Large-Scale Optimization

Description

Spectral projected gradient method for large-scale optimization with simple constraints.

Usage

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     spg(par, fn, gr=NULL, method=3, lower=-Inf, upper=Inf, 
           project=NULL, projectArgs=NULL, 
	   control=list(), quiet=FALSE, alertConvergence=TRUE, ...)
 

Arguments

par

A real vector argument to fn, indicating the initial guess for the optimization of nonlinear objective function fn.

fn

Nonlinear objective function that is to be optimized. A scalar function that takes a real vector as argument and returns a scalar that is the value of the function at that point (see details).

gr

The gradient of the objective function fn evaluated at the argument. This is a vector-function that takes a real vector as argument and returns a real vector of the same length. It defaults to "NULL", which means that gradient is evaluated numerically. Computations are dramatically faster in high-dimensional problems when the exact gradient is provided. See *Example*.

method

An integer (1, 2, or 3) specifying which Barzilai-Borwein steplength to use. The default is 3. See *Details*.

upper

An upper bound for box constraints.

lower

An lower bound for box constraints.

project

A projection function or character string indicating its name. The projection function takes a point in R^n and projects it onto a region that defines the constraints of the problem. This is a vector-function that takes a real vector as argument and returns a real vector of the same length. See *Details*. If a projection function is not supplied, arguments lower and upper will cause the use of an internally defined function that enforces the implied box constraints.

projectArgs

A list with arguments to the project function. 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 and gr. (Both must accept any specified arguments, either explicitly or by having a ... argument, but they do not need to use them all.)

Details

R adaptation, with significant modifications, by Ravi Varadhan, Johns Hopkins University (March 25, 2008), from the original FORTRAN code of Birgin, Martinez, and Raydan (2001). The original is available at the TANGO project http://www.ime.usp.br/~egbirgin/tango/downloads.php

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 Birgin, Martinez and Raydan (2000); method = 2 is the other steplength proposed in Barzilai and Borwein's (1988) original paper. Finally, method = 3, is a new steplength, which was 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 = 3 as the "default" method. This method may be slightly slower than the other 2 BB steplength schemes, but it generally exhibited more reliable convergence to a better optimum in our experiments. We recommend that the user try the other steplength schemes if the default method does not perform well in their problem.

Box constraints can be imposed by vectors lower and upper. Scalar values for lower and upper are expanded to apply to all parameters. The default lower is -Inf and upper is +Inf, which imply no constraints.

The project argument provides a way to implement more general constraints to be imposed on the parameters in spg. projectArgs is passed to the project function if one is specified. The first argument of any project function should be par and any other arguments should be passed using its argument projectArgs. To avoid confusion it is suggested that user defined project functions should not use arguments lower and upper.

The function projectLinear incorporates linear equalities and inequalities. This function also provides an example of how other projections might be implemented.

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. The list items are as follows:

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. The default is M = 10.

maxit

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

ftol

Convergence tolerance on the absolute change in objective function between successive iterations. Convergence is declared when the change is less than ftol. Default is ftol = 1.e-10.

gtol

Convergence tolerance on the infinity-norm of projected gradient gr evaluated at the current parameter. Convergence is declared when the infinity-norm of projected gradient is less than gtol. Default is gtol = 1.e-05.

maxfeval

Maximum limit on the number of function evaluations. Default is maxfeval = 10000.

maximize

A logical variable indicating whether the objective function is to be maximized. Default is maximize = FALSE indicating minimization. For maximization (e.g. log-likelihood maximization in statistical modeling), this may be set to TRUE.

trace

A logical variable (TRUE/FALSE). If TRUE, information on the progress of optimization is printed. Default is trace = TRUE.

triter

An integer that controls the frequency of tracing when trace=TRUE. Default is triter=10, which means that the objective fn and the infinity-norm of its projected gradient are printed at every 10-th iteration.

eps

A small positive increment used in the finite-difference approximation of gradient. Default is 1.e-07.

checkGrad

NULL or a logical variable TRUE/FALSE indicating whether to check the provided analytical gradient against a numerical approximation. With the default NULL the gradient is checked if it is estimated to take less than about ten seconds. A warning will be issued in the case it takes longer. The default can be overridden by specifying TRUE or FALSE. It is recommended that this be set to FALSE for high-dimensional problems, after making sure that the gradient is correctly specified, possibly by running once with TRUE specified.

checkGrad.tol

A small positive value use to compare the maximum relative difference between a user supplied gradient gr and the numerical approximation calculated by grad from package numDeriv. The default is 1.e-06. If this value is exceeded then an error message is issued, as it is a reasonable indication of a problem with the user supplied gr. The user can either fix the gr function, remove it so the finite-difference approximation is used, or increase the tolerance so the check passes.

Value

A list with the following components:

par

Parameters that optimize the nonlinear objective function, if convergence is successful.

value

The value of the objective function at termination.

gradient

L-infinity norm of the projected gradient of the objective function at termination. If convergence is successful, this should be less than gtol.

fn.reduction

Reduction in the value of the function from its initial value. This is negative in maximization.

iter

Number of iterations taken by the algorithm. The gradient is evaluated once each iteration, so the number of gradient evaluations will also be equal to iter, plus any evaluations necessary for checkGrad.

feval

Number of times the objective fn was evaluated.

convergence

An integer code indicating type of convergence. 0 indicates successful convergence, in which case the projcted gradient is smaller than pgtol or the change in objective function is smaller than ftol. Error codes are: 1 indicates that the maximum limit for iterations maxit has been reached. 2 indicates that maximum limit on function evals has been exceeded. 3 indicates failure due to error in function evaluation. 4 indicates failure due to error in gradient evaluation. 5 indicates failure due to error in projection.

message

A text message explaining which termination criterion was used.

References

Birgin EG, Martinez JM, and Raydan M (2000): Nonmonotone spectral projected gradient methods on convex sets, SIAM J Optimization, 10, 1196-1211.

Birgin EG, Martinez JM, and Raydan M (2001): SPG: software for convex-constrained optimization, ACM Transactions on Mathematical Software.

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.

M Raydan (1997), Barzilai-Borwein gradient method for large-scale unconstrained minimization problem, SIAM J of Optimization, 7, 26-33.

R Varadhan and C Roland (2008), Simple and globally-convergent methods for accelerating the convergence of any EM algorithm, Scandinavian J Statistics, doi: 10.1111/j.1467-9469.2007.00585.x.

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

projectLinear, BBoptim, optim, nlm, sane, dfsane, grad

Examples

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sc2.f <- function(x){ sum((1:length(x)) * (exp(x) - x)) / 10}

sc2.g <- function(x){ (1:length(x)) * (exp(x) - 1) / 10}

p0 <- rnorm(50)
ans.spg1 <- spg(par=p0, fn=sc2.f)  # Default is method=3
ans.spg2 <- spg(par=p0, fn=sc2.f, method=1)
ans.spg3 <- spg(par=p0, fn=sc2.f, method=2)
ans.cg <- optim(par=p0, fn=sc2.f, method="CG")  #Uses conjugate-gradient method in "optim"
ans.lbfgs <- optim(par=p0, fn=sc2.f, method="L-BFGS-B")  #Uses low-memory BFGS method in "optim"

# Now we use exact gradient.  
# Computation is much faster compared to when using numerical gradient.
ans.spg1 <- spg(par=p0, fn=sc2.f, gr=sc2.g)

############
# Another example illustrating use of additional parameters to objective function 
valley.f <- function(x, cons) {
  n <- length(x)
  f <- rep(NA, n)
  j <- 3 * (1:(n/3))
  jm2 <- j - 2
  jm1 <- j - 1
  f[jm2] <- (cons[2] * x[jm2]^3 + cons[1] * x[jm2]) * exp(-(x[jm2]^2)/100) - 1
  f[jm1] <- 10 * (sin(x[jm2]) - x[jm1])
  f[j] <- 10 * (cos(x[jm2]) - x[j])
  sum(f*f)
  }

k <- c(1.003344481605351, -3.344481605351171e-03)
p0 <- rnorm(30)  # number of parameters should be a multiple of 3 for this example
ans.spg2 <- spg(par=p0, fn=valley.f, cons=k, method=2)  
ans.cg <- optim(par=p0, fn=valley.f, cons=k, method="CG")  
ans.lbfgs <- optim(par=p0, fn=valley.f, cons=k, method="L-BFGS-B")  

####################################################################
# Here is a statistical example illustrating log-likelihood maximization.

poissmix.loglik <- function(p,y) {
  # Log-likelihood for a binary Poisson mixture
  i <- 0:(length(y)-1)
  loglik <- y*log(p[1]*exp(-p[2])*p[2]^i/exp(lgamma(i+1)) + 
        (1 - p[1])*exp(-p[3])*p[3]^i/exp(lgamma(i+1)))
  return (sum(loglik) )
  }

# Data from Hasselblad (JASA 1969)
poissmix.dat <- data.frame(death=0:9, freq=c(162,267,271,185,111,61,27,8,3,1))
y <- poissmix.dat$freq

# Lower and upper bounds on parameters
lo <- c(0.001,0,0)  
hi <- c(0.999, Inf, Inf)

p0 <- runif(3,c(0.2,1,1),c(0.8,5,8))  # randomly generated starting values

ans.spg <- spg(par=p0, fn=poissmix.loglik, y=y, lower=lo, upper=hi, 
     control=list(maximize=TRUE))

# how to compute hessian at the MLE
  require(numDeriv)
  hess <- hessian(x=ans.spg$par, poissmix.loglik, y=y)
  se <- sqrt(-diag(solve(hess)))  # approximate standard errors