[Ipopt] question on formulation of the objective function and IPOPT performance

[Greg Horn]

Hi Damien,

Since no one has answered I'll take a shot at it, though it's not my
expertise. The case you described is:

minimize f(x) w.r.t {x}
vs
minimize y w.r.t {x,y}, subject to y == f(x)

I'm not sure if there will be any difference in this case. However, I have
had personal experience with:

minimize f(g(x)) w.r.t {x}      [prob1]
vs
minimize f(y) w.r.t. {x, y} subject to y == g(x)    [prob2]

In this case you can have significantly better convergence by "lifting"
these variables out. [prob2] will compute a better search direction than
[prob1]. This is why people use direct multiple shooting instead of direct
single shooting sometimes. There are even specialized lifting solvers which
can do linear algebra in the space of [prob1] and obtain the search
direction in the space of [prob2], for example:
http://num.math.uni-bayreuth.de/en/conferences/ompc_2013/program/download/friday/Diehl_ompc2013.pdf

Hope this helps,
Greg


On Wed, Jun 11, 2014 at 8:52 PM, Damien <damien at khubla.com> wrote:

> All,
>
> I'm looking at two ways to formulate the objective function in a new
> optimisation model.  The first way is what you'd call the conventional way
> I suppose, where you have a variety of variables contributing to the
> objective value and you calculate the gradient and return that to IPOPT.
>  The other way I'm considering is to equate the objective function to a new
> variable in an extra equality constraint, and have the new variable as the
> only variable in the objective, with a gradient of 1.0.  The new equality
> constraint then contributes first partial derivatives like any other
> equation or constraint.
>
> Mathematically (and I think algorithmically) these are equivalent, but I
> was wondering if anyone who's done this before has seen a performance
> difference between the two.
>
> Cheers,
>
> Damien
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