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 ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.