optim
General-purpose Optimization

### Description

Provides general-purpose optimization based on Nelder-Mead, quasi-Newton, simulated annealing, and conjugate-gradient algorithms. Includes an option for box-constrained optimization.

### Usage

    optim(par, fn, gr = NULL, ..., method = c("Nelder-Mead",
        "BFGS", "CG", "L-BFGS-B", "SANN"), lower = -Inf,
        upper = Inf, control = list(), hessian = FALSE)

### Arguments

par | initial values for the parameters to be optimized.
fn | a function to be minimized (or maximized). Its first argument is the vector of parameters over which minimization is to take place. The function should return a scalar result.
gr | a function to return the gradient. Not needed for the "Nelder-Mead". If it is NULL and it is needed, a finite-difference approximation is used.
... | further arguments to pass to fn and gr.
method | the method to use. See Details.
lower, upper | the bounds on the variables for the method "L-BFGS-B".
control | a list of control parameters. See Details.
hessian | a logical value. If TRUE, a numerically differentiated Hessian matrix is returned. The default is FALSE.

### Details

By default this function performs minimization, but it maximizes if control$fnscale is negative.
The default method is an implementation of that of Nelder and Mead (1965), that uses only function values and is robust but relatively slow. It works reasonably well for non-differentiable functions.

- Method "BFGS" is a quasi-Newton method (also known as a variable metric algorithm, specifically) that published simultaneously in 1970 by Broyden, Fletcher, Goldfarb and Shanno. This uses function values and gradients to build up a picture of the surface to be optimized.
- Method "CG" is a conjugate gradients method based on that by Fletcher and Reeves (1964) (but with the option of Polak--Ribiere or Beale--Sorenson updates). Conjugate gradient methods generally are more fragile than the BFGS method, but because they do not store a matrix they can be successful in much larger optimization problems.
- Method "L-BFGS-B" is that of Byrd et. al. (1994) which allows box constraints: that is, each variable can be given a lower and/or upper bound. The initial value must satisfy the constraints. This uses a limited-memory modification of the BFGS quasi-Newton method. If non-trivial bounds are supplied, this method is selected with a warning.

Nocedal and Wright (1999) is a comprehensive reference for the previous three methods.

- Method "SANN" is a variant of simulated annealing given in Belisle (1992). Simulated-annealing belongs to the class of stochastic global optimization methods. It uses only function values but is relatively slow. It also works for non-differentiable functions. This implementation uses the Metropolis function for the acceptance probability. The next candidate point is generated from a Gaussian Markov kernel with scale proportional to the actual temperature. Temperatures are decreased according to the logarithmic cooling schedule as given in Belisle (1992, p. 890). Note that the "SANN" method depends critically on the settings of the control parameters. It is not a general-purpose method but can be very useful in getting to a good value on a very rough surface.

The control argument is a list that can supply any of the following components:

trace | an integer. If positive, tracing information on the progress of the optimization is produced. Higher values may produce more tracing information: for method "L-BFGS-B" there are six levels of tracing. (To understand exactly what these do see the source code: higher levels give more detail.)
fnscale | an overall scaling to be applied to the value of fn and gr during optimization. If negative, turns the problem into a maximization problem. Optimization is performed on fn(par)/fnscale.
parscale | a vector of scaling values for the parameters. Optimization is performed on par/parscale and these should be comparable in the sense that a unit change in any element produces about a unit change in the scaled value.
ndeps | a vector of step sizes for the finite-difference approximation to the gradient, on par/parscale scale. Defaults to 1e-3.
maxit | the maximum number of iterations. Defaults to 100 for the derivative-based methods, and 500 for "Nelder-Mead". For "SANN" maxitgives the total number of function evaluations. There is no other stopping criterion. Defaults to 10000.
abstol | the absolute convergence tolerance. Only useful for non-negative functions, as a tolerance for reaching zero.
reltol | the relative convergence tolerance. The algorithm stops if it is unable to reduce the value by a factor of reltol * (abs(val) + reltol) at a step. Defaults to sqrt(.Machine\$double.eps), typically about 1e-8.
alpha, beta, gamma | the scaling parameters for the "Nelder-Mead" method. alpha is the reflection factor (default 1.0), beta the contraction factor (0.5) and gamma the expansion factor (2.0).
REPORT | the frequency of reports for the "BFGS" and "L-BFGS-B" methods if control\$trace is positive. Defaults to every 10 iterations. For "SANN" is 100.
type | for the conjugate-gradients method. Takes value 1 for the Fletcher-Reeves update, 2 for Polak-Ribiere and 3 for Beale-Sorenson.
lmm | an integer giving the number of BFGS updates retained in the "L-BFGS-B" method, It defaults to 5.
factr | controls the convergence of the "L-BFGS-B" method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default is 1e7, that is a tolerance of about 1e-8.
pgtol | helps control the convergence of the "L-BFGS-B" method. It is a tolerance on the projected gradient in the current search direction. This defaults to zero when the check is suppressed.
temp | controls the "SANN" method. It is the starting temperature for the cooling schedule. Defaults to 10.
tmax | is the number of function evaluations at each temperature for the "SANN" method. Defaults to 10.

Value
returns a list with components:

par | the best set of parameters found.
value | the value of fn corresponding to par.
counts | a two-element integer vector giving the number of calls to fn and gr respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient.
convergence | an integer code. 0 indicates successful convergence. Error codes are 1 indicates that the iteration limit maxit had been reached. 10 indicates degeneracy of the Nelder-Mead simplex. 51 indicates a warning from the "L-BFGS-B" method; see component message for further details. 52 indicates an error from the "L-BFGS-B" method; see component message for further details.
message | a character string giving any additional information returned by the optimizer, or NULL.
hessian | only if argument hessian is TRUE. A symmetric matrix giving an estimate of the Hessian at the solution found. Note that this is the Hessian of the unconstrained problem even if the box constraints are active.

Differences between Spotfire Enterprise Runtime for R and Open-source R
Spotfire Enterprise Runtime for R does not support the "Brent" optimization method.
Note
The code for methods "Nelder-Mead", "BFGS" and "CG" was based originally on Pascal code in Nash (1990) that was translated by p2c and then re-crafted by B.D. Ripley. Dr Nash has agreed that the code can be made freely available.
The code for method "L-BFGS-B" is based on the reference: [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, "A limited memory algorithm for bound constrained optimization",
SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208.
[2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, "L-BFGS-B: a limited memory FORTRAN code for solving bound constrained optimization problems", Tech. Report, NAM-11, EECS Department, Northwestern University, 1994.
[3] R. Byrd, J. Nocedal and R. Schnabel "Representations of Quasi-Newton Matrices and their use in Limited Memory Methods", Mathematical Programming 63 (1994), no. 4, pp. 129-156.
The code for method "SANN" was contributed by A. Trapletti.
References
Belisle, C. J. P. 1992. Convergence theorems for a class of simulated annealing algorithms on Rd. J Applied Probability. Volume 29. 885-895.
Byrd, R. H., et al. 1995. A limited memory algorithm for bound constrained optimization. SIAM J. Scientific Computing. Volume 16. 1190-1208.
Fletcher, R. and Reeves, C. M. 1964. Function minimization by conjugate gradients. Computer Journal. Volume 7. 148-154.
Nelder, J. A. and Mead, R. 1965. A simplex algorithm for function minimization. Computer Journal. Volume 7. 308-313.
Nash, J. C. 1990. Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation. Bristol, NY: Adam Hilger.
Nocedal, J. and Wright, S. J. 1999. Numerical Optimization. New York, NY: Springer.
See Also
nlminb.
Examples

    fr <- function(x) {   ## Rosenbrock Banana function
        x1 <- x[1]
        x2 <- x[2]
        100 * (x2 - x1 * x1)^2 + (1 - x1)^2
    }
    grr <- function(x) { ## Gradient of 'fr'
        x1 <- x[1]
        x2 <- x[2]
        c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1),
           200 *      (x2 - x1 * x1))
    }
    optim(par=c(-1.2,1), fn=fr)
    optim(par=c(-1.2,1), fn=fr, gr=grr, method = "BFGS")
    optim(par=c(-1.2,1), fn=fr, method = "BFGS", hessian = TRUE)
    optim(par=c(-1.2,1), fn=fr, gr=grr, method = "CG")
    optim(par=c(-1.2,1), fn=fr, gr=grr, method = "CG", control=list(type=2))
    optim(par=c(-1.2,1), fn=fr, gr=grr, method = "L-BFGS-B")
    flb <- function(x) {
        p <- length(x)
        sum(c(1, rep(4, p-1)) * (x - c(1, x[-p])^2)^2)
    }
    ## 25-dimensional box constrained
    optim(par=rep(3, 25), fn=flb, method="L-BFGS-B",
          lower=rep(2, 25), upper=rep(4, 25)) # par[24] is *not* at boundary
    ## "wild" function , global minimum at about -15.81515
    fw <- function (x) {
        10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80
    }
    res1 <- optim(par=50, fn=fw, method="SANN",
                  control=list(maxit=20000, temp=20, parscale=20))
    res1$par
    ## Now improve locally
    res2 <- optim(res1$par, fw, method="BFGS")
    res2$par
    ## objective function has extra argument, "phase"
    res3 <- optim(par=c(4,5)*pi, function(x, phase) sum(sin(x-phase)^2),
        method="L-BFGS-B", lower=c(3,4)*pi, upper=c(5,6)*pi, phase=c(0.25,0.75))
    res3$par %% pi

Package stats version 6.1.5-18
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