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Flexible Optimization of Complex Loss Functions with State and Parameter Space Constraints

Description

Flexible optimizer with numerous input specifications for detailed parameterisation. Designed for complex loss functions with state and parameter space constraints. Visualization tools for validation and analysis of the convergence are included.

Details

Package: optimization Type: Package Version: 1.0-6 Date: 2017-09-23 License: GPL-2

Author(s)

NA

Maintainer: NA

References

Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987), Minimizing Multimodal Functions of Continuous Variables with the 'Simulated Annealing' Algorithm. ACM Transactions on Mathematical Software, 13(3):262-280.

Gao, F. and Han, L. (2012). Implementing the nelder-mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, 51(1):259 277.

Geiger, C. and Kanzow, C. (1999). Das nelder-mead-verfahren. Numerische Verfahren zur Loesung unregestrierter Optimierungsaufgaben.

Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983). Optimization by Simulated Annealing. Science, 220(4598): 671-680.

Nelder, J. and Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7(4).

Pronzato, L., Walter, E., Venot, A. and Lebruchec, J.-F. (1984). A general-purpose global optimizer: Implementation and applications. Mathematics and Computers in Simulation, 26(5):412-422.

See Also

[optim_nm](#topic+optim%5Fnm), [optim_sa](#topic+optim%5Fsa), [optim](../../../../doc/manuals/r-patched/packages/stats/refman/stats.html#topic+optim), [plot](../../../../doc/manuals/r-patched/packages/base/refman/base.html#topic+plot)

Examples

hi <- function(x){(x[1]**2 + x[2] - 11)**2 + (x[1] + x[2]**2 -7)**2}
optim_nm(fun = hi, k = 2)
optim_sa(fun = hi, start = c(runif(2, min = -1, max = 1)),
  trace = FALSE,
  lower = c(-4, -4),
  upper = c(4, 4),
  control = list(dyn_rf = FALSE,
    rf = 1.2,
    t0 = 10,
    nlimit = 100,
    r = 0.6,
    t_min = 0.1
  )
)

Optimization with Nelder-Mead

Description

This function contains a direct search algorithm, to minimize or maximize an objective function with respect to their input parameters.

Usage

  optim_nm(fun, k = 0, start, maximum = FALSE, trace = FALSE,
           alpha = 1, beta = 2, gamma = 1/2, delta = 1/2,
           tol = 0.00001, exit = 500, edge = 1)

Arguments

Details

The Nelder-Mead method is a comparatively simple heuristic optimization algorithm. It is, However, useful for relatively simple optimization problems without many local minima and low dimensions(n < 10). Nevertheless, the speed and accuracy are rather useful for simple problems. Moreover, the Nelder-Mead is able to optimize functions without derivatives. The handling of the optimization function is quite easy, because there are only few parameters to adjust.

Value

The output is a nmsa_optim object with following entries:

par

Function parameters after optimization.

function_value

Function response after optimization.

trace

Matrix with interim results. NULL if trace was not activated.

fun

The loss function.

start

The initial function parameters.

lower

The lower boundaries of the function parameters.

upper

The upper boundaries of the function parameters.

control

The number of parameters and iterations of the algorithm.

Author(s)

Alexander Lange

References

Gao, F. and Han, L. (2012). Implementing the nelder-mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, 51(1):259 277.

Geiger, C. and Kanzow, C. (1999). Das Nelder-Mead-Verfahren. Numerische Verfahren zur Loesung unregestrierter Optimierungsaufgaben.

Nelder, J. and Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7(4).

See Also

[optim_sa](#topic+optim%5Fsa), [optim](../../../../doc/manuals/r-patched/packages/stats/refman/stats.html#topic+optim), [plot.optim_nmsa](#topic+plot.optim%5Fnmsa)

Examples

#####  Rosenbrock function
# minimum at f(1,1) = 0
   B <- function(x){
    100*(x[2]-x[1]^2)^2+(1-x[1])^2
  }

##### Minimization with an initial guess at c(-2.048, 2.048)
  optim_nm(B, start = c(-2.048, 2.048))

#####  Himmelblau's function
# minimum at f(3,2) = 0
# f(-2.805, -3.1313) = 0
# f(-3.779, -3.283) = 0
#f(3.5844, -1.848) = 0
  H <- function(x){
    (x[1]^2+x[2]-11)^2+(x[1]+x[2]^2-7)^2
  }

##### Minimization with defined number of parameters
  optim_nm(fun = H, k = 2)

##### Colville function with 4 parameters
  co <- function(x){
    x1 <- x[1]
    x2 <- x[2]
    x3 <- x[3]
    x4 <- x[4]

    term1 <- 100 * (x1^2 - x2)^2
    term2 <- (x1 - 1)^2
    term3 <- (x3-1)^2
    term4 <- 90 * (x3^2 - x4)^2
    term5 <- 10.1 * ((x2 - 1)^2 + (x4 - 1)^2)
    term6 <- 19.8 * (x2 - 1)*(x4-1)

     y <- term1 + term2 + term3 + term4 + term5 + term6
  }

  optim_nm(co, k = 4)

#### Minimization with trace
  Output <- optim_nm(H, k = 2, trace = TRUE)
  plot(Output)
  plot(Output, 'contour')

Flexible Optimization with Simulated Annealing

Description

Random search optimization method with systematic component that searches the global optimum. The loss function is allowed to be non-linear, non-differentiable and multimodal. Undefined responses are allowed as well.

Usage

optim_sa(fun, start, maximization = FALSE, trace = FALSE,
         lower, upper, control = list())

Arguments

Details

Simulated Annealing is an optimization algorithm for solving complex functions that may have several optima. The method is composed of a random and a systematic component. Basically, it randomly modifies the variables combination n_limit times to compare their response values. Depending on the temperature and the constant k, there is also a likelihood of choosing variables combinations with worse response. There is thus a time-decreasing likelihood of leaving local optima. The Simulated Annealing Optimization method is therefore advantageous for multimodal functions. Undefined response values (NA) are allowed as well. This can be useful for loss functions with variables restrictions. The high number of parameters allows a very flexible parameterization. optim_sa is able to solve mathematical formulas as well as complex rule sets.

The performance therefore highly depends on the settings. It is indispensable to parameterize the algorithm carefully. The control list is pre-parameterized for loss functions of medium complexity. To improve the performance, the settings should be changed when solving relatively simple functions (e. g. three dimensional multimodal functions). For complex functions the settings should be changed to improve the accuracy. Most important parameters are nlimit, r and t0.

The dynamic rf adjustment depends on the number of loss function calls which are out of the variables boundaries as well as the temperature of the current iteration. The obligatory decreasing rf ensures a relatively wide search grid at the beginning of the optimization process that shrinks over time. It thus automatically adjusts for the trade-off between range of the search grid and accuracy. See Pronzato (1984) for more details. It is sometimes useful to disable the dynamic rf changing when the most performant rf are known. As dyn_rf usually improves the performance as well as the accuracy, the default is TRUE.

Value

The output is a nmsa_optim list object with following entries:

par

Function variables after optimization.

function_value

Loss function response after optimization.

trace

Matrix with interim results. NULL if trace was not activated.

fun

The loss function.

start

The initial function variables.

lower

The lower boundaries of the function variables.

upper

The upper boundaries of the function variables.

control

Control arguments, see 'details'.

Author(s)

Kai Husmann

References

Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987), Minimizing Multimodal Functions of Continuous Variables with the 'Simulated Annealing' Algorithm. ACM Transactions on Mathematical Software, 13(3):262-280.

Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983). Optimization by Simulated Annealing. Science, 220(4598):671-680.

Pronzato, L., Walter, E., Venot, A. and Lebruchec, J.-F. (1984). A general-purpose global optimizer: Implementation and applications. Mathematics and Computers in Simulation, 26(5):412-422.

See Also

[optim_nm](#topic+optim%5Fnm), [optim](../../../../doc/manuals/r-patched/packages/stats/refman/stats.html#topic+optim), [plot.optim_nmsa](#topic+plot.optim%5Fnmsa)

Examples

##### Rosenbrock function
# minimum at f(1,1) = 0
ro <- function(x){
  100*(x[2]-x[1]^2)^2+(1-x[1])^2
}

# Random start values. Example arguments for the relatively simple Rosenbrock function.
ro_sa <- optim_sa(fun = ro,
                  start = c(runif(2, min = -1, max = 1)),
                  lower = c(-5, -5),
                  upper = c(5, 5),
                  trace = TRUE,
                  control = list(t0 = 100,
                            nlimit = 550,
                            t_min = 0.1,
                            dyn_rf = FALSE,
                            rf = 1,
                            r = 0.7
                  )
         )


# Visual inspection.
plot(ro_sa)
plot(ro_sa, type = "contour")


##### Holder table function

# 4 minima at
  #f(8.055, 9.665) = -19.2085
  #f(-8.055, 9.665) = -19.2085
  #f(8.055, -9.665) = -19.2085
  #f(-8.055, -9.665) = -19.2085

ho <- function(x){
  x1 <- x[1]
  x2 <- x[2]

  fact1 <- sin(x1) * cos(x2)
  fact2 <- exp(abs(1 - sqrt(x1^2 + x2^2) / pi))
  y <- -abs(fact1 * fact2)
}

# Random start values. Example arguments for the relatively complex Holder table function.
optim_sa(fun = ho,
         start = c(1, 1),
         lower = c(-10, -10),
         upper = c(10, 10),
         trace = TRUE,
         control = list(dyn_rf = FALSE,
                        rf = 1.6,
                        t0 = 10,
                        nlimit = 200,
                        r = 0.6,
                        t_min = 0.1
         )
)

Plot an optim_nmsa Object

Description

Creates convergence or contour plots for visual inspection of the optimization result. Note that 'trace' must be activated for this function.
In case of a bivariate optimization, the 'contour' plot gives an overview of the parameter development over time in the entire state space. This is useful for the evaluation of the algorithm settings and therefore helps improving the performance. The development of the response can be visualized via the 'convergence' plot.

Usage

## S3 method for class 'optim_nmsa'
plot(x, type = 'convergence', lower = NA, upper = NA, ...)

Arguments

Author(s)

Kai Husmann, Alexander Lange

See Also

[optim_nm](#topic+optim%5Fnm), [optim_sa](#topic+optim%5Fsa)

Examples

# S3 method for class 'optim_nlme'

# Himmelblau's function
hi <- function(x){(x[1]**2 + x[2] - 11)**2 + (x[1] + x[2]**2 -7)**2}

out_nm <- optim_nm(hi, k = 2, trace = TRUE)
out_sa <- optim_sa(fun = hi, start = c(runif(2, min = -1, max = 1)),
                   trace = TRUE, lower = c(-4, -4) ,upper=c(4, 4),
                   control = list(t0 = 1000, nlimit = 1500,r = 0.8))

# Examples for optimization results via 'Nelder-Mead' method.
plot(out_nm)
plot(out_nm, type = "contour", lower = c(-4, -4), upper = c(4, 4))

# Examples for optimization results via 'Simulated Annealing' method.
plot(out_sa)
plot(out_sa, type = "contour")