I am a new user of CVX.
I have a minimization problem that requires the incorporation of a total variation (TV) constraint into the objective function (i.e. in addition to the linear model Ax = b).
This TV constraint is represented by |Grad x|_1; the L1 norm of the gradient of the desired solution x.
To be precise, I am working on an inverse problem using 2D images. The TV constraint amounts to say that the desired solution should not exhibit rapid oscillations at neighboring pixels.
I have read the user manual but did not quite figure out how to do this.
I’d appreciate if someone could point in the right direction.
As for applying it to the gradient, you will have to analytically determine a suitable expression for the gradient. if the Total Variation of the gradient is to be convex, I believe that gradient will need to be affine in the CVX (optimization) variables.
In the bigger picture, i don’t know what you are talking about. Gradient of the solution x, as opposed to the gradient of the objective function? A “constraint” being incorporated into the objective function? Moreover, that includes a gradient? But incorporation of the constraint or function of gradient into the objective changes the problem, so I have no idea what you’re doing.
If you are talking about imposing a total variation norm constraint on the gradient of the objective function, then because convexity would require the gradient to be affine, that would mean that the objective would need to be quadratic.
Perhaps, I should have stated my problem as an instance of regularized optimization problem.
The total variation term (can have different implementation and different meanings depending on the field) that I was referring to is k*|Grad x|_L1 term (the regularizer) added to the original |Ax-b|_L2 objective function, where k is a scalar (a Lagrangian multiplier).
It seems to me that I can approximate the Grad x using finite differencing.
If you want help on this forum, you should be much more specific as to your problem formulation. You should clearly specify the convex optimization problem you are trying to solve, and ask for any needed help in implementing it in CVX.