R help - May 2021 - Testing optimization solvers with equality constraints

Hi Hans: I think  that you are missing minus signs in the 2nd and 3rd
elements of your gradient.
Also, I don't know how all of the optimixation functions work as far as
their arguments but it's best to supply
the gradient when possible. I hope it helps.






On Fri, May 21, 2021 at 11:01 AM Hans W <hwborchers at gmail.com> wrote:
> Just by chance I came across the following example of minimizing
> a simple function
>
>     (x,y,z) --> 2 (x^2 - y z)
>
> on the unit sphere, the only constraint present.
> I tried it with two starting points, x1 = (1,0,0) and x2 = (0,0,1).
>
>     #-- Problem definition in R
>     f = function(x)  2 * (x[1]^2 - x[2]*x[3])   # (x,y,z) |-> 2(x^2 -yz)
>     g = function(x)  c(4*x[1], 2*x[3], 2*x[2])  # its gradient
>
>     x0 = c(1, 0, 0); x1 = c(0, 0, 1)            # starting points
>     xmin = c(0, 1/sqrt(2), 1/sqrt(2))           # true minimum -1
>
>     heq = function(x)  1-x[1]^2-x[2]^2-x[3]^2   # staying on the sphere
>     conf = function(x) {                        # constraint function
>         fun = x[1]^2 + x[2]^2 + x[3]^2 - 1
>         return(list(ceq = fun, c = NULL))
>     }
>
> I tried all the nonlinear optimization solvers in R packages that
> allow for equality constraints: 'auglag()' in alabama,
'solnl()' in
> NlcOptim, 'auglag()' in nloptr, 'solnp()' in Rsolnp, or
even 'donlp2()'
> from the Rdonlp2 package (on R-Forge).
>
> None of them worked from both starting points:
>
>     # alabama
>     alabama::auglag(x0, fn = f, gr = g, heq = heq)  # right (inaccurate)
>     alabama::auglag(x1, fn = f, gr = g, heq = heq)  # wrong
>
>     # NlcOptim
>     NlcOptim::solnl(x0, objfun = f, confun = conf)  # wrong
>     NlcOptim::solnl(x1, objfun = f, confun = conf)  # right
>
>     # nloptr
>     nloptr::auglag(x0, fn = f, heq = heq)           # wrong
>     # nloptr::auglag(x1, fn = f, heq = heq)         # not returning
>
>     # Rsolnp
>     Rsolnp::solnp(x0, fun = f, eqfun = heq)         # wrong
>     Rsolnp::solnp(x1, fun = f, eqfun = heq)         # wrong
>
>     # Rdonlp2
>     Rdonlp2::donlp2(x0, fn = f, nlin = list(heq),   # wrong
>            nlin.lower = 0, nlin.upper = 0)
>     Rdonlp2::donlp2(x1, fn = f, nlin = list(heq),   # right
>            nlin.lower = 0, nlin.upper = 0)          # (fast and exact)
>
> The problem with starting point x0 appears to be that the gradient at
> that point, projected onto the unit sphere, is zero. Only alabama is
> able to handle this somehow.
>
> I do not know what problem most solvers have with starting point x1.
> The fact that Rdonlp2 is the fastest and most accurate is no surprise.
>
> If anyone with more experience with one or more of these packages can
> give a hint of what I made wrong, or how to change calling the solver
> to make it run correctly, please let me know.
>
> Thanks  -- HW
>
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	[[alternative HTML version deleted]]

Mark, you're right, and it's a bit embarrassing as I thought I had
looked at it closely enough.

This solves the problem for 'alabama::auglag()' in both cases, but NOT
for

  * NlcOptim::solnl     -- with x0
  * nloptr::auglag      -- both x0, x1
  * Rsolnp::solnp       -- with x0
  * Rdonlp::donlp2      -- with x0

as for these solver calls the gradient function g was *not* used.

Actually, 'solnl()' and 'solnp()' do not allow a gradient
argument,
'nloptr::auglag()' says it does not use a supplied gradient, and
'donlp2' again does not provide it.
Gradients, if needed, are computed internally which in most cases is
sufficient, anyway.

So the question remains:
Is the fact that the projection of the gradient onto the constraint is
zero, is this the reason for the solvers not finding the minimum?

And how to avoid this? Except, maybe, checking the gradient for all
the given constraints

Thanks  --HW



On Fri, 21 May 2021 at 17:58, Mark Leeds <markleeds2 at gmail.com>
wrote:
>
> Hi Hans: I think  that you are missing minus signs in the 2nd and 3rd
elements of your gradient.
> Also, I don't know how all of the optimixation functions work as far as
their arguments but it's best to supply
> the gradient when possible. I hope it helps.
>