Sorry, this might sound like a poor question: But by "on the unit sphere", do you mean on the ***surface*** of the sphere?
In which case, can't the surface of a sphere be projected onto a pair of circles? Where the cost function is reformulated as a function of two (rather than three) variables. On Sat, May 22, 2021 at 3:01 AM Hans W <hwborch...@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 > > ______________________________________________ > 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. ______________________________________________ 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.