# [R] Testing optimization solvers with equality constraints

Hans W hwborcher@ @end|ng |rom gm@||@com
Fri May 21 17:03:19 CEST 2021

```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

```