everyone.

It has been well known that the following minimization problem of a Bregman divergence with linear inequality
can be solved by *successively projecting the current point to each constraint $\mathbf{a}_i^\top\mathbf{x}\leq b_i$ with a correction step*, where $A=\begin{bmatrix} \mathbf{a}_1 & \ldots & \mathbf{a}_n \end{bmatrix}$, $\mathbf{b}=\begin{bmatrix} b_1 & \ldots & b_n \end{bmatrix}^\top$ and $n$ is the number of the constraints.
\begin{align}
\min_\mathbf{x}&D_\varphi(\mathbf{x},\mathbf{y})\\
\mathrm{s.t.}&A^\top\mathbf{x}\leq\mathbf{b}
\end{align}

Since each linear inequality constraint defines a half-space and the intersection of these half-spaces is a polyhedron, it is indeed an optimization problem on a polyhedron.

However, if the polyhedron is given by the **convex hull description** spanned by a set of vectors $\{\mathbf{v}_i\}_{i=1}^n$, i.e.
\begin{align}
\min_{\mathbf{x},\alpha}&D_\varphi(\mathbf{x},\mathbf{y})\\
\mathrm{s.t.}&\mathbf{x}=\sum_{i=1}^n{\alpha_i\mathbf{v}_i}\\
&0\leq\alpha_i\leq 1,\;\forall i=1,\ldots,n\\
&\sum_{i=1}^n{\alpha_i}=1
\end{align}
how can it be solved?

It seems that the problem can be solved by first converting the convex hull constraint to its *half-space description*, which is composed of a set of linear inequality constraints, and then applying the successive projection algorithm. However, it is not simple to obtain such a representation except in some special cases. Then, how can I solve this problem?

Any suggestion is welcome and I appreciate your help. Thank you very much!