Priorities for pure Linear Tasks

[EnricoMingo]

If two Linear tasks are stacked to have Hard priorites, these are not respected since in the solver is missing the computation of the optimality constraint if the task is purely Linear. This means that, at the moment, pure Linear tasks can be added only at the lowest priority. I am not sure if this feature is useful or not but should be very easy to implement and test.

[alaurenzi]

I'm under the impression that doing this would be very problematic. Consider the least squares problem as we usually do: the objective function is ||Ax - b||^2, and so the optimality constraint can be a linear constraint (Ax = Ay, where y is the previous solution). Now consider the least squares plus linear term, whose objective function is ||Ax-b||^2 + c'x: now the optimality constraint is ||Ax-b||^2 + c'x = ||Ay-b||^2 + c'y which is quadratic and non-convex.

[EnricoMingo]

Is it non convex? I thought convexity was given by H matrix in the cost function which in our case is always SPD, we should have basically three cases:

1. c = 0 -> F = ||Ax - b||^2 where the optimality is given by Ax = Ay, which is ok
2. A = 0 -> F = c'x where the optimality is given by c'x = c'y, which is ok
3. Bot different from 0, then we have ||Ax-b||^2 + c'x = x'Hx + (g' + c')x and the optimality should be again in the form Ax = Ay right?

[EnricoMingo]

After some discussion we end up with the following optimality condition:
[A; c'] x = [A; c']y
which is a linearization of the real optimality condition which indeed is not linear.