Is it a convex problem? Could it be solved by cvx?

This is non-convex. Why isn't CVX accepting my model? READ THIS FIRST!

0.99 <= sum(max(var_x,0)) and 0.99 <= sum(max(-var_x,0) are both non-convex.

Depending on matrix_a , unless it is all zeros, at least one side, and maybe both sides of the last two constraints are non-convex.