[petsc-dev] Semismooth Newton

[Jed Brown]

On Wed, Aug 1, 2012 at 11:10 AM, Todd Munson <tmunson at mcs.anl.gov> wrote:

>
> Yes, I am all in favor of actually specifying a VI externally and then
> internally formulating the appropriate augmented system.  This is what
> I do in other contexts (i.e. PATH for GAMS/AMPL).  Some things you
> probably want to do with such an interface is differentiate between
> linear and nonlinear constraints -- you do need second derivatives
> for nonlinear constraints and they should be convex.
>

Why not treat it as always nonlinear with the trivial second derivative in
the linear case?


>
> You can start getting very general (i.e. linear and nonlinear cone
> constraints, which can appear in contact problems with friction),
> but I would advocate for simplicity rather than generality.
>

I like simplicity, but I also like not having to change interfaces. I would
rather write an interface for a more general case and even temporarily have
an implementation that can't do the fully general thing, eventually perhaps
doing something more general. We put a rather large amount of effort into
making TS work with a convenient (solver-friendly) representation for DAEs,
even though lots of methods cannot be used for DAEs. If the general case is
not too complicated for the user, I'd rather do it that way and internally
reformulate.


>
> The addition and deletion of nonlinear constraints in your example is fine.
> For an active set method, the constraints can be implicitly defined and you
> only need to bring in the new active nonlinear constraints and add them to
> the constraint set for the linear algebra.  You can also do this for other
> methods.  So the active nonlinear constraints can change from one Newton
> iteration to the next.  The only place where the inactive constraints
> matter is in the evaluation of the residual for the line search and
> termination tests.
>

Cool, I'm glad you're happy with the math. Now we need a suitable
interface, which I think means something a fair amount different from what
we have now. We need constraint function, its Jacobian, and a Hessian, each
of which can change size?


>
> For the contact problems with friction, you typically do an explicit
> method though.  In this setting, you know the subset of the possible
> active constraints at each time step and can keep them constant in
> the VI solver.  They only change as you step forward through time.
> So you can optimize for this case and not have to worry about the
> constraints changing within each Newton iteration.
>
> I'm not sure how you want to design the variable constraint functionality,
> but there is nothing wrong with it technically.
>
> I can explain the semismooth method with slacks thing in a few lines.
> The simple problem is:
>
>   0 <= x  perp  f(x) >= 0
>
> The compact semismooth formulation is:
>
>   Phi(x,f(x)) = 0
>
> where Phi is the reformulation used.
>
> The semismooth formulation with slacks is:
>
>   f(x) - s = 0
>   Phi(x,s) = 0
>
> The linear algebra for the semismooth reformulation with slacks
> is very similar to that for the compact formulation, but there
> are subtle differences in the right hand side and the
> subdifferential.
>

Do you mean that the slacks are eliminated again in the linear algebra?
What are the "subtle differences"?
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