The ao package implements a numerical optimization algorithm called alternating optimization in R.
Alternating optimization is an iterative procedure which optimizes a function jointly over all parameters by alternately performing restricted optimization over individual parameter subsets.
For additional details on the method, please refer to the package vignette.
Example
The following is a simple example to perform alternating optimization of the Himmelblau’s function, separately for x1 and x2, with the parameter restrictions − 5 ≤ x1, x2 ≤ 5.
Step 2: Define the function to be optimized
himmelblau <- function(x, a, b) (x[1]^2 + x[2] + a)^2 + (x[1] + x[2]^2 + b)^2
The function is optimized over its first argument (x
), which needs to be a numeric
vector
. Other function arguments (a
and b
in this case) remain fixed during the optimization. The function should return a single numeric
value.
Step 3: Define a base optimizer
Alternating optimization requires a base optimizer that numerically solves the optimization problems in the partitions of the parameter vector. Such an optimizer must be defined through the framework provided by the optimizeR package, please see its documentation for details.
base_optimizer <- optimizeR::Optimizer$new(which = "stats::optim", lower = -5, upper = 5, method = "L-BFGS-B")
Step 4: Call the ao()
function
Despite f
and base_optimizer
, which have been defined above, the ao()
function requires the following arguments:
p
defines the starting parameter values,a
andb
are fixed function arguments,partition
defines the parameter subsets (here, the first entry ofx
and the second are optimized separately).
ao(f = himmelblau, p = c(0, 0), a = -11, b = -7, partition = list(1, 2), base_optimizer = base_optimizer)
#> $value
#> [1] 1.940035e-12
#>
#> $estimate
#> [1] 3.584428 -1.848126
#>
#> $sequence
#> iteration partition value seconds p1 p2
#> 1 0 NA 1.700000e+02 0.0000000000 0.000000 0.000000
#> 2 1 1 1.327270e+01 0.0090060234 3.395691 0.000000
#> 3 1 2 1.743666e+00 0.0009770393 3.395691 -1.803183
#> 4 2 1 2.847292e-02 0.0007698536 3.581412 -1.803183
#> 5 2 2 4.687472e-04 0.0006618500 3.581412 -1.847412
#> 6 3 1 7.368063e-06 0.0011827946 3.584381 -1.847412
#> 7 3 2 1.157612e-07 0.0004611015 3.584381 -1.848115
#> 8 4 1 1.900153e-09 0.0004670620 3.584427 -1.848115
#> 9 4 2 4.221429e-11 0.0003750324 3.584427 -1.848126
#> 10 5 1 3.598278e-12 0.0004618168 3.584428 -1.848126
#> 11 5 2 1.940035e-12 0.0003538132 3.584428 -1.848126
#>
#> $seconds
#> [1] 0.01471639
The output contains:
the function
value
at convergence,the parameter value
estimate
at convergence,sequence
provides information about the updates in the single iterations and partitions,and the optimization time in
seconds
.
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