You are not showing the entirety of your code, Specifically, where is your code for making v_delta sum of absolute value of (x_i - x_i-1) ?
Also, your code not an LP. The one-norm regularization term can be formulated as linear, but not the two-norms. CVX will reformulate this as an SOCP, and call a solver to solve the problem you have specified.
If lambda = 0 or is small enough, or if the optimal x without regularization has all components equal to each other (or close enough to equal), the optimal x may be the same with or without the regularization term.
In addition to that, because you have not shown the entirety of your code, I don’t know whether you have correctly implemented what you intended. Hint: if you place the expression for v_delta after the objective function rather than before, it will be treated as zero, and therefore have no effect.
If you are calculating v_delta as the difference of a previously caclulated optimal (without regularization) value of x, then it will be just a numerical constant as far as CVX is concerned. So it has no effect on the optimal x.
Maybe there’s a nice vectorized way to get v_delta, but in any event, you can build it up using a for loop
I greatly appreciate your insight. It makes sense what you are saying but I don’t understand how to incorporate the for loop into the minimization for this regularization term. Could you give me some advice in it?