Question: How to use CVX to solve some discontinuous problems, is this a problem CVX can solve or is it just a mathematical problem?

The object to be optimized is a discontinuous function, which can lead to serious drags in the optimization results, this picture shows the function I need to get from CVX.
the red line is ORIGIN and the blue one is RECONSTRUCTION , We can see that there is a drag at the break point.

variable x(n);
subject to

and for X is the discontinuous function that I want to restore, S and T is what I know.

I doubt anyone can tell you unless you specify the mathematical optimization problem you want to solve.

Recall the problem should be a (mixed-integer) convex optimization problem.

Thank you for your answer and suggestion!
the problem is very simple there is S=T * I ,We know T and S so want to get the I, always the T is not a n*n matrix so I can’t solve it by T inversion .

is that a linear least squares problem? I don’t know, because I don’t understand what all your symbols are. But how would a linear least squares model match the curves you plotted? So you need to clarify what all the symbols are.

Thank you for your answer and suggestion!
Maybe I need to clear my question again, in the beginning with T (m * n) and I (n * 1), we get S = T * I (this is usually a unknown physical process, but we know T matrix, and then we can get S in some ways ). so in fact, we know that S(m * 1) and T(m * n), need to work out the I(n * 1), we use CVX tools, but because of I is a discontinuous vector, The resulting I has some drags at the discontinuity point(The part that comes out of the green coil).8a4f4160b70062b428b715d1cc8d808175e6d4a2
the red line is ORIGIN and the blue one is RECONSTRUCTION

I’m not sure if I understand your question, or if you can understand my subsequent explanation.

It is still not clear to me what you want. Maybe some type of mixed-integer least squares which will result in MIQP or MISOCP. See for instance, or . But if you do it in CVX, you will have to “manually” implement the logic, rather than relying on higher level constructs such as YALMIP’s implies. You can see some “manual” modeling in “9.2.3 Convex piecewise linear regression” of .

The preceding will handle piecewise linear models, which are MIDCP representable. But not piecewise nonlinear models, which are not MIDCP representable (and for which you would need an MINLP solver).