I currently deals with optimization problems with a convex constraint function by cvx.
I consider approximating the function by point-wise maximum function composed of the tangent lines that can be obtained at optimal solutions in each iteration.
However, I am not sure how I can use the previous information and add the constraint in each iteration.

I tried to preserve the previous information, such as optimal solutions.
For example,

X = [];
cvx_begin
variable opt_x(n,1)

minimize ( 'objective function' )
subject to
      'constraints'
X = [X, opt_x];

cvx_end

However, X contains only last optimal solutions, and doesn’t contain the optimal solutions in each iteration.