Solving Large-Scale Nonlinear System of Equations

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Description

Non-Monotone spectral approach for Solving Large-Scale Nonlinear Systems of Equations

Usage

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  sane(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 sane implements a non-monotone spectral residual method for finding a root of nonlinear systems. It stands for "spectral approach for nonlinear equations". It differs from the function dfsane in that it 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 and Raydan (2003).

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 and Raydan (2003); 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 equivalent steplength schemes in their EM acceleration work. Here, we have chosen method = 2 as the "default" method, as it generally performed 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 will not work. Argument control has the following components:

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 sane. 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 evaluated at par, 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 indicates error in function evaluation; 3 is failure due to exceeding 100 steplength reductions in line-search; 4 denotes failure due to an anomalous iteration; 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, and M Raydan (2003), Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18, 583-599.

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/

See Also

BBsolve, dfsane, spg, grad

Examples

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  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)
    sane(par=p0, fn=trigexp)
    sane(par=p0, fn=trigexp, method=1)    
######################################
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, 10))
  sane(par=p0, fn=brent, control=list(trace=FALSE))
  sane(par=p0, fn=brent, control=list(M=200, trace=FALSE))