I have the following linear programming equation and want to add a regularization term.

My problem is as follows:

echo on
cvx_begin
variable x(n)
dual variables y
minimize (norm((A_matrixA*x-b_vectorA),2)+norm((A_matrixD*x-b_vectorD),2)+lambda*norm((v_delta),1))
subject to
cvx_end
echo off

Where v_delta is the sum of absolute value of (x_i - x_i-1)

But it seems that CVX does not recognize this v_delta term as it does not affect the result.

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.

Yet the result/graph of x the second time around is the same as the first one. When changing lambda, the optimal value does increase but the result of x is still the same.

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?