Projected Gradient Method with Multiple Constraints (Projection of Each Row to the Unit Simplex) I am trying to minimize convex objective $f(X)$, for matrix $X$
s.t. $X\ge 0$ component-wise, and $X1^T = 1^T$.
I want to use projected gradient descent. However, I only know how to project on feasible set of a single constraint, but not both at the same time (i.e. their intersection). 
But projecting on boundary of one feasible set, may bring the solution out of the feasible set for the other one. What is the standard method to overcome this?
 A: Projecting onto the simplex is straightforward, solutions crop up from time to time. Here is an example: C. Michelot, "A finite algorithm for finding the projection of a point onto the canonical simplex of $\mathbb{R}^n$". JOTA, 50(1):195–200, 1986.
It is also a simple linear programming problem.
This is an elaboration that belongs in the comments but is too long:
Perhaps I am missing something here, but from what you have above, my understanding is that if you write $X^T = \begin{bmatrix}x_1 & \cdots & x_n \end{bmatrix}$, then you want $[x_i]_j \geq 0$ for all $i,j$, and $\sum_j [x_i]_j = 1$, for all $i$. If you let $\Sigma = \{\sigma \in \mathbb{R}^n | \sigma_i \geq 0,\ \sum \sigma_i = 1 \} $, then the space of matrices $X$ satisfying the constraints is basically $\Sigma^n$.
So, if you are at some point $X$ and have a direction $\Delta$ and some step size $\lambda$, then you are trying to project $X+\lambda \Delta$ onto the feasible set. You would do so by projecting each row of $X+\lambda \Delta$ separately onto $\Sigma$ and then recombining to get the projected step.
