BBoptim: Large=Scale Nonlinear Optimization - A Wrapper for spg()
In BB: Solving and Optimizing Large-Scale Nonlinear Systems
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
A strategy using different Barzilai-Borwein steplengths to
optimize a nonlinear objective function subject to box constraints.
Usage
1
2
3
4
Arguments
par
A real vector argument to fn, indicating the initial
guess for the root of the nonliinear system of equations fn.
fn
Nonlinear objective function that is to be optimized. A scalar
function that takes a real vector as argument and returns a scalar
that is the value of the function at that point (see details).
gr
The gradient of the objective function fn evaluated at the
argument. This is a vector-function that takes a real vector as argument
and returns a real vector of the same length. It defaults to
NULL, which means that gradient is evaluated numerically.
Computations are dramatically faster in high-dimensional problems when
the exact gradient is provided. See *Example*.
method
A vector of integers specifying which Barzilai-Borwein
steplengths should be used in a consecutive manner. The methods will
be used in the order specified.
upper
An upper bound for box constraints. See spg
lower
An lower bound for box constraints. See spg
project
The projection function that takes a point in $R^n$ and
projects it onto a region that defines the constraints of the problem.
This is a vector-function that takes a real vector as argument and
returns a real vector of the same length.
See spg for more details.
projectArgs
list of arguments to project. See spg()
for more details.
control
A list of parameters governing the algorithm behaviour.
This list is the same as that for spg (excepting
the default for trace). See details for
important special features of control parameters.
quiet
logical indicating if messages about convergence success or
failure should be suppressed
...
arguments passed fn (via the optimization algorithm).
Details
This wrapper is especially useful in problems where (spg is likely
to experience convergence difficulties. When spg() fails, i.e.
when convergence > 0 is obtained, a user might attempt various strategies
to find a local optimizer. The function BBoptim tries the following
sequential strategy:
Try a different BB steplength. Since the default is method = 2
for dfsane, BBoptim wrapper tries method = c(2, 3, 1).
Try a different non-monotonicity parameter M for each method,
i.e. BBoptim wrapper tries M = c(50, 10) for each BB steplength.
The argument control defaults to a list with values
maxit = 1500, M = c(50, 10), ftol=1.e-10, gtol = 1e-05, maxfeval = 10000,
maximize = FALSE, trace = FALSE, triter = 10, eps = 1e-07, checkGrad=NULL.
It is recommended that checkGrad be set to FALSE for high-dimensional
problems, after making sure that the gradient is correctly specified. See
spg for additional details about the default.
If control is specified as an argument, only values which are different
need to be given in the list. See spg for more details.
Value
A list with the same elements as returned by spg. One additional
element returned is cpar which contains the control parameter settings
used to obtain successful convergence, or to obtain the best solution in
case of failure.
References
R Varadhan and PD Gilbert (2009), BB: An R Package for Solving a Large
System of Nonlinear Equations and for Optimizing a High-Dimensional
Nonlinear Objective Function, J. Statistical Software, 32:4,
http://www.jstatsoft.org/v32/i04/
See Also
BBsolve,
spg,
multiStart
optim
grad
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
# Use a preset seed so test values are reproducable.
require("setRNG")
old.seed <- setRNG(list(kind="Mersenne-Twister", normal.kind="Inversion",
seed=1234))
rosbkext <- function(x){
# Extended Rosenbrock function
n <- length(x)
j <- 2 * (1:(n/2))
jm1 <- j - 1
sum(100 * (x[j] - x[jm1]^2)^2 + (1 - x[jm1])^2)
}
p0 <- rnorm(50)
spg(par=p0, fn=rosbkext)
BBoptim(par=p0, fn=rosbkext)
# compare the improvement in convergence when bounds are specified
BBoptim(par=p0, fn=rosbkext, lower=0)
# identical to spg() with defaults
BBoptim(par=p0, fn=rosbkext, method=3, control=list(M=10, trace=TRUE))
Example output
Loading required package: setRNG
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 39.77528 pgrad: 27.99759
iter: 20 f-value: 30.22168 pgrad: 10.38169
iter: 30 f-value: 26.85768 pgrad: 40.95957
iter: 40 f-value: 24.2456 pgrad: 2.925454
iter: 50 f-value: 18.21567 pgrad: 3.079435
iter: 60 f-value: 14.91738 pgrad: 2.334929
iter: 70 f-value: 13.44534 pgrad: 11.99571
iter: 80 f-value: 11.40815 pgrad: 1.799464
iter: 90 f-value: 9.712975 pgrad: 4.29862
iter: 100 f-value: 8.392164 pgrad: 11.32991
iter: 110 f-value: 7.311477 pgrad: 1.565696
iter: 120 f-value: 6.639707 pgrad: 3.421991
iter: 130 f-value: 6.193688 pgrad: 22.20667
iter: 140 f-value: 4.66529 pgrad: 1.574616
iter: 150 f-value: 3.937451 pgrad: 8.75098
iter: 160 f-value: 3.153241 pgrad: 6.673114
iter: 170 f-value: 2.208422 pgrad: 1.53933
iter: 180 f-value: 1.669985 pgrad: 0.8507576
iter: 190 f-value: 1.449601 pgrad: 1.953953
iter: 200 f-value: 1.371948 pgrad: 0.6711463
iter: 210 f-value: 1.114267 pgrad: 1.776885
iter: 220 f-value: 0.9583621 pgrad: 1.568243
iter: 230 f-value: 0.8269146 pgrad: 3.633741
iter: 240 f-value: 0.6304826 pgrad: 0.3339276
iter: 250 f-value: 0.5107885 pgrad: 0.2684569
iter: 260 f-value: 0.4613847 pgrad: 0.2476938
iter: 270 f-value: 0.3270633 pgrad: 11.01763
iter: 280 f-value: 0.1742687 pgrad: 0.444439
iter: 290 f-value: 0.1625566 pgrad: 0.1235645
iter: 300 f-value: 0.101921 pgrad: 1.459339
iter: 310 f-value: 0.09256554 pgrad: 0.08955312
iter: 320 f-value: 0.04521948 pgrad: 0.05963155
iter: 330 f-value: 0.02634377 pgrad: 0.04566414
iter: 340 f-value: 0.01193873 pgrad: 0.0904557
iter: 350 f-value: 0.008038826 pgrad: 0.1842721
iter: 360 f-value: 0.01248124 pgrad: 3.643765
iter: 370 f-value: 0.001487418 pgrad: 0.1724186
iter: 380 f-value: 0.0009728912 pgrad: 0.008121104
iter: 390 f-value: 0.0001587139 pgrad: 0.003207078
iter: 400 f-value: 1.091916e-07 pgrad: 4.596227e-05
$par
[1] 0.9999292 0.9998582 0.9999353 0.9998704 0.9999426 0.9998850 0.9999344
[8] 0.9998686 0.9999350 0.9998698 0.9999351 0.9998700 0.9999316 0.9998630
[15] 0.9999404 0.9998806 0.9999350 0.9998699 0.9999125 0.9998247 0.9999356
[22] 0.9998711 0.9999310 0.9998619 0.9999353 0.9998704 0.9999356 0.9998710
[29] 0.9999353 0.9998705 0.9999378 0.9998755 0.9999341 0.9998680 0.9999349
[36] 0.9998695 0.9999350 0.9998697 0.9999347 0.9998693 0.9999360 0.9998718
[43] 0.9999330 0.9998659 0.9999348 0.9998694 0.9999351 0.9998700 0.9999344
[50] 0.9998686
$value
[1] 1.091439e-07
$gradient
[1] 4.597844e-05
$fn.reduction
[1] 10824.77
$iter
[1] 401
$feval
[1] 493
$convergence
[1] 0
$message
[1] "Successful convergence"
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 54.40324 pgrad: 52.26724
iter: 20 f-value: 33.13841 pgrad: 2.110108
iter: 30 f-value: 26.40889 pgrad: 2.109646
iter: 40 f-value: 18.79585 pgrad: 2.063969
iter: 50 f-value: 16.09957 pgrad: 2.000223
iter: 60 f-value: 13.9462 pgrad: 2.576766
iter: 70 f-value: 12.70453 pgrad: 2.179719
iter: 80 f-value: 10.25791 pgrad: 2.176441
iter: 90 f-value: 8.034416 pgrad: 12.35637
iter: 100 f-value: 7.331731 pgrad: 2.065253
iter: 110 f-value: 6.600971 pgrad: 1.61036
iter: 120 f-value: 4.842992 pgrad: 1.517246
iter: 130 f-value: 4.273633 pgrad: 1.502324
iter: 140 f-value: 3.519309 pgrad: 17.5416
iter: 150 f-value: 2.483051 pgrad: 2.053522
iter: 160 f-value: 3.74858 pgrad: 35.27227
iter: 170 f-value: 1.095609 pgrad: 0.8585015
iter: 180 f-value: 1.002489 pgrad: 0.4675539
iter: 190 f-value: 0.9069942 pgrad: 0.4363785
iter: 200 f-value: 0.4666699 pgrad: 1.127691
iter: 210 f-value: 0.3889428 pgrad: 0.2377373
iter: 220 f-value: 0.07270047 pgrad: 2.09311
iter: 230 f-value: 0.01169285 pgrad: 0.0282269
iter: 240 f-value: 0.01127181 pgrad: 1.139276
Successful convergence.
$par
[1] 0.9999683 0.9999366 0.9999685 0.9999369 0.9999687 0.9999373 0.9999684
[8] 0.9999368 0.9999685 0.9999369 0.9999685 0.9999369 0.9999684 0.9999367
[15] 0.9999686 0.9999372 0.9999685 0.9999369 0.9999682 0.9999363 0.9999685
[22] 0.9999369 0.9999684 0.9999367 0.9999685 0.9999369 0.9999685 0.9999369
[29] 0.9999685 0.9999369 0.9999685 0.9999370 0.9999684 0.9999368 0.9999685
[36] 0.9999369 0.9999685 0.9999369 0.9999685 0.9999369 0.9999685 0.9999369
[43] 0.9999684 0.9999368 0.9999685 0.9999369 0.9999685 0.9999369 0.9999684
[50] 0.9999368
$value
[1] 2.488589e-08
$gradient
[1] 1.617719e-06
$fn.reduction
[1] 10824.77
$iter
[1] 242
$feval
[1] 345
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
iter: 0 f-value: 10824.77 pgrad: 93589969068
iter: 10 f-value: 80.17689 pgrad: 89.45276
iter: 20 f-value: 18.6482 pgrad: 2.331853
iter: 30 f-value: 16.57026 pgrad: 14.70272
iter: 40 f-value: 14.53976 pgrad: 1.620273
iter: 50 f-value: 13.00756 pgrad: 14.99384
iter: 60 f-value: 12.46553 pgrad: 1.191981
iter: 70 f-value: 11.5946 pgrad: 1.075108
iter: 80 f-value: 11.05444 pgrad: 1.007026
iter: 90 f-value: 10.2394 pgrad: 0.9117447
iter: 100 f-value: 9.546816 pgrad: 0.8072167
iter: 110 f-value: 9.009637 pgrad: 0.7865717
iter: 120 f-value: 8.107801 pgrad: 1.592373
iter: 130 f-value: 7.803584 pgrad: 0.6117142
iter: 140 f-value: 7.656692 pgrad: 0.5939561
iter: 150 f-value: 4.065993 pgrad: 0.4956877
iter: 160 f-value: 3.959986 pgrad: 0.8255063
iter: 170 f-value: 3.887696 pgrad: 0.4946903
iter: 180 f-value: 3.701328 pgrad: 2.09004
iter: 190 f-value: 3.632759 pgrad: 0.4825317
iter: 200 f-value: 3.098259 pgrad: 1.007279
iter: 210 f-value: 3.039008 pgrad: 1.593319
iter: 220 f-value: 3.018869 pgrad: 0.488434
iter: 230 f-value: 2.667001 pgrad: 4.129558
iter: 240 f-value: 2.554116 pgrad: 0.4780626
iter: 250 f-value: 2.207891 pgrad: 1.591027
iter: 260 f-value: 1.998948 pgrad: 0.5785394
iter: 270 f-value: 1.96601 pgrad: 0.4639743
iter: 280 f-value: 1.934402 pgrad: 0.462478
iter: 290 f-value: 1.849995 pgrad: 0.5029136
iter: 300 f-value: 1.834994 pgrad: 0.4596853
iter: 310 f-value: 1.814697 pgrad: 0.4329975
iter: 320 f-value: 1.100727 pgrad: 0.6330166
iter: 330 f-value: 1.091672 pgrad: 0.4246225
iter: 340 f-value: 1.034675 pgrad: 0.4189956
iter: 350 f-value: 1.017113 pgrad: 0.4180038
iter: 360 f-value: 1.00267 pgrad: 0.4166663
iter: 370 f-value: 0.8821494 pgrad: 7.53588
iter: 380 f-value: 0.7199736 pgrad: 0.3883554
iter: 390 f-value: 0.5232619 pgrad: 0.3573914
iter: 400 f-value: 0.5201436 pgrad: 0.355807
iter: 410 f-value: 0.4125356 pgrad: 0.3006682
iter: 420 f-value: 0.4067102 pgrad: 0.3310747
iter: 430 f-value: 0.4018838 pgrad: 0.3295781
iter: 440 f-value: 0.3468103 pgrad: 0.3148879
iter: 450 f-value: 0.2720018 pgrad: 0.2911181
iter: 460 f-value: 0.2599182 pgrad: 0.2865981
iter: 470 f-value: 0.258163 pgrad: 0.2861937
iter: 480 f-value: 0.0674882 pgrad: 0.3424147
iter: 490 f-value: 0.002635549 pgrad: 0.0397948
iter: 500 f-value: 0.0003608076 pgrad: 0.01779499
iter: 510 f-value: 2.282022e-08 pgrad: 0.0006876697
Successful convergence.
$par
[1] 0.9999699 0.9999399 0.9999700 0.9999399 0.9999700 0.9999399 0.9999699
[8] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399
[15] 0.9999700 0.9999399 0.9999699 0.9999399 0.9999700 0.9999399 0.9999699
[22] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999700 0.9999399
[29] 0.9999699 0.9999399 0.9999700 0.9999399 0.9999699 0.9999399 0.9999699
[36] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999698 0.9999395
[43] 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699
[50] 0.9999399
$value
[1] 2.259115e-08
$gradient
[1] 1.583901e-07
$fn.reduction
[1] 10824.77
$iter
[1] 513
$feval
[1] 638
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 39.77528 pgrad: 27.99759
iter: 20 f-value: 30.22168 pgrad: 10.38169
iter: 30 f-value: 26.85768 pgrad: 40.95957
iter: 40 f-value: 24.2456 pgrad: 2.925454
iter: 50 f-value: 18.21567 pgrad: 3.079435
iter: 60 f-value: 14.91738 pgrad: 2.334929
iter: 70 f-value: 13.44534 pgrad: 11.99571
iter: 80 f-value: 11.40815 pgrad: 1.799464
iter: 90 f-value: 9.712975 pgrad: 4.29862
iter: 100 f-value: 8.392164 pgrad: 11.32991
iter: 110 f-value: 7.311477 pgrad: 1.565696
iter: 120 f-value: 6.639707 pgrad: 3.421991
iter: 130 f-value: 6.193688 pgrad: 22.20667
iter: 140 f-value: 4.66529 pgrad: 1.574616
iter: 150 f-value: 3.937451 pgrad: 8.75098
iter: 160 f-value: 3.153241 pgrad: 6.673114
iter: 170 f-value: 2.208422 pgrad: 1.53933
iter: 180 f-value: 1.669985 pgrad: 0.8507576
iter: 190 f-value: 1.449601 pgrad: 1.953953
iter: 200 f-value: 1.371948 pgrad: 0.6711463
iter: 210 f-value: 1.114267 pgrad: 1.776885
iter: 220 f-value: 0.9583621 pgrad: 1.568243
iter: 230 f-value: 0.8269146 pgrad: 3.633741
iter: 240 f-value: 0.6304826 pgrad: 0.3339276
iter: 250 f-value: 0.5107885 pgrad: 0.2684569
iter: 260 f-value: 0.4613847 pgrad: 0.2476938
iter: 270 f-value: 0.3270633 pgrad: 11.01763
iter: 280 f-value: 0.1742687 pgrad: 0.444439
iter: 290 f-value: 0.1625566 pgrad: 0.1235645
iter: 300 f-value: 0.101921 pgrad: 1.459339
iter: 310 f-value: 0.09256554 pgrad: 0.08955312
iter: 320 f-value: 0.04521948 pgrad: 0.05963155
iter: 330 f-value: 0.02634377 pgrad: 0.04566414
iter: 340 f-value: 0.01193873 pgrad: 0.0904557
iter: 350 f-value: 0.008038826 pgrad: 0.1842721
iter: 360 f-value: 0.01248124 pgrad: 3.643765
iter: 370 f-value: 0.001487418 pgrad: 0.1724186
iter: 380 f-value: 0.0009728912 pgrad: 0.008121104
iter: 390 f-value: 0.0001587139 pgrad: 0.003207078
iter: 400 f-value: 1.091916e-07 pgrad: 4.596227e-05
Successful convergence.
$par
[1] 0.9999292 0.9998582 0.9999353 0.9998704 0.9999426 0.9998850 0.9999344
[8] 0.9998686 0.9999350 0.9998698 0.9999351 0.9998700 0.9999316 0.9998630
[15] 0.9999404 0.9998806 0.9999350 0.9998699 0.9999125 0.9998247 0.9999356
[22] 0.9998711 0.9999310 0.9998619 0.9999353 0.9998704 0.9999356 0.9998710
[29] 0.9999353 0.9998705 0.9999378 0.9998755 0.9999341 0.9998680 0.9999349
[36] 0.9998695 0.9999350 0.9998697 0.9999347 0.9998693 0.9999360 0.9998718
[43] 0.9999330 0.9998659 0.9999348 0.9998694 0.9999351 0.9998700 0.9999344
[50] 0.9998686
$value
[1] 1.091439e-07
$gradient
[1] 4.597844e-05
$fn.reduction
[1] 10824.77
$iter
[1] 401
$feval
[1] 493
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
3 10
BB documentation built on Oct. 30, 2019, 11:41 a.m.
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Description Usage Arguments Details Value References See Also Examples
Description
A strategy using different Barzilai-Borwein steplengths to optimize a nonlinear objective function subject to box constraints.
Usage
1 2 3 4 |
Arguments
par |
A real vector argument to |
fn |
Nonlinear objective function that is to be optimized. A scalar function that takes a real vector as argument and returns a scalar that is the value of the function at that point (see details). |
gr |
The gradient of the objective function |
method |
A vector of integers specifying which Barzilai-Borwein steplengths should be used in a consecutive manner. The methods will be used in the order specified. |
upper |
An upper bound for box constraints. See |
lower |
An lower bound for box constraints. See |
project |
The projection function that takes a point in $R^n$ and
projects it onto a region that defines the constraints of the problem.
This is a vector-function that takes a real vector as argument and
returns a real vector of the same length.
See |
projectArgs |
list of arguments to |
control |
A list of parameters governing the algorithm behaviour.
This list is the same as that for |
quiet |
logical indicating if messages about convergence success or failure should be suppressed |
... |
arguments passed fn (via the optimization algorithm). |
Details
This wrapper is especially useful in problems where (spg is likely
to experience convergence difficulties. When spg() fails, i.e.
when convergence > 0 is obtained, a user might attempt various strategies
to find a local optimizer. The function BBoptim tries the following
sequential strategy:
Try a different BB steplength. Since the default is
method = 2fordfsane, BBoptim wrapper triesmethod = c(2, 3, 1).Try a different non-monotonicity parameter
Mfor each method, i.e. BBoptim wrapper triesM = c(50, 10)for each BB steplength.
The argument control defaults to a list with values
maxit = 1500, M = c(50, 10), ftol=1.e-10, gtol = 1e-05, maxfeval = 10000,
maximize = FALSE, trace = FALSE, triter = 10, eps = 1e-07, checkGrad=NULL.
It is recommended that checkGrad be set to FALSE for high-dimensional
problems, after making sure that the gradient is correctly specified. See
spg for additional details about the default.
If control is specified as an argument, only values which are different
need to be given in the list. See spg for more details.
Value
A list with the same elements as returned by spg. One additional
element returned is cpar which contains the control parameter settings
used to obtain successful convergence, or to obtain the best solution in
case of failure.
References
R Varadhan and PD Gilbert (2009), BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function, J. Statistical Software, 32:4, http://www.jstatsoft.org/v32/i04/
See Also
BBsolve,
spg,
multiStart
optim
grad
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | # Use a preset seed so test values are reproducable.
require("setRNG")
old.seed <- setRNG(list(kind="Mersenne-Twister", normal.kind="Inversion",
seed=1234))
rosbkext <- function(x){
# Extended Rosenbrock function
n <- length(x)
j <- 2 * (1:(n/2))
jm1 <- j - 1
sum(100 * (x[j] - x[jm1]^2)^2 + (1 - x[jm1])^2)
}
p0 <- rnorm(50)
spg(par=p0, fn=rosbkext)
BBoptim(par=p0, fn=rosbkext)
# compare the improvement in convergence when bounds are specified
BBoptim(par=p0, fn=rosbkext, lower=0)
# identical to spg() with defaults
BBoptim(par=p0, fn=rosbkext, method=3, control=list(M=10, trace=TRUE))
|
Example output
Loading required package: setRNG
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 39.77528 pgrad: 27.99759
iter: 20 f-value: 30.22168 pgrad: 10.38169
iter: 30 f-value: 26.85768 pgrad: 40.95957
iter: 40 f-value: 24.2456 pgrad: 2.925454
iter: 50 f-value: 18.21567 pgrad: 3.079435
iter: 60 f-value: 14.91738 pgrad: 2.334929
iter: 70 f-value: 13.44534 pgrad: 11.99571
iter: 80 f-value: 11.40815 pgrad: 1.799464
iter: 90 f-value: 9.712975 pgrad: 4.29862
iter: 100 f-value: 8.392164 pgrad: 11.32991
iter: 110 f-value: 7.311477 pgrad: 1.565696
iter: 120 f-value: 6.639707 pgrad: 3.421991
iter: 130 f-value: 6.193688 pgrad: 22.20667
iter: 140 f-value: 4.66529 pgrad: 1.574616
iter: 150 f-value: 3.937451 pgrad: 8.75098
iter: 160 f-value: 3.153241 pgrad: 6.673114
iter: 170 f-value: 2.208422 pgrad: 1.53933
iter: 180 f-value: 1.669985 pgrad: 0.8507576
iter: 190 f-value: 1.449601 pgrad: 1.953953
iter: 200 f-value: 1.371948 pgrad: 0.6711463
iter: 210 f-value: 1.114267 pgrad: 1.776885
iter: 220 f-value: 0.9583621 pgrad: 1.568243
iter: 230 f-value: 0.8269146 pgrad: 3.633741
iter: 240 f-value: 0.6304826 pgrad: 0.3339276
iter: 250 f-value: 0.5107885 pgrad: 0.2684569
iter: 260 f-value: 0.4613847 pgrad: 0.2476938
iter: 270 f-value: 0.3270633 pgrad: 11.01763
iter: 280 f-value: 0.1742687 pgrad: 0.444439
iter: 290 f-value: 0.1625566 pgrad: 0.1235645
iter: 300 f-value: 0.101921 pgrad: 1.459339
iter: 310 f-value: 0.09256554 pgrad: 0.08955312
iter: 320 f-value: 0.04521948 pgrad: 0.05963155
iter: 330 f-value: 0.02634377 pgrad: 0.04566414
iter: 340 f-value: 0.01193873 pgrad: 0.0904557
iter: 350 f-value: 0.008038826 pgrad: 0.1842721
iter: 360 f-value: 0.01248124 pgrad: 3.643765
iter: 370 f-value: 0.001487418 pgrad: 0.1724186
iter: 380 f-value: 0.0009728912 pgrad: 0.008121104
iter: 390 f-value: 0.0001587139 pgrad: 0.003207078
iter: 400 f-value: 1.091916e-07 pgrad: 4.596227e-05
$par
[1] 0.9999292 0.9998582 0.9999353 0.9998704 0.9999426 0.9998850 0.9999344
[8] 0.9998686 0.9999350 0.9998698 0.9999351 0.9998700 0.9999316 0.9998630
[15] 0.9999404 0.9998806 0.9999350 0.9998699 0.9999125 0.9998247 0.9999356
[22] 0.9998711 0.9999310 0.9998619 0.9999353 0.9998704 0.9999356 0.9998710
[29] 0.9999353 0.9998705 0.9999378 0.9998755 0.9999341 0.9998680 0.9999349
[36] 0.9998695 0.9999350 0.9998697 0.9999347 0.9998693 0.9999360 0.9998718
[43] 0.9999330 0.9998659 0.9999348 0.9998694 0.9999351 0.9998700 0.9999344
[50] 0.9998686
$value
[1] 1.091439e-07
$gradient
[1] 4.597844e-05
$fn.reduction
[1] 10824.77
$iter
[1] 401
$feval
[1] 493
$convergence
[1] 0
$message
[1] "Successful convergence"
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 54.40324 pgrad: 52.26724
iter: 20 f-value: 33.13841 pgrad: 2.110108
iter: 30 f-value: 26.40889 pgrad: 2.109646
iter: 40 f-value: 18.79585 pgrad: 2.063969
iter: 50 f-value: 16.09957 pgrad: 2.000223
iter: 60 f-value: 13.9462 pgrad: 2.576766
iter: 70 f-value: 12.70453 pgrad: 2.179719
iter: 80 f-value: 10.25791 pgrad: 2.176441
iter: 90 f-value: 8.034416 pgrad: 12.35637
iter: 100 f-value: 7.331731 pgrad: 2.065253
iter: 110 f-value: 6.600971 pgrad: 1.61036
iter: 120 f-value: 4.842992 pgrad: 1.517246
iter: 130 f-value: 4.273633 pgrad: 1.502324
iter: 140 f-value: 3.519309 pgrad: 17.5416
iter: 150 f-value: 2.483051 pgrad: 2.053522
iter: 160 f-value: 3.74858 pgrad: 35.27227
iter: 170 f-value: 1.095609 pgrad: 0.8585015
iter: 180 f-value: 1.002489 pgrad: 0.4675539
iter: 190 f-value: 0.9069942 pgrad: 0.4363785
iter: 200 f-value: 0.4666699 pgrad: 1.127691
iter: 210 f-value: 0.3889428 pgrad: 0.2377373
iter: 220 f-value: 0.07270047 pgrad: 2.09311
iter: 230 f-value: 0.01169285 pgrad: 0.0282269
iter: 240 f-value: 0.01127181 pgrad: 1.139276
Successful convergence.
$par
[1] 0.9999683 0.9999366 0.9999685 0.9999369 0.9999687 0.9999373 0.9999684
[8] 0.9999368 0.9999685 0.9999369 0.9999685 0.9999369 0.9999684 0.9999367
[15] 0.9999686 0.9999372 0.9999685 0.9999369 0.9999682 0.9999363 0.9999685
[22] 0.9999369 0.9999684 0.9999367 0.9999685 0.9999369 0.9999685 0.9999369
[29] 0.9999685 0.9999369 0.9999685 0.9999370 0.9999684 0.9999368 0.9999685
[36] 0.9999369 0.9999685 0.9999369 0.9999685 0.9999369 0.9999685 0.9999369
[43] 0.9999684 0.9999368 0.9999685 0.9999369 0.9999685 0.9999369 0.9999684
[50] 0.9999368
$value
[1] 2.488589e-08
$gradient
[1] 1.617719e-06
$fn.reduction
[1] 10824.77
$iter
[1] 242
$feval
[1] 345
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
iter: 0 f-value: 10824.77 pgrad: 93589969068
iter: 10 f-value: 80.17689 pgrad: 89.45276
iter: 20 f-value: 18.6482 pgrad: 2.331853
iter: 30 f-value: 16.57026 pgrad: 14.70272
iter: 40 f-value: 14.53976 pgrad: 1.620273
iter: 50 f-value: 13.00756 pgrad: 14.99384
iter: 60 f-value: 12.46553 pgrad: 1.191981
iter: 70 f-value: 11.5946 pgrad: 1.075108
iter: 80 f-value: 11.05444 pgrad: 1.007026
iter: 90 f-value: 10.2394 pgrad: 0.9117447
iter: 100 f-value: 9.546816 pgrad: 0.8072167
iter: 110 f-value: 9.009637 pgrad: 0.7865717
iter: 120 f-value: 8.107801 pgrad: 1.592373
iter: 130 f-value: 7.803584 pgrad: 0.6117142
iter: 140 f-value: 7.656692 pgrad: 0.5939561
iter: 150 f-value: 4.065993 pgrad: 0.4956877
iter: 160 f-value: 3.959986 pgrad: 0.8255063
iter: 170 f-value: 3.887696 pgrad: 0.4946903
iter: 180 f-value: 3.701328 pgrad: 2.09004
iter: 190 f-value: 3.632759 pgrad: 0.4825317
iter: 200 f-value: 3.098259 pgrad: 1.007279
iter: 210 f-value: 3.039008 pgrad: 1.593319
iter: 220 f-value: 3.018869 pgrad: 0.488434
iter: 230 f-value: 2.667001 pgrad: 4.129558
iter: 240 f-value: 2.554116 pgrad: 0.4780626
iter: 250 f-value: 2.207891 pgrad: 1.591027
iter: 260 f-value: 1.998948 pgrad: 0.5785394
iter: 270 f-value: 1.96601 pgrad: 0.4639743
iter: 280 f-value: 1.934402 pgrad: 0.462478
iter: 290 f-value: 1.849995 pgrad: 0.5029136
iter: 300 f-value: 1.834994 pgrad: 0.4596853
iter: 310 f-value: 1.814697 pgrad: 0.4329975
iter: 320 f-value: 1.100727 pgrad: 0.6330166
iter: 330 f-value: 1.091672 pgrad: 0.4246225
iter: 340 f-value: 1.034675 pgrad: 0.4189956
iter: 350 f-value: 1.017113 pgrad: 0.4180038
iter: 360 f-value: 1.00267 pgrad: 0.4166663
iter: 370 f-value: 0.8821494 pgrad: 7.53588
iter: 380 f-value: 0.7199736 pgrad: 0.3883554
iter: 390 f-value: 0.5232619 pgrad: 0.3573914
iter: 400 f-value: 0.5201436 pgrad: 0.355807
iter: 410 f-value: 0.4125356 pgrad: 0.3006682
iter: 420 f-value: 0.4067102 pgrad: 0.3310747
iter: 430 f-value: 0.4018838 pgrad: 0.3295781
iter: 440 f-value: 0.3468103 pgrad: 0.3148879
iter: 450 f-value: 0.2720018 pgrad: 0.2911181
iter: 460 f-value: 0.2599182 pgrad: 0.2865981
iter: 470 f-value: 0.258163 pgrad: 0.2861937
iter: 480 f-value: 0.0674882 pgrad: 0.3424147
iter: 490 f-value: 0.002635549 pgrad: 0.0397948
iter: 500 f-value: 0.0003608076 pgrad: 0.01779499
iter: 510 f-value: 2.282022e-08 pgrad: 0.0006876697
Successful convergence.
$par
[1] 0.9999699 0.9999399 0.9999700 0.9999399 0.9999700 0.9999399 0.9999699
[8] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399
[15] 0.9999700 0.9999399 0.9999699 0.9999399 0.9999700 0.9999399 0.9999699
[22] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999700 0.9999399
[29] 0.9999699 0.9999399 0.9999700 0.9999399 0.9999699 0.9999399 0.9999699
[36] 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999698 0.9999395
[43] 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699 0.9999399 0.9999699
[50] 0.9999399
$value
[1] 2.259115e-08
$gradient
[1] 1.583901e-07
$fn.reduction
[1] 10824.77
$iter
[1] 513
$feval
[1] 638
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
iter: 0 f-value: 10824.77 pgrad: 5320.046
iter: 10 f-value: 39.77528 pgrad: 27.99759
iter: 20 f-value: 30.22168 pgrad: 10.38169
iter: 30 f-value: 26.85768 pgrad: 40.95957
iter: 40 f-value: 24.2456 pgrad: 2.925454
iter: 50 f-value: 18.21567 pgrad: 3.079435
iter: 60 f-value: 14.91738 pgrad: 2.334929
iter: 70 f-value: 13.44534 pgrad: 11.99571
iter: 80 f-value: 11.40815 pgrad: 1.799464
iter: 90 f-value: 9.712975 pgrad: 4.29862
iter: 100 f-value: 8.392164 pgrad: 11.32991
iter: 110 f-value: 7.311477 pgrad: 1.565696
iter: 120 f-value: 6.639707 pgrad: 3.421991
iter: 130 f-value: 6.193688 pgrad: 22.20667
iter: 140 f-value: 4.66529 pgrad: 1.574616
iter: 150 f-value: 3.937451 pgrad: 8.75098
iter: 160 f-value: 3.153241 pgrad: 6.673114
iter: 170 f-value: 2.208422 pgrad: 1.53933
iter: 180 f-value: 1.669985 pgrad: 0.8507576
iter: 190 f-value: 1.449601 pgrad: 1.953953
iter: 200 f-value: 1.371948 pgrad: 0.6711463
iter: 210 f-value: 1.114267 pgrad: 1.776885
iter: 220 f-value: 0.9583621 pgrad: 1.568243
iter: 230 f-value: 0.8269146 pgrad: 3.633741
iter: 240 f-value: 0.6304826 pgrad: 0.3339276
iter: 250 f-value: 0.5107885 pgrad: 0.2684569
iter: 260 f-value: 0.4613847 pgrad: 0.2476938
iter: 270 f-value: 0.3270633 pgrad: 11.01763
iter: 280 f-value: 0.1742687 pgrad: 0.444439
iter: 290 f-value: 0.1625566 pgrad: 0.1235645
iter: 300 f-value: 0.101921 pgrad: 1.459339
iter: 310 f-value: 0.09256554 pgrad: 0.08955312
iter: 320 f-value: 0.04521948 pgrad: 0.05963155
iter: 330 f-value: 0.02634377 pgrad: 0.04566414
iter: 340 f-value: 0.01193873 pgrad: 0.0904557
iter: 350 f-value: 0.008038826 pgrad: 0.1842721
iter: 360 f-value: 0.01248124 pgrad: 3.643765
iter: 370 f-value: 0.001487418 pgrad: 0.1724186
iter: 380 f-value: 0.0009728912 pgrad: 0.008121104
iter: 390 f-value: 0.0001587139 pgrad: 0.003207078
iter: 400 f-value: 1.091916e-07 pgrad: 4.596227e-05
Successful convergence.
$par
[1] 0.9999292 0.9998582 0.9999353 0.9998704 0.9999426 0.9998850 0.9999344
[8] 0.9998686 0.9999350 0.9998698 0.9999351 0.9998700 0.9999316 0.9998630
[15] 0.9999404 0.9998806 0.9999350 0.9998699 0.9999125 0.9998247 0.9999356
[22] 0.9998711 0.9999310 0.9998619 0.9999353 0.9998704 0.9999356 0.9998710
[29] 0.9999353 0.9998705 0.9999378 0.9998755 0.9999341 0.9998680 0.9999349
[36] 0.9998695 0.9999350 0.9998697 0.9999347 0.9998693 0.9999360 0.9998718
[43] 0.9999330 0.9998659 0.9999348 0.9998694 0.9999351 0.9998700 0.9999344
[50] 0.9998686
$value
[1] 1.091439e-07
$gradient
[1] 4.597844e-05
$fn.reduction
[1] 10824.77
$iter
[1] 401
$feval
[1] 493
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
3 10
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