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README.md
In ao: Alternating Optimization
Alternating optimization 
The {ao} package implements alternating optimization (AO) in
R.
Why?
AO is an iterative process that optimizes a function by alternately
performing restricted optimization over parameter subsets. Instead of
joint optimization, it breaks the optimization problem down into simpler
sub-problems. This approach can make optimization feasible when joint
optimization is too difficult. See the package
vignette for more
details.
How?
You can install the released package version from
CRAN with:
install.packages("ao")
Then load the package via library("ao") and you should be ready to go.
Here is a simple example of alternating minimization of the Rosenbrock
function:
rosenbrock <- function(x) (1 - x[1])^2 + (x[2] - x[1]^2)^2
The resulting optimization path …
… is obtained as follows:
ao(f = rosenbrock, initial = c(2, 2), add_details = TRUE)
#> $estimate
#> [1] 1.000895 1.001791
#>
#> $value
#> [1] 8.016137e-07
#>
#> $details
#> iteration value p1 p2 b1 b2 seconds
#> 1 0 5.000000e+00 2.000000 2.000000 0 0 0.000000000
#> 2 1 1.519238e-01 1.366025 2.000000 1 0 0.006363869
#> 3 1 1.339744e-01 1.366025 1.866024 0 1 0.002068996
#> 4 2 1.176778e-01 1.320824 1.866024 1 0 0.002699137
#> 5 2 1.029278e-01 1.320824 1.744575 0 1 0.002110958
#> 6 3 8.966402e-02 1.278883 1.744575 1 0 0.002901077
#> 7 3 7.777546e-02 1.278883 1.635540 0 1 0.001904011
#> 8 4 6.719114e-02 1.240415 1.635540 1 0 0.003144026
#> 9 4 5.779955e-02 1.240415 1.538630 0 1 0.001903057
#> 10 5 4.952339e-02 1.205560 1.538630 1 0 0.009890079
#> 11 5 4.225482e-02 1.205560 1.453374 0 1 0.002016068
#> 12 6 3.591491e-02 1.174366 1.453374 1 0 0.002801895
#> 13 6 3.040344e-02 1.174366 1.379135 0 1 0.001818895
#> 14 7 2.564430e-02 1.146792 1.379135 1 0 0.003333092
#> 15 7 2.154801e-02 1.146792 1.315133 0 1 0.002017021
#> 16 8 1.804492e-02 1.122712 1.315133 1 0 0.003165960
#> 17 8 1.505832e-02 1.122712 1.260483 0 1 0.002074957
#> 18 9 1.252724e-02 1.101923 1.260483 1 0 0.003380060
#> 19 9 1.038836e-02 1.101923 1.214235 0 1 0.001863956
#> 20 10 8.590837e-03 1.084167 1.214235 1 0 0.006909132
#> 21 10 7.084101e-03 1.084167 1.175418 0 1 0.001886129
#> 22 11 5.827377e-03 1.069149 1.175418 1 0 0.006983995
#> 23 11 4.781578e-03 1.069149 1.143079 0 1 0.001940012
#> 24 12 3.915156e-03 1.056558 1.143079 1 0 0.006923914
#> 25 12 3.198754e-03 1.056558 1.116314 0 1 0.003469944
#> 26 13 2.608707e-03 1.046082 1.116314 1 0 0.018738031
#> 27 13 2.123531e-03 1.046082 1.094287 0 1 0.001847029
#> 28 14 1.725945e-03 1.037424 1.094287 1 0 0.002201080
#> 29 14 1.400576e-03 1.037424 1.076249 0 1 0.001825809
#> 30 15 1.135093e-03 1.030310 1.076249 1 0 0.002213955
#> 31 15 9.187038e-04 1.030310 1.061539 0 1 0.001797915
#> 32 16 7.427825e-04 1.024492 1.061539 1 0 0.002194881
#> 33 16 5.998755e-04 1.024492 1.049585 0 1 0.001878977
#> 34 17 4.840462e-04 1.019754 1.049585 1 0 0.002230883
#> 35 17 3.902161e-04 1.019754 1.039898 0 1 0.001816034
#> 36 18 3.143566e-04 1.015907 1.039898 1 0 0.002327919
#> 37 18 2.530454e-04 1.015907 1.032068 0 1 0.002080917
#> 38 19 2.035803e-04 1.012794 1.032068 1 0 0.002326012
#> 39 19 1.636760e-04 1.012794 1.025751 0 1 0.001852989
#> 40 20 1.315375e-04 1.010279 1.025751 1 0 0.002521038
#> 41 20 1.056496e-04 1.010279 1.020663 0 1 0.002076864
#> 42 21 8.482978e-05 1.008251 1.020663 1 0 0.002562046
#> 43 21 6.807922e-05 1.008251 1.016570 0 1 0.001961946
#> 44 22 5.462405e-05 1.006619 1.016570 1 0 0.002355099
#> 45 22 4.380882e-05 1.006619 1.013281 0 1 0.001914978
#> 46 23 3.513011e-05 1.005307 1.013281 1 0 0.002474070
#> 47 23 2.815916e-05 1.005307 1.010641 0 1 0.002257109
#> 48 24 2.257018e-05 1.004252 1.010641 1 0 0.002510071
#> 49 24 1.808332e-05 1.004252 1.008523 0 1 0.002375126
#> 50 25 1.448872e-05 1.003406 1.008523 1 0 0.002344847
#> 51 25 1.160399e-05 1.003406 1.006825 0 1 0.001878977
#> 52 26 9.294548e-06 1.002728 1.006825 1 0 0.002475977
#> 53 26 7.441548e-06 1.002728 1.005463 0 1 0.001852989
#> 54 27 5.959072e-06 1.002184 1.005463 1 0 0.002201080
#> 55 27 4.769667e-06 1.002184 1.004373 0 1 0.001419067
#> 56 28 3.818729e-06 1.001748 1.004373 1 0 0.002189875
#> 57 28 3.055717e-06 1.001748 1.003499 0 1 0.001791000
#> 58 29 2.446111e-06 1.001399 1.003499 1 0 0.002210140
#> 59 29 1.956863e-06 1.001399 1.002800 0 1 0.001793861
#> 60 30 1.566279e-06 1.001119 1.002800 1 0 0.004199028
#> 61 30 1.252688e-06 1.001119 1.002240 0 1 0.002015114
#> 62 31 1.002554e-06 1.000895 1.002240 1 0 0.002316952
#> 63 31 8.016137e-07 1.000895 1.001791 0 1 0.001415968
#>
#> $seconds
#> [1] 0.1820159
#>
#> $stopping_reason
#> [1] "change in function value between 1 iteration is < 1e-06"
Contact?
If you have any questions, found a bug, need a feature, just file an
issue on GitHub.
Try the ao package in your browser
Any scripts or data that you put into this service are public.
ao documentation built on June 8, 2025, 11:20 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Alternating optimization
The {ao} package implements alternating optimization (AO) in
R.
Why?
AO is an iterative process that optimizes a function by alternately performing restricted optimization over parameter subsets. Instead of joint optimization, it breaks the optimization problem down into simpler sub-problems. This approach can make optimization feasible when joint optimization is too difficult. See the package vignette for more details.
How?
You can install the released package version from CRAN with:
install.packages("ao")
Then load the package via library("ao") and you should be ready to go.
Here is a simple example of alternating minimization of the Rosenbrock
function:
rosenbrock <- function(x) (1 - x[1])^2 + (x[2] - x[1]^2)^2
The resulting optimization path …
… is obtained as follows:
ao(f = rosenbrock, initial = c(2, 2), add_details = TRUE)
#> $estimate
#> [1] 1.000895 1.001791
#>
#> $value
#> [1] 8.016137e-07
#>
#> $details
#> iteration value p1 p2 b1 b2 seconds
#> 1 0 5.000000e+00 2.000000 2.000000 0 0 0.000000000
#> 2 1 1.519238e-01 1.366025 2.000000 1 0 0.006363869
#> 3 1 1.339744e-01 1.366025 1.866024 0 1 0.002068996
#> 4 2 1.176778e-01 1.320824 1.866024 1 0 0.002699137
#> 5 2 1.029278e-01 1.320824 1.744575 0 1 0.002110958
#> 6 3 8.966402e-02 1.278883 1.744575 1 0 0.002901077
#> 7 3 7.777546e-02 1.278883 1.635540 0 1 0.001904011
#> 8 4 6.719114e-02 1.240415 1.635540 1 0 0.003144026
#> 9 4 5.779955e-02 1.240415 1.538630 0 1 0.001903057
#> 10 5 4.952339e-02 1.205560 1.538630 1 0 0.009890079
#> 11 5 4.225482e-02 1.205560 1.453374 0 1 0.002016068
#> 12 6 3.591491e-02 1.174366 1.453374 1 0 0.002801895
#> 13 6 3.040344e-02 1.174366 1.379135 0 1 0.001818895
#> 14 7 2.564430e-02 1.146792 1.379135 1 0 0.003333092
#> 15 7 2.154801e-02 1.146792 1.315133 0 1 0.002017021
#> 16 8 1.804492e-02 1.122712 1.315133 1 0 0.003165960
#> 17 8 1.505832e-02 1.122712 1.260483 0 1 0.002074957
#> 18 9 1.252724e-02 1.101923 1.260483 1 0 0.003380060
#> 19 9 1.038836e-02 1.101923 1.214235 0 1 0.001863956
#> 20 10 8.590837e-03 1.084167 1.214235 1 0 0.006909132
#> 21 10 7.084101e-03 1.084167 1.175418 0 1 0.001886129
#> 22 11 5.827377e-03 1.069149 1.175418 1 0 0.006983995
#> 23 11 4.781578e-03 1.069149 1.143079 0 1 0.001940012
#> 24 12 3.915156e-03 1.056558 1.143079 1 0 0.006923914
#> 25 12 3.198754e-03 1.056558 1.116314 0 1 0.003469944
#> 26 13 2.608707e-03 1.046082 1.116314 1 0 0.018738031
#> 27 13 2.123531e-03 1.046082 1.094287 0 1 0.001847029
#> 28 14 1.725945e-03 1.037424 1.094287 1 0 0.002201080
#> 29 14 1.400576e-03 1.037424 1.076249 0 1 0.001825809
#> 30 15 1.135093e-03 1.030310 1.076249 1 0 0.002213955
#> 31 15 9.187038e-04 1.030310 1.061539 0 1 0.001797915
#> 32 16 7.427825e-04 1.024492 1.061539 1 0 0.002194881
#> 33 16 5.998755e-04 1.024492 1.049585 0 1 0.001878977
#> 34 17 4.840462e-04 1.019754 1.049585 1 0 0.002230883
#> 35 17 3.902161e-04 1.019754 1.039898 0 1 0.001816034
#> 36 18 3.143566e-04 1.015907 1.039898 1 0 0.002327919
#> 37 18 2.530454e-04 1.015907 1.032068 0 1 0.002080917
#> 38 19 2.035803e-04 1.012794 1.032068 1 0 0.002326012
#> 39 19 1.636760e-04 1.012794 1.025751 0 1 0.001852989
#> 40 20 1.315375e-04 1.010279 1.025751 1 0 0.002521038
#> 41 20 1.056496e-04 1.010279 1.020663 0 1 0.002076864
#> 42 21 8.482978e-05 1.008251 1.020663 1 0 0.002562046
#> 43 21 6.807922e-05 1.008251 1.016570 0 1 0.001961946
#> 44 22 5.462405e-05 1.006619 1.016570 1 0 0.002355099
#> 45 22 4.380882e-05 1.006619 1.013281 0 1 0.001914978
#> 46 23 3.513011e-05 1.005307 1.013281 1 0 0.002474070
#> 47 23 2.815916e-05 1.005307 1.010641 0 1 0.002257109
#> 48 24 2.257018e-05 1.004252 1.010641 1 0 0.002510071
#> 49 24 1.808332e-05 1.004252 1.008523 0 1 0.002375126
#> 50 25 1.448872e-05 1.003406 1.008523 1 0 0.002344847
#> 51 25 1.160399e-05 1.003406 1.006825 0 1 0.001878977
#> 52 26 9.294548e-06 1.002728 1.006825 1 0 0.002475977
#> 53 26 7.441548e-06 1.002728 1.005463 0 1 0.001852989
#> 54 27 5.959072e-06 1.002184 1.005463 1 0 0.002201080
#> 55 27 4.769667e-06 1.002184 1.004373 0 1 0.001419067
#> 56 28 3.818729e-06 1.001748 1.004373 1 0 0.002189875
#> 57 28 3.055717e-06 1.001748 1.003499 0 1 0.001791000
#> 58 29 2.446111e-06 1.001399 1.003499 1 0 0.002210140
#> 59 29 1.956863e-06 1.001399 1.002800 0 1 0.001793861
#> 60 30 1.566279e-06 1.001119 1.002800 1 0 0.004199028
#> 61 30 1.252688e-06 1.001119 1.002240 0 1 0.002015114
#> 62 31 1.002554e-06 1.000895 1.002240 1 0 0.002316952
#> 63 31 8.016137e-07 1.000895 1.001791 0 1 0.001415968
#>
#> $seconds
#> [1] 0.1820159
#>
#> $stopping_reason
#> [1] "change in function value between 1 iteration is < 1e-06"
Contact?
If you have any questions, found a bug, need a feature, just file an issue on GitHub.
Try the ao package in your browser
Any scripts or data that you put into this service are public.
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