[R] Testing optimization solvers with equality constraints

[Hans W]

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