G'day Yingfu, >>>>> "YX" == Yingfu Xie <[EMAIL PROTECTED]> writes:
YX> Thanks a lot for answering! I prefer the first method, which YX> seems easier to carry out. So, what I need now is some method YX> to solve the non-linear equation [...] I searched the R YX> archive somehow, but didn't find anything valuable. Is there YX> any function in R available for this problem? I am grateful of YX> any hints. You have to code the function f(lam) = c'(M_11-lam*I)^(-2)c-1 in R. Note that lam is univariate. If you could easily find lam1 and lam2 with f(lam1)<0 and f(lam2)>0, then you could use uniroot(). But optim() could be easier to use for finding lam s.t. f(lam)=0, just try to minimise f(lam)^2. YX> When c=0, my problem reduces to the typical minimization of YX> b'M_11b w.r.t b'b=1. The solution is the normalized YX> eigenvector associated with the minimum eigenvalue of M_11, YX> right? Depends on your M_11, if M_11 is the identity matrix, then you have infinitely many solutions. Intuitively, I would have thought that the solution is the normalised eigenvector (and its negation, so there are at least two solutions) of the maximum eigenvalue of M_11, but I might be drawing the wrong picture in my mind. It should be either the maximum or minimum. Note, if that eigenvalue has geometric multiplicity > 1, then there will be infinitely many solutions. YX> PS: There is a type error in the first condition for b: the YX> '+' should write to '-'. I believe both versions are correct. We are enforcing an equality constraint, not an inequality constraint. So it doesn't matter whether we write the Lagrangian as objective function + lam *(b'b-1) or objective function - lam *(b'b-1) In the KKT conditions, we will only have the complimentary condition that lam*(b'b-1)=0, no condition that lam>=0. When you enforce inequality constraints, then you have to take care with your signs and how you write the Lagrangian. Cheers, Berwin ______________________________________________ R-help@stat.math.ethz.ch mailing list 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.