamma_kG}(\mathbf{x}_k-\gamma_k\nabla_1F(\mathbf{x}_k))$$
As we know, a proximal operator for function $f$ at $\mathbf{x}$ is formulated as $$\mathrm{prox}_f(\mathbf{x}) = arg\min_\limits{\mathbf{u}}\frac{1}{2}||\mathbf{x}-\mathbf{u}||^2+\gamma f(\mathbf{u})$$
so $$\mathbf{x}_{k+1} = \mathrm{prox}_{\mu_k\gamma_kG}(\mathbf{x}_k-\gamma_k\nabla_1F(\mathbf{x}_k)) \\ = arg\min_\limits{\mathbf{u}}\frac{1}{2}||\mathbf{u}-\mathbf{x}_k-\gamma_k\nabla_1F(\mathbf{x}_k)||^2 -\mu_k\sum_\limits{i=1}^{M}\mathrm{ln}(\mathbf{b}_i - \mathbf{B}_{i,:}\mathbf{u}) $$
There is a solution associated above problem's subproblem
while my question is that the solution can't be applied to the sum of several affine constraints. So what should I do?