convergence=0 in optim and nlminb is real?
As indicated, if optimizers check Hessians on every occasion, R would enrich all the computer manufacturers. In this case it is not too large a problem, so worth doing. However, for this problem, the Hessian is being evaluated by doing numerical approximations to second partial derivatives, so the Hessian may be almost a fiction of the analytic Hessian. I've seen plenty of Hessian approximations that are not positive definite, when the answers were OK. That Inf is allowed does not mean that it is recommended. R is very tolerant of many things that are not generally good ideas. That can be helpful for some computations, but still cause trouble. It seems that it is not the problem here. I did not look at all the results for this problem from optimx, but it appeared that several results were lower than the optim(BFGS) one. Is any of the optimx results acceptable? Note that optimx DOES offer to check the KKT conditions, and defaults to doing so unless the problem is large. That was included precisely because the optimizers generally avoid this very expensive computation. But given the range of results from the optimx answers using "all methods", I'd still want to do a lot of testing of the results. This may be a useful case to point out that nonlinear optimization is not a calculation that should be taken for granted. It is much less reliable than most users think. I rarely find ANY problem for which all the optimx methods return the same answer. You really do need to look at the answers and make sure that they are meaningful. JN
On 13-12-17 11:32 AM, Adelchi Azzalini wrote:
On Tue, 17 Dec 2013 08:27:36 -0500, Prof J C Nash (U30A) wrote: PJCN> If you run all methods in package optimx, you will see results PJCN> all over the western hemisphere. I suspect a problem with some PJCN> nasty computational issues. Possibly the replacement of the PJCN> function with Inf when any eigenvalues < 0 or nu < 0 is one PJCN> source of this. A value Inf is allowed, as indicated in this passage from the documentation of optim: Function fn can return NA or Inf if the function cannot be evaluated at the supplied value, but the initial value must have a computable finite value of fn. Incidentally, the documentation of optimx includes the same sentence. However, this aspect is not crucial anyway, since the point selected by optim is within the feasible space (by a good margin), and evaluation of the Hessian matrix occurs at this point. PJCN> PJCN> Note that Hessian eigenvalues are not used to determine PJCN> convergence in optimization methods. If they did, nobody would PJCN> ever get promoted from junior lecturer who was under 100 if they PJCN> needed to do this, because determining the Hessian from just the PJCN> function requires two levels of approximate derivatives. At the end of the optimization process, when a point is going to be declared a minimum point, I expect that an optimizer checks that it really *is* a minimum. It may do this in other ways other than computing the eigenvalues, but it must be done somehow. Actually, I first realized the problem by attempting inversion (to get standard errors) under the assumption of positive definiteness, and it failed. For instance mnormt:::pd.solve(opt$hessian) says "x appears to be not positive definite". This check does not involve a further level of approximation. PJCN> PJCN> If you want to get this problem reliably solved, I think you will PJCN> need to PJCN> 1) sort out a way to avoid the Inf values -- can you constrain PJCN> the parameters away from such areas, or at least not use Inf. PJCN> This messes up the gradient computation and hence the optimizers PJCN> and also the final Hessian. PJCN> 2) work out an analytic gradient function. PJCN> In my ealier message, I have indicated that this is a semplified version of the real thing, which is function mst.mle of pkg 'sn'. What mst.mle does is exactly what you indicated, that is, it re-parameterizes the problem so that we always stay within the feasible region and works with analytic gradient function (of the transformed parameters). The final outcome is the same: we land on the same point. However, once the (supposed) point of minimum has been found, the Hessian matrix must be computed on the original parameterization, to get standard errors. Adelchi Azzalini PJCN> PJCN> PJCN> > Date: Mon, 16 Dec 2013 16:09:46 +0100 PJCN> > From: Adelchi Azzalini <azzalini at stat.unipd.it> PJCN> > To: r-help at r-project.org PJCN> > Subject: [R] convergence=0 in optim and nlminb is real? PJCN> > Message-ID: PJCN> > <20131216160946.91858ff279db26bd65e187bc at stat.unipd.it> PJCN> > Content-Type: text/plain; charset=US-ASCII PJCN> > PJCN> > It must be the case that this issue has already been rised PJCN> > before, but I did not manage to find it in past posting. PJCN> > PJCN> > In some cases, optim() and nlminb() declare a successful PJCN> > convergence, but the corresponding Hessian is not PJCN> > positive-definite. A simplified version of the original PJCN> > problem is given in the code which for readability is placed PJCN> > below this text. The example is built making use of package PJCN> > 'sn', but this is only required to set-up the example: the PJCN> > question is about the outcome of the optimizers. At the end of PJCN> > the run, a certain point is declared to correspont to a minimum PJCN> > since 'convergence=0' is reported, but the eigenvalues of the PJCN> > (numerically evaluated) Hessian matrix at that point are not PJCN> > all positive. PJCN> > PJCN> > Any views on the cause of the problem? (i) the point does not PJCN> > correspong to a real minimum, (ii) it does dive a minimum but PJCN> > the Hessian matrix is wrong, (iii) the eigenvalues are not PJCN> > right. ...and, in case, how to get the real solution. PJCN> > PJCN> > PJCN> > Adelchi Azzalini PJCN>