Alternating optimization is an iterative procedure for optimizing some function jointly over all parameters by alternating restricted optimization over individual parameter subsets.
More precisely, consider minimizing or maximizing \(f:\mathbb{R}^n \to \mathbb{R}\) over the set of feasible points \(x \in X \subseteq \mathbb{R}^n\). The underlying algorithm of alternating optimization is as follows:
Assign an initial value for \(x\).
Optimize \(f\) with respect to a subset of parameters \(\tilde{x}\) while holding the other parameters constant. (Note that alternating optimization is a generalization of joint optimization, where the only parameter subset would be the whole set of parameters.)
Replace the values in \(x\) by the optimal values for \(\tilde{x}\) found in step 2.
Repeat from step 2 with another parameter subset.
Stop when the process has converged or reached an iteration limit.
When the joint optimization is (numerically) difficult (or not feasible).
When there is a natural division of the parameters. That is the case, e.g., for likelihood functions, where the parameter space naturally divides into parameter subsets corresponding to linear effects, variances and covariances with different influence on the likelihood value.
To improve optimization time in some cases, see (Hu and Hathaway 2002) for an example.
Compared to joint optimization, alternating optimization may be better in bypassing local optima, see (James C. Bezdek and Hathaway 2002).
Alternating optimization (under certain conditions on \(f\)) can convergence to the global optimum. However, the set of possible solutions also contains saddle points (James C. Bezdek et al. 1987).
J. Bezdek and Hathaway (2003) shows that alternating optimization under reasonable assumptions is locally \(q\)-linearly convergent.
As an application, we consider minimizing the Himmelblau’s function in \(n = 2\) dimensions with parameter constraints \(-5 \leq x_1, x_2 \leq 5\):
library("ao")
#> Lade nötiges Paket: optimizeR
himmelblau <- function(x) (x[1]^2 + x[2] - 11)^2 + (x[1] + x[2]^2 - 7)^2
ao(f = himmelblau, p = c(0,0), partition = list(1, 2),
optimizer = set_optimizer_optim(lower = -5, upper = 5, method = "L-BFGS-B"))
#> $optimum
#> [1] 1.940035e-12
#>
#> $estimate
#> [1] 3.584428 -1.848126
#>
#> $sequence
#> iteration partition time p1 p2
#> 1 0 0 0.0000000000 0.000000 0.000000
#> 2 1 1 0.0081338882 3.395691 0.000000
#> 3 1 2 0.0002748966 3.395691 -1.803183
#> 4 2 1 0.0002350807 3.581412 -1.803183
#> 5 2 2 0.0002110004 3.581412 -1.847412
#> 6 3 1 0.0003659725 3.584381 -1.847412
#> 7 3 2 0.0001718998 3.584381 -1.848115
#> 8 4 1 0.0001800060 3.584427 -1.848115
#> 9 4 2 0.0001559258 3.584427 -1.848126
#> 10 5 1 0.0001559258 3.584428 -1.848126
#> 11 5 2 0.0001590252 3.584428 -1.848126
#>
#> $time
#> Time difference of 0.01508212 secsThe call minimizes f by alternating optimizing with
respect to each parameter separately, where the parameters all are
initialized at the value 0.
The ao() function has the following arguments:
f: The function to be optimized, returning a single
numeric value. The first argument of f must be a numeric
vector of the length of p followed by any other arguments
specified by the ... argument.p: Starting parameter values for the optimization....: Additional arguments to be passed to
f.partition: A list of vectors of indices of
p, specifying the partition of the alternating
optimization. The default is as.list(1:length(p)),
i.e. each parameter is optimized separately. Parameter indices can be
members of multiple subsets.optimizer: An object of class optimizer,
which can be specified via optimizeR::set_optimizer(). The
default optimizer is stats::optim().iterations: The number of iterations through the
partitions. The default is 10.tolerance: A non-negative numeric value. The function
terminates prematurely if the euclidean distance between the current
solution and the one from the last iteration is smaller than
tolerance. The default is 1e-6.print.level: This argument determines the level of
printing which is done during the optimization process. Three values
(analogue to stats::nlm()) can be specified:
0 (the default): no printing1: initial and final details are printed2: full tracing information is printedplot: If TRUE, the parameter updates are
plotted. The default is FALSE.