Hi Gerard & Pavel Isn't this the proviso I was referring to, that one cannot in practice use an infinite weight because of rounding errors in the target function. The weight just has to be 'big enough' such that the restraint residual becomes sufficiently small that it's no longer significant.
In numerical constrained optimisation the method of increasing the constraint weights (a.k.a. 'penalty coefficients') until the constraint violations are sufficiently small is called the 'penalty method', see http://en.wikipedia.org/wiki/Penalty_method . The method where you substitute some of the parameters using the constraint equations is called (you guessed it!) the 'substitution method', see http://people.ucsc.edu/~rgil/Optimization.pdf . There are several other methods, e.g. the 'augmented Lagrangian method' is very popular, see http://www.ualberta.ca/CNS/RESEARCH/NAG/FastfloDoc/Tutorial/html/node112.html . As in the penalty method, the AL method adds additional parameters to be determined (the Lagrange multipliers, one per constraint) instead of eliminating some parameters using the constraint equations; however the advantage is that it removes the requirement that the penalty coefficient be very big. The point about all these methods of constrained optimisation is that they are in principle only different ways of achieving the same result, at least that's what the textbooks say! And now after the penalties and substitutions it's time to blow the whistle ... Cheers -- Ian On Wed, Sep 22, 2010 at 10:00 PM, Pavel Afonine <[email protected]> wrote: > I agree with Gerard. Example: it's unlikely to achieve a result of > rigid-body refinement (when you refine six rotation/translation parameters) > by replacing it with refining individual coordinates using infinitely large > weights for restraints. > Pavel. > > > On 9/22/10 1:46 PM, Gerard DVD Kleywegt wrote: >> >> Hi Ian, >> >>> First, constraints are just a special case of restraints in the limit >>> of infinite weights, in fact one way of getting constraints is simply >>> to use restraints with very large weights (though not too large that >>> you get rounding problems). These 'pseudo-constraints' will be >>> indistinguishable in effect from the 'real thing'. So why treat >>> restraints and constraints differently as far as the statistics are >>> concerned: the difference is purely one of implementation. >> >> In practice this is not true, of course. If you impose "infinitely strong" >> NCS restraints, any change to a thusly restrained parameter by the >> refinement program will make the target function infinite, so effectively >> your model will never change. This is very different from the behaviour >> under NCS constraints and the resulting models in these two cases will in >> fact be very easily distinguishable. >> >> --Gerard >> >> ****************************************************************** >> Gerard J. Kleywegt >> Dept. of Cell & Molecular Biology University of Uppsala >> Biomedical Centre Box 596 >> SE-751 24 Uppsala SWEDEN >> >> http://xray.bmc.uu.se/gerard/ mailto:[email protected] >> ****************************************************************** >> The opinions in this message are fictional. Any similarity >> to actual opinions, living or dead, is purely coincidental. >> ****************************************************************** >
