Hi, I think "quadratic programming" is the way to go. Look at "solve.QP" or "limSolve" package.
Here is a toy example that I had worked out some time back for a linear least squares problem with simple box constraints: # Problem: minimize ||Ax - y||, subject to low <= x <= upp require(limSolve) nc <- 7 # 7 unknown parameters nr <- 20 # 20 equations # Bounds on the parameters: 0 < x < 1, for all x # set.seed(123) A <- matrix(rnorm(nr*nc), nr, nc) x <- c(runif(nc-1), 1.5) # Note: the last component is out of bounds! y <- A %*% x + rnorm(nr, sd=0.1) qr.solve(A, y) # unconstrained least-squares low <- rep(0, nc) # lower bounds upp <- rep(1, nc) # upper bounds # Implementing the bounds (there is probably a simpler way to do this) # c1 <- matrix(0, nc, nc) diag(c1) <- 1 c2 <- matrix(0, nc, nc) diag(c2) <- -1 cmat <- rbind(c1, c2) vec <- rep(0, 10) vec[seq(1, 2*nc, by=2)] <- 1:nc vec[seq(2, 2*nc, by=2)] <- (nc+1):(2*nc) Cmat <- rbind(c1, c2)[vec, ] # Constraint matrix G b0 <- c(low, -upp)[vec] ans <- lsei(A = A, B = y, G = Cmat, H = b0) ans Hope this helps, Ravi. ---------------------------------------------------------------------------- ------- Ravi Varadhan, Ph.D. Assistant Professor, The Center on Aging and Health Division of Geriatric Medicine and Gerontology Johns Hopkins University Ph: (410) 502-2619 Fax: (410) 614-9625 Email: rvarad...@jhmi.edu Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html ---------------------------------------------------------------------------- -------- -----Original Message----- From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On Behalf Of spencerg Sent: Friday, May 15, 2009 2:22 PM To: avraham.ad...@guycarp.com Cc: r-help@r-project.org; Douglas Bates Subject: Re: [R] Optimization algorithm to be applied to S4 classes - specifically sparse matrices I suggest you try to translate your constraints into an unconstrained constrained problem using logarithms, then do "nonlinear mixed effects modeling" as described in chapters 6-8 of Pinheiro and Bates (2000). To do this, I would first start with the simpler linear estimation problem to get starting values for the nonlinear estimation. You should be able to do this using the "nlme" function in the "nlme" package. If you have trouble with this, you might consider the "nlmer" function in the "lme4" package. The latter is newer and better in many ways but not as well documented. Hope this helps. Spencer Graves avraham.ad...@guycarp.com wrote: > Thank you both very much for your replies. What makes this a little > less straightforward, at least to me, is that there needs to be > constraints on the solved parameters. They most certainly need to be > positive and there may be an upper limit as well. The true best linear > fit would have negative entries for some of the parameters. > > > Originally, I was using the L-BFGS-B method of optim which both allows > for box constraints and has the limited memory advantage useful when > dealing with large matrices. Having the analytic gradient, I thought > of using BFGS and having a statement in the function returning "Inf" > for any parameters outside the allowable constraints. > > > I do /not/ know how to apply parameter constraints when using linear > models. I looked around at the various manuals and help features, and > outside of package "glmc" I did not find anything I could use. Perhaps > I overlooked something. If there is something I missed, please let me know. > > > If there truly is no standard optimization routine that works on > sparse matrices, my next step may be to use the normal equations to > shrink the size of the matrix, recast it as a dense matrix (it would > only be 1173x1173 > then) and then hand it off to optim. > > > Any further suggestions or corrections would be very much appreciated. > > > Thank you, > > > --Avraham Adler > > > > Douglas Bates > <ba...@stat.wisc. > edu> To > Sent by: avraham.ad...@guycarp.com > dmba...@gmail.com cc > r-help@r-project.org > Subject > 05/15/2009 11:57 Re: [R] Optimization algorithm to > AM be applied to S4 classes - > specifically sparse matrices > > > > > > > > > > > On Wed, May 13, 2009 at 5:21 PM, <avraham.ad...@guycarp.com> wrote: > >> Hello. >> >> I am trying to optimize a set of parameters using /optim/ in which >> the actual function to be minimized contains matrix multiplication >> and is of the form: >> >> SUM ((A%*%X - B)^2) >> >> where A is a matrix and X and B are vectors, with X as parameter vector. >> > > As Spencer Graves pointed out, what you are describing here is a > linear least squares problem, which has a direct (i.e. non-iterative) > solution. A comparison of the speed of various ways of solving such a > system is given in one of the vignettes in the Matrix package. > > >> This has worked well so far. Recently, I was given a data set A of >> size 360440 x 1173, which could not be handled as a normal matrix. I >> brought >> > it > >> into 'R' as a sparse matrix (dgCMatrix - using sparseMatrix from the >> > Matrix > >> package), and the formulæ and gradient work, but /optim/ returns an >> error of the form "no method for coercing this S4 class to a vector". >> > > If you just want the least squares solution X then > > X <- solve(crossprod(A), crossprod(A, B)) > > will likely be the fastest method where A is the sparse matrix. > > I do feel obligated to point out that the least squares solution for > such large systems is rarely a sensible solution to the underlying > problem. If you have over 1000 columns in A and it is very sparse > then likely at least parts of A are based on indicator columns for a > categorical variable. In such situations a model with random effects > for the category is often preferable to the fixed-effects model you > are fitting. > > > >> After briefly looking into methods and classes, I realize I am in way >> > over > >> my head. Is there any way I could use /optim/ or another optimization >> algorithm, on sparse matrices? >> >> Thank you very much, >> >> --Avraham Adler >> ______________________________________________ >> R-help@r-project.org 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. >> >> > > > > ______________________________________________ R-help@r-project.org 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. ______________________________________________ R-help@r-project.org 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.
