dfsane: Solving Large-Scale Nonlinear System of Equations In BB: Solving and Optimizing Large-Scale Nonlinear Systems

Description

Derivative-Free Spectral Approach for solving nonlinear systems of equations

Usage

 ```1 2 3``` ``` dfsane(par, fn, method=2, control=list(), quiet=FALSE, alertConvergence=TRUE, ...) ```

Arguments

 `fn` a function that takes a real vector as argument and returns a real vector of same length (see details). `par` A real vector argument to `fn`, indicating the initial guess for the root of the nonlinear system. `method` An integer (1, 2, or 3) specifying which Barzilai-Borwein steplength to use. The default is 2. See *Details*. `control` A list of control parameters. See *Details*. `quiet` A logical variable (TRUE/FALSE). If `TRUE` warnings and some additional information printing are suppressed. Default is `quiet = FALSE` Note that the `control` variable `trace` and `quiet` affect different printing, so if `trace` is not set to `FALSE` there will be considerable printed output. `alertConvergence` A logical variable. With the default `TRUE` a warning is issued if convergence is not obtained. When set to `FALSE` the warning is suppressed. `...` Additional arguments passed to `fn`.

Details

The function `dfsane` is another algorithm for implementing non-monotone spectral residual method for finding a root of nonlinear systems, by working without gradient information. It stands for "derivative-free spectral approach for nonlinear equations". It differs from the function `sane` in that `sane` requires an approximation of a directional derivative at every iteration of the merit function F(x)^t F(x).

R adaptation, with significant modifications, by Ravi Varadhan, Johns Hopkins University (March 25, 2008), from the original FORTRAN code of La Cruz, Martinez, and Raydan (2006).

A major modification in our R adaptation of the original FORTRAN code is the availability of 3 different options for Barzilai-Borwein (BB) steplengths: `method = 1` is the BB steplength used in LaCruz, Martinez and Raydan (2006); `method = 2` is equivalent to the other steplength proposed in Barzilai and Borwein's (1988) original paper. Finally, `method = 3`, is a new steplength, which is equivalent to that first proposed in Varadhan and Roland (2008) for accelerating the EM algorithm. In fact, Varadhan and Roland (2008) considered 3 similar steplength schemes in their EM acceleration work. Here, we have chosen `method = 2` as the "default" method, since it generally performe better than the other schemes in our numerical experiments.

Argument `control` is a list specifing any changes to default values of algorithm control parameters. Note that the names of these must be specified completely. Partial matching does not work.

M

A positive integer, typically between 5-20, that controls the monotonicity of the algorithm. `M=1` would enforce strict monotonicity in the reduction of L2-norm of `fn`, whereas larger values allow for more non-monotonicity. Global convergence under non-monotonicity is ensured by enforcing the Grippo-Lampariello-Lucidi condition (Grippo et al. 1986) in a non-monotone line-search algorithm. Values of `M` between 5 to 20 are generally good, although some problems may require a much larger M. The default is `M = 10`.

maxit

The maximum number of iterations. The default is `maxit = 1500`.

tol

The absolute convergence tolerance on the residual L2-norm of `fn`. Convergence is declared when sqrt(sum(F(x)^2) / npar) < tol. Default is `tol = 1.e-07`.

trace

A logical variable (TRUE/FALSE). If `TRUE`, information on the progress of solving the system is produced. Default is `trace = TRUE`.

triter

An integer that controls the frequency of tracing when `trace=TRUE`. Default is `triter=10`, which means that the L2-norm of `fn` is printed at every 10-th iteration.

noimp

An integer. Algorithm is terminated when no progress has been made in reducing the merit function for `noimp` consecutive iterations. Default is `noimp=100`.

NM

A logical variable that dictates whether the Nelder-Mead algorithm in `optim` will be called upon to improve user-specified starting value. Default is `NM=FALSE`.

BFGS

A logical variable that dictates whether the low-memory L-BFGS-B algorithm in `optim` will be called after certain types of unsuccessful termination of `dfsane`. Default is `BFGS=FALSE`.

Value

A list with the following components:

 `par` The best set of parameters that solves the nonlinear system. `residual` L2-norm of the function at convergence, divided by `sqrt(npar)`, where "npar" is the number of parameters. `fn.reduction` Reduction in the L2-norm of the function from the initial L2-norm. `feval` Number of times `fn` was evaluated. `iter` Number of iterations taken by the algorithm. `convergence` An integer code indicating type of convergence. `0` indicates successful convergence, in which case the `resid` is smaller than `tol`. Error codes are `1` indicates that the iteration limit `maxit` has been reached. `2` is failure due to stagnation; `3` indicates error in function evaluation; `4` is failure due to exceeding 100 steplength reductions in line-search; and `5` indicates lack of improvement in objective function over `noimp` consecutive iterations. `message` A text message explaining which termination criterion was used.

References

J Barzilai, and JM Borwein (1988), Two-point step size gradient methods, IMA J Numerical Analysis, 8, 141-148.

L Grippo, F Lampariello, and S Lucidi (1986), A nonmonotone line search technique for Newton's method, SIAM J on Numerical Analysis, 23, 707-716.

W LaCruz, JM Martinez, and M Raydan (2006), Spectral residual mathod without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75, 1429-1448.

R Varadhan and C Roland (2008), Simple and globally-convergent methods for accelerating the convergence of any EM algorithm, Scandinavian J Statistics.

R Varadhan and PD Gilbert (2009), BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function, J. Statistical Software, 32:4, http://www.jstatsoft.org/v32/i04/

`BBsolve`, `sane`, `spg`, `grad`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ``` trigexp <- function(x) { # Test function No. 12 in the Appendix of LaCruz and Raydan (2003) n <- length(x) F <- rep(NA, n) F[1] <- 3*x[1]^2 + 2*x[2] - 5 + sin(x[1] - x[2]) * sin(x[1] + x[2]) tn1 <- 2:(n-1) F[tn1] <- -x[tn1-1] * exp(x[tn1-1] - x[tn1]) + x[tn1] * ( 4 + 3*x[tn1]^2) + 2 * x[tn1 + 1] + sin(x[tn1] - x[tn1 + 1]) * sin(x[tn1] + x[tn1 + 1]) - 8 F[n] <- -x[n-1] * exp(x[n-1] - x[n]) + 4*x[n] - 3 F } p0 <- rnorm(50) dfsane(par=p0, fn=trigexp) # default is method=2 dfsane(par=p0, fn=trigexp, method=1) dfsane(par=p0, fn=trigexp, method=3) dfsane(par=p0, fn=trigexp, control=list(triter=5, M=5)) ###################################### brent <- function(x) { n <- length(x) tnm1 <- 2:(n-1) F <- rep(NA, n) F[1] <- 3 * x[1] * (x[2] - 2*x[1]) + (x[2]^2)/4 F[tnm1] <- 3 * x[tnm1] * (x[tnm1+1] - 2 * x[tnm1] + x[tnm1-1]) + ((x[tnm1+1] - x[tnm1-1])^2) / 4 F[n] <- 3 * x[n] * (20 - 2 * x[n] + x[n-1]) + ((20 - x[n-1])^2) / 4 F } p0 <- sort(runif(50, 0, 20)) dfsane(par=p0, fn=brent, control=list(trace=FALSE)) dfsane(par=p0, fn=brent, control=list(M=200, trace=FALSE)) ```