multiStart: Nonlinear Optimization or Root-Finding with Multiple Starting...
In BB: Solving and Optimizing Large-Scale Nonlinear Systems
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
Start BBsolve or BBoptim from multiple starting
points to obtain multiple solutions and to test sensitivity to starting values.
Usage
1
2
3
4
5
Arguments
par
A real matrix, each row of which is an argument to fn,
indicating initial guesses for solving a nonlinear system fn = 0
or for optimizing the objective function fn.
fn
see BBsolve or BBoptim.
gr
Only required for optimization. See BBoptim.
action
A character string indicating whether to solve a nonlinear
system or to optimize. Default is “solve”.
method
see BBsolve or BBoptim.
upper
An upper bound for box constraints. See spg
lower
An lower bound for box constraints. See spg
project
A projection
function or character string indicating its name. 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
A list with arguments to the project
function.
control
See BBsolve and BBoptim.
quiet
A logical variable (TRUE/FALSE). If TRUE warnings
and some additional information printing are suppressed.
Default is quiet = FALSE
Note that the control variable trace and quiet
affect different printing, so if trace is not set to FALSE
there will be considerable printed output.
details
Logical indicating if the result should include the full
result from BBsolve or BBoptim for each starting value.
...
arguments passed fn (via the optimization algorithm).
Details
The optimization or root-finder is run with each row of par indicating
initial guesses.
Value
list with elements par, values, and converged.
It optionally returns an attribute called “details”, which is a list as long as
the number of starting values, which contains the complete object returned
by dfsane or spg for each starting value.
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
Examples
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# Use a preset seed so the example is reproducable.
require("setRNG")
old.seed <- setRNG(list(kind="Mersenne-Twister", normal.kind="Inversion",
seed=1234))
# Finding multiple roots of a nonlinear system
brownlin <- function(x) {
# Brown's almost linear system(A.P. Morgan, ACM 1983)
# two distinct solutions if n is even
# three distinct solutions if n is odd
n <- length(x)
f <- rep(NA, n)
nm1 <- 1:(n-1)
f[nm1] <- x[nm1] + sum(x) - (n+1)
f[n] <- prod(x) - 1
f
}
p <- 9
n <- 50
p0 <- matrix(rnorm(n*p), n, p) # n starting values, each of length p
ans <- multiStart(par=p0, fn=brownlin)
pmat <- ans$par[ans$conv, 1:p] # selecting only converged solutions
ord1 <- order(abs(pmat[,1]))
round(pmat[ord1, ], 3) # all 3 roots can be seen
# An optimization example
rosbkext <- function(x){
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)
pmat <- matrix(rnorm(100), 20, 5) # 20 starting values each of length 5
ans <- multiStart(par=pmat, fn=rosbkext, action="optimize")
ans
attr(ans, "details")[[1]] #
pmat <- ans$par[ans$conv, 1:5] # selecting only converged solutions
round(pmat, 3)
Example output
Loading required package: setRNG
Parameter set : 1 ...
Successful convergence.
Parameter set : 2 ...
Successful convergence.
Parameter set : 3 ...
Successful convergence.
Parameter set : 4 ...
Successful convergence.
Parameter set : 5 ...
Successful convergence.
Parameter set : 6 ...
Successful convergence.
Parameter set : 7 ...
Successful convergence.
Parameter set : 8 ...
Successful convergence.
Parameter set : 9 ...
Successful convergence.
Parameter set : 10 ...
Successful convergence.
Parameter set : 11 ...
Successful convergence.
Parameter set : 12 ...
Successful convergence.
Parameter set : 13 ...
Successful convergence.
Parameter set : 14 ...
Successful convergence.
Parameter set : 15 ...
Successful convergence.
Parameter set : 16 ...
Successful convergence.
Parameter set : 17 ...
Successful convergence.
Parameter set : 18 ...
Successful convergence.
Parameter set : 19 ...
Successful convergence.
Parameter set : 20 ...
Successful convergence.
Parameter set : 21 ...
Successful convergence.
Parameter set : 22 ...
Successful convergence.
Parameter set : 23 ...
Successful convergence.
Parameter set : 24 ...
Successful convergence.
Parameter set : 25 ...
Successful convergence.
Parameter set : 26 ...
Successful convergence.
Parameter set : 27 ...
Successful convergence.
Parameter set : 28 ...
Successful convergence.
Parameter set : 29 ...
Successful convergence.
Parameter set : 30 ...
Successful convergence.
Parameter set : 31 ...
Successful convergence.
Parameter set : 32 ...
Successful convergence.
Parameter set : 33 ...
Successful convergence.
Parameter set : 34 ...
Successful convergence.
Parameter set : 35 ...
Successful convergence.
Parameter set : 36 ...
Successful convergence.
Parameter set : 37 ...
Successful convergence.
Parameter set : 38 ...
Successful convergence.
Parameter set : 39 ...
Successful convergence.
Parameter set : 40 ...
Successful convergence.
Parameter set : 41 ...
Successful convergence.
Parameter set : 42 ...
Successful convergence.
Parameter set : 43 ...
Successful convergence.
Parameter set : 44 ...
Successful convergence.
Parameter set : 45 ...
Successful convergence.
Parameter set : 46 ...
Successful convergence.
Parameter set : 47 ...
Successful convergence.
Parameter set : 48 ...
Successful convergence.
Parameter set : 49 ...
Successful convergence.
Parameter set : 50 ...
Successful convergence.
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 16.347
[2,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[3,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[4,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[5,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[6,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[7,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[8,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[9,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[10,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[11,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[12,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[13,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[14,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[15,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[16,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[17,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[18,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[19,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[20,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[21,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[22,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[23,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[24,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[25,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[26,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[27,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[28,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[29,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[30,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[31,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[32,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[33,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[34,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[35,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[36,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[37,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[38,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[39,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[40,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[41,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[42,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[43,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[44,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[45,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[46,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[47,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[48,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[49,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[50,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
iter: 0 f-value: 22648.33 pgrad: 9061.209
iter: 10 f-value: 44.83298 pgrad: 168.1783
iter: 20 f-value: 29.42533 pgrad: 2.073708
iter: 30 f-value: 26.56617 pgrad: 52.06237
iter: 40 f-value: 23.01367 pgrad: 6.263308
iter: 50 f-value: 21.83825 pgrad: 2.040631
iter: 60 f-value: 18.25856 pgrad: 2.166934
iter: 70 f-value: 16.78601 pgrad: 2.00985
iter: 80 f-value: 14.67511 pgrad: 1.701071
iter: 90 f-value: 11.62743 pgrad: 2.440629
iter: 100 f-value: 9.914879 pgrad: 3.168448
iter: 110 f-value: 8.882793 pgrad: 31.73547
iter: 120 f-value: 7.576571 pgrad: 1.906156
iter: 130 f-value: 6.864912 pgrad: 1.99973
iter: 140 f-value: 6.328198 pgrad: 20.60001
iter: 150 f-value: 5.582651 pgrad: 2.100997
iter: 160 f-value: 4.727286 pgrad: 1.871804
iter: 170 f-value: 4.134273 pgrad: 1.568214
iter: 180 f-value: 2.987975 pgrad: 20.52218
iter: 190 f-value: 2.623606 pgrad: 0.689513
iter: 200 f-value: 2.461139 pgrad: 5.85653
iter: 210 f-value: 2.211638 pgrad: 1.01064
iter: 220 f-value: 2.13492 pgrad: 0.513771
iter: 230 f-value: 1.799606 pgrad: 6.160031
iter: 240 f-value: 1.699531 pgrad: 1.723531
iter: 250 f-value: 1.638155 pgrad: 0.421163
iter: 260 f-value: 1.346872 pgrad: 4.516073
iter: 270 f-value: 1.295948 pgrad: 0.3287437
iter: 280 f-value: 1.241569 pgrad: 0.3423485
iter: 290 f-value: 1.003999 pgrad: 0.4740795
iter: 300 f-value: 0.9668864 pgrad: 0.3245561
iter: 310 f-value: 0.7856029 pgrad: 0.2984773
iter: 320 f-value: 0.7358545 pgrad: 0.2307154
iter: 330 f-value: 0.7062176 pgrad: 6.145852
iter: 340 f-value: 0.7024254 pgrad: 11.1443
iter: 350 f-value: 0.6192393 pgrad: 0.2122829
iter: 360 f-value: 0.590852 pgrad: 0.2078623
iter: 370 f-value: 0.1914484 pgrad: 0.3003456
iter: 380 f-value: 0.1832165 pgrad: 0.1156638
iter: 390 f-value: 0.1676437 pgrad: 0.114445
iter: 400 f-value: 0.1374987 pgrad: 0.106399
iter: 410 f-value: 0.1414682 pgrad: 4.007048
iter: 420 f-value: 0.1207254 pgrad: 0.3420549
iter: 430 f-value: 0.1175114 pgrad: 0.10141
iter: 440 f-value: 0.08686819 pgrad: 0.1062418
iter: 450 f-value: 0.08254599 pgrad: 0.3201123
iter: 460 f-value: 0.07800522 pgrad: 1.575407
iter: 470 f-value: 0.07062682 pgrad: 2.791684
iter: 480 f-value: 0.0628413 pgrad: 0.1299277
iter: 490 f-value: 0.05238513 pgrad: 7.42971
iter: 500 f-value: 0.01946227 pgrad: 0.1119249
iter: 510 f-value: 0.01749382 pgrad: 0.04822032
iter: 520 f-value: 0.01299227 pgrad: 0.04074298
iter: 530 f-value: 0.01173887 pgrad: 0.09360855
iter: 540 f-value: 0.01129308 pgrad: 0.9992779
iter: 550 f-value: 0.0102092 pgrad: 0.5334276
iter: 560 f-value: 0.006627701 pgrad: 0.07300872
iter: 570 f-value: 0.004353487 pgrad: 0.2634308
iter: 580 f-value: 0.004004913 pgrad: 0.02492965
iter: 590 f-value: 0.003009892 pgrad: 0.02181283
iter: 600 f-value: 0.0017318 pgrad: 0.01683172
iter: 610 f-value: 0.001331618 pgrad: 0.01480588
iter: 620 f-value: 0.0002272057 pgrad: 0.00631634
iter: 630 f-value: 2.340518e-05 pgrad: 0.002222858
iter: 640 f-value: 3.057962e-07 pgrad: 0.01303558
$par
[1] 0.9999672 0.9999343 0.9999576 0.9999151 0.9999566 0.9999131 0.9999560
[8] 0.9999119 0.9999532 0.9999063 0.9999562 0.9999123 0.9999585 0.9999169
[15] 0.9999559 0.9999117 0.9999560 0.9999118 0.9999553 0.9999105 0.9999711
[22] 0.9999421 0.9999564 0.9999126 0.9999559 0.9999118 0.9999531 0.9999062
[29] 0.9999563 0.9999125 0.9999522 0.9999043 0.9999532 0.9999064 1.0000130
[36] 1.0000261 0.9999555 0.9999108 0.9999817 0.9999635 0.9999562 0.9999123
[43] 0.9999562 0.9999123 0.9999547 0.9999093 0.9999675 0.9999349 0.9999852
[50] 0.9999704
$value
[1] 4.133372e-08
$gradient
[1] 3.443613e-05
$fn.reduction
[1] 22648.33
$iter
[1] 642
$feval
[1] 778
$convergence
[1] 0
$message
[1] "Successful convergence"
iter: 0 f-value: 22648.33 pgrad: 9061.209
iter: 10 f-value: 40.13237 pgrad: 29.92377
iter: 20 f-value: 29.65503 pgrad: 2.084342
iter: 30 f-value: 24.15897 pgrad: 2.028767
iter: 40 f-value: 22.05032 pgrad: 2.059338
iter: 50 f-value: 19.63944 pgrad: 2.044971
iter: 60 f-value: 17.54232 pgrad: 2.083767
iter: 70 f-value: 16.73025 pgrad: 2.004452
iter: 80 f-value: 14.23287 pgrad: 1.897868
iter: 90 f-value: 13.45627 pgrad: 1.792503
iter: 100 f-value: 10.14956 pgrad: 2.049154
iter: 110 f-value: 11.60104 pgrad: 25.25272
iter: 120 f-value: 5.646377 pgrad: 2.106924
iter: 130 f-value: 4.698865 pgrad: 1.920606
iter: 140 f-value: 3.825852 pgrad: 1.852113
iter: 150 f-value: 3.586715 pgrad: 1.232265
iter: 160 f-value: 1.74063 pgrad: 0.6242387
iter: 170 f-value: 1.686722 pgrad: 0.4283466
iter: 180 f-value: 1.481776 pgrad: 0.3898767
iter: 190 f-value: 1.280082 pgrad: 0.3417461
iter: 200 f-value: 0.6583132 pgrad: 0.2120633
iter: 210 f-value: 0.5228338 pgrad: 1.904417
iter: 220 f-value: 0.4583797 pgrad: 0.165238
iter: 230 f-value: 0.3408936 pgrad: 2.653384
iter: 240 f-value: 0.3160754 pgrad: 0.132524
iter: 250 f-value: 0.213777 pgrad: 0.132754
iter: 260 f-value: 0.1566734 pgrad: 1.005709
iter: 270 f-value: 0.1375318 pgrad: 0.1962938
iter: 280 f-value: 0.1276687 pgrad: 0.115445
iter: 290 f-value: 0.06870504 pgrad: 0.08940269
iter: 300 f-value: 0.05673515 pgrad: 0.08451464
iter: 310 f-value: 0.009130962 pgrad: 0.2581156
iter: 320 f-value: 0.005917028 pgrad: 0.03364445
iter: 330 f-value: 8.363714e-06 pgrad: 0.003680363
Successful convergence.
$par
[1] 0.9999700 0.9999400 0.9999703 0.9999405 0.9999703 0.9999406 0.9999703
[8] 0.9999406 0.9999704 0.9999408 0.9999703 0.9999406 0.9999703 0.9999405
[15] 0.9999703 0.9999406 0.9999703 0.9999406 0.9999703 0.9999407 0.9999699
[22] 0.9999398 0.9999703 0.9999406 0.9999703 0.9999406 0.9999704 0.9999408
[29] 0.9999703 0.9999406 0.9999704 0.9999408 0.9999704 0.9999408 0.9999687
[36] 0.9999374 0.9999703 0.9999406 0.9999697 0.9999394 0.9999703 0.9999406
[43] 0.9999703 0.9999406 0.9999704 0.9999407 0.9999700 0.9999400 0.9999697
[50] 0.9999393
$value
[1] 2.223557e-08
$gradient
[1] 1.429335e-06
$fn.reduction
[1] 22648.33
$iter
[1] 337
$feval
[1] 338
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
Parameter set : 1 ...
iter: 0 f-value: 104.1477 pgrad: 195.7775
iter: 10 f-value: 0.8165461 pgrad: 1.691339
iter: 20 f-value: 0.3660929 pgrad: 0.6701369
iter: 30 f-value: 0.1948056 pgrad: 3.295041
iter: 40 f-value: 0.03397907 pgrad: 0.2969536
iter: 50 f-value: 0.03081283 pgrad: 0.1332274
iter: 60 f-value: 0.006901353 pgrad: 0.05083775
iter: 70 f-value: 0.0005085973 pgrad: 0.02791898
iter: 80 f-value: 3.46663e-05 pgrad: 0.003696932
Successful convergence.
Parameter set : 2 ...
iter: 0 f-value: 1043.794 pgrad: 1446.276
iter: 10 f-value: 2.737732 pgrad: 1.673578
iter: 20 f-value: 1.707397 pgrad: 2.12588
iter: 30 f-value: 0.7376359 pgrad: 1.054303
iter: 40 f-value: 0.6052184 pgrad: 14.46225
iter: 50 f-value: 0.1062305 pgrad: 0.2269856
iter: 60 f-value: 0.129 pgrad: 10.58344
iter: 70 f-value: 3.347751e-06 pgrad: 0.00101306
Successful convergence.
Parameter set : 3 ...
iter: 0 f-value: 118.5023 pgrad: 226.2174
iter: 10 f-value: 1.544505 pgrad: 2.058662
iter: 20 f-value: 1.042624 pgrad: 1.982083
iter: 30 f-value: 0.5748645 pgrad: 6.803917
iter: 40 f-value: 0.2663864 pgrad: 0.5238955
iter: 50 f-value: 0.09414201 pgrad: 0.2876652
iter: 60 f-value: 0.01393715 pgrad: 0.1003782
iter: 70 f-value: 0.005687909 pgrad: 0.06244017
iter: 80 f-value: 0.01253125 pgrad: 4.223294
iter: 90 f-value: 8.144976e-06 pgrad: 0.002220145
Successful convergence.
Parameter set : 4 ...
iter: 0 f-value: 3.438234 pgrad: 21.56433
iter: 10 f-value: 1.160692 pgrad: 4.846107
iter: 20 f-value: 0.6238095 pgrad: 4.638383
iter: 30 f-value: 0.3261322 pgrad: 5.303616
iter: 40 f-value: 0.1370613 pgrad: 6.31439
iter: 50 f-value: 0.05995255 pgrad: 5.840455
iter: 60 f-value: 0.001563663 pgrad: 0.7089548
iter: 70 f-value: 0.0002575129 pgrad: 0.009598437
Successful convergence.
Parameter set : 5 ...
iter: 0 f-value: 1504.554 pgrad: 2749.706
iter: 10 f-value: 3.520653 pgrad: 1.61283
iter: 20 f-value: 2.877953 pgrad: 1.580511
iter: 30 f-value: 2.164019 pgrad: 1.611121
iter: 40 f-value: 4.529342 pgrad: 32.99485
iter: 50 f-value: 1.224356 pgrad: 2.11142
iter: 60 f-value: 0.8293356 pgrad: 1.754804
iter: 70 f-value: 0.4338205 pgrad: 1.227703
iter: 80 f-value: 0.3383105 pgrad: 0.6689532
iter: 90 f-value: 0.2322107 pgrad: 9.774733
iter: 100 f-value: 0.01032914 pgrad: 0.08242389
iter: 110 f-value: 0.01756212 pgrad: 4.194925
iter: 120 f-value: 0.003351996 pgrad: 0.04558846
iter: 130 f-value: 0.0005658241 pgrad: 0.0183096
iter: 140 f-value: 0.01210931 pgrad: 4.355415
Successful convergence.
Parameter set : 6 ...
iter: 0 f-value: 89.85157 pgrad: 181.4924
iter: 10 f-value: 4.993377 pgrad: 1.613387
iter: 20 f-value: 3.852569 pgrad: 1.620627
iter: 30 f-value: 0.6696012 pgrad: 1.338925
iter: 40 f-value: 0.3486178 pgrad: 0.6238029
iter: 50 f-value: 0.253394 pgrad: 0.471531
iter: 60 f-value: 0.1569684 pgrad: 0.3468646
iter: 70 f-value: 0.05717632 pgrad: 0.194028
iter: 80 f-value: 0.002223493 pgrad: 0.03174228
iter: 90 f-value: 0.01307329 pgrad: 3.935886
Successful convergence.
Parameter set : 7 ...
iter: 0 f-value: 2553.425 pgrad: 3161.945
iter: 10 f-value: 3.287716 pgrad: 2.100978
iter: 20 f-value: 2.323605 pgrad: 2.061737
iter: 30 f-value: 1.22322 pgrad: 2.440811
iter: 40 f-value: 0.7073892 pgrad: 0.8351592
iter: 50 f-value: 0.1609366 pgrad: 0.7610845
iter: 60 f-value: 0.007220043 pgrad: 2.257647
iter: 70 f-value: 5.743478e-07 pgrad: 0.0004048528
Successful convergence.
Parameter set : 8 ...
iter: 0 f-value: 7.301386 pgrad: 50.49826
iter: 10 f-value: 0.4876734 pgrad: 0.6070551
iter: 20 f-value: 0.2424819 pgrad: 3.883128
iter: 30 f-value: 0.05690639 pgrad: 0.1565022
iter: 40 f-value: 9.563579e-05 pgrad: 0.005619649
Successful convergence.
Parameter set : 9 ...
iter: 0 f-value: 97.45441 pgrad: 234.7565
iter: 10 f-value: 0.4877573 pgrad: 1.020726
iter: 20 f-value: 0.1865057 pgrad: 0.5415518
iter: 30 f-value: 0.1315637 pgrad: 0.3539517
iter: 40 f-value: 0.07136122 pgrad: 0.2486395
iter: 50 f-value: 0.0427114 pgrad: 0.1848895
iter: 60 f-value: 0.03879811 pgrad: 0.1765162
iter: 70 f-value: 0.01747809 pgrad: 0.2651659
iter: 80 f-value: 0.01392243 pgrad: 0.1011329
iter: 90 f-value: 0.01635814 pgrad: 2.888483
iter: 100 f-value: 0.003261127 pgrad: 0.04737148
iter: 110 f-value: 0.01097115 pgrad: 4.169896
Successful convergence.
Parameter set : 10 ...
iter: 0 f-value: 176.7632 pgrad: 334.1628
iter: 10 f-value: 1.817845 pgrad: 2.074538
iter: 20 f-value: 1.041073 pgrad: 1.298292
iter: 30 f-value: 0.3539918 pgrad: 0.7796873
iter: 40 f-value: 0.0745093 pgrad: 6.304562
iter: 50 f-value: 0.0003067417 pgrad: 0.009971132
Successful convergence.
Parameter set : 11 ...
iter: 0 f-value: 558.4307 pgrad: 1202.867
iter: 10 f-value: 10.22346 pgrad: 53.95344
iter: 20 f-value: 1.957312 pgrad: 2.139882
iter: 30 f-value: 1.132162 pgrad: 1.63763
iter: 40 f-value: 0.6903289 pgrad: 1.115213
iter: 50 f-value: 0.3360284 pgrad: 0.4902542
iter: 60 f-value: 0.06463214 pgrad: 0.1659278
iter: 70 f-value: 2.2056e-05 pgrad: 0.007548476
Successful convergence.
Parameter set : 12 ...
iter: 0 f-value: 936.7399 pgrad: 1381.4
iter: 10 f-value: 3.791033 pgrad: 10.22923
iter: 20 f-value: 2.481176 pgrad: 1.565749
iter: 30 f-value: 1.848268 pgrad: 1.817987
iter: 40 f-value: 1.539006 pgrad: 2.018057
iter: 50 f-value: 1.147663 pgrad: 2.088939
iter: 60 f-value: 0.8899576 pgrad: 1.848375
iter: 70 f-value: 0.3315418 pgrad: 4.199062
iter: 80 f-value: 0.2255326 pgrad: 1.4914
iter: 90 f-value: 0.2123317 pgrad: 0.4528963
iter: 100 f-value: 0.04973951 pgrad: 0.1968087
iter: 110 f-value: 0.0124454 pgrad: 0.09117443
iter: 120 f-value: 0.0001859101 pgrad: 0.01256614
Successful convergence.
Parameter set : 13 ...
iter: 0 f-value: 116.0656 pgrad: 467.5044
iter: 10 f-value: 1.945429 pgrad: 2.08734
iter: 20 f-value: 0.7791629 pgrad: 1.02835
iter: 30 f-value: 0.4993475 pgrad: 0.6020762
iter: 40 f-value: 0.1458184 pgrad: 0.3209832
iter: 50 f-value: 0.0305428 pgrad: 5.139955
iter: 60 f-value: 4.704206e-07 pgrad: 0.0003640162
Successful convergence.
Parameter set : 14 ...
iter: 0 f-value: 158.3783 pgrad: 310.6516
iter: 10 f-value: 1.233519 pgrad: 2.103569
iter: 20 f-value: 0.8147884 pgrad: 17.53057
iter: 30 f-value: 0.5689345 pgrad: 1.090678
iter: 40 f-value: 0.2180282 pgrad: 0.3935711
iter: 50 f-value: 0.1540431 pgrad: 4.261513
iter: 60 f-value: 0.1201162 pgrad: 0.2838095
iter: 70 f-value: 0.07007922 pgrad: 0.8969035
iter: 80 f-value: 0.06464419 pgrad: 0.1881796
iter: 90 f-value: 0.04395621 pgrad: 0.1466053
iter: 100 f-value: 0.03089696 pgrad: 0.457613
iter: 110 f-value: 0.02483754 pgrad: 0.102058
iter: 120 f-value: 0.01280933 pgrad: 0.3240663
iter: 130 f-value: 0.008748824 pgrad: 0.0543076
iter: 140 f-value: 0.0005616496 pgrad: 0.6241075
iter: 150 f-value: 4.384569e-05 pgrad: 0.004445839
iter: 160 f-value: 1.805938e-09 pgrad: 1.906689e-05
Successful convergence.
Parameter set : 15 ...
iter: 0 f-value: 102.838 pgrad: 190.781
iter: 10 f-value: 2.725574 pgrad: 2.147047
iter: 20 f-value: 1.684569 pgrad: 2.149739
iter: 30 f-value: 0.8545071 pgrad: 4.062114
iter: 40 f-value: 0.5186857 pgrad: 0.5901075
iter: 50 f-value: 0.08392794 pgrad: 0.370899
iter: 60 f-value: 0.005359188 pgrad: 2.108803
Successful convergence.
Parameter set : 16 ...
iter: 0 f-value: 513.5289 pgrad: 684.5052
iter: 10 f-value: 1.109876 pgrad: 1.229066
iter: 20 f-value: 0.2733764 pgrad: 0.3807037
iter: 30 f-value: 0.1347278 pgrad: 9.811595
iter: 40 f-value: 8.692025e-06 pgrad: 0.001645287
Successful convergence.
Parameter set : 17 ...
iter: 0 f-value: 5781.69 pgrad: 8216.68
iter: 10 f-value: 2958.348 pgrad: 2595.746
iter: 20 f-value: 2.205485 pgrad: 16.35224
iter: 30 f-value: 0.3015938 pgrad: 0.3812378
iter: 40 f-value: 0.004954546 pgrad: 0.04162067
iter: 50 f-value: 0.009318093 pgrad: 2.72599
Successful convergence.
Parameter set : 18 ...
iter: 0 f-value: 10769.93 pgrad: 11885.55
iter: 10 f-value: 1.086693 pgrad: 1.944855
iter: 20 f-value: 0.5642495 pgrad: 3.709624
iter: 30 f-value: 0.3503109 pgrad: 0.5451841
iter: 40 f-value: 0.1757639 pgrad: 0.3478917
iter: 50 f-value: 0.06314237 pgrad: 3.063494
iter: 60 f-value: 0.01987791 pgrad: 0.09774254
iter: 70 f-value: 1.594943e-06 pgrad: 0.003916945
Successful convergence.
Parameter set : 19 ...
iter: 0 f-value: 213.8751 pgrad: 509.3601
iter: 10 f-value: 1.233618 pgrad: 1.574912
iter: 20 f-value: 0.4427763 pgrad: 1.342841
iter: 30 f-value: 0.1468199 pgrad: 0.7130403
iter: 40 f-value: 0.009736555 pgrad: 0.05829691
iter: 50 f-value: 5.906892e-08 pgrad: 0.0003150453
Successful convergence.
Parameter set : 20 ...
iter: 0 f-value: 381.5038 pgrad: 717.5239
iter: 10 f-value: 1.041809 pgrad: 1.821658
iter: 20 f-value: 0.7302731 pgrad: 4.555018
iter: 30 f-value: 0.5153623 pgrad: 0.8002466
iter: 40 f-value: 0.1290155 pgrad: 0.4946614
iter: 50 f-value: 0.04967761 pgrad: 0.156249
iter: 60 f-value: 0.001800669 pgrad: 0.02640828
iter: 70 f-value: 0.001448272 pgrad: 1.101689
Successful convergence.
$par
[,1] [,2] [,3] [,4] [,5]
[1,] 0.9999871 0.9999743 0.9999736 0.9999471 -0.96211903
[2,] 0.9999692 0.9999383 0.9999692 0.9999383 0.47463463
[3,] 0.9999735 0.9999470 0.9999585 0.9999169 0.35175427
[4,] 0.9999541 0.9999080 0.9999541 0.9999082 -0.10702036
[5,] 0.9999700 0.9999398 0.9999699 0.9999398 0.53751726
[6,] 0.9999431 0.9998860 0.9999395 0.9998788 -0.60210490
[7,] 0.9999699 0.9999398 0.9999699 0.9999398 -2.02329765
[8,] 0.9999149 0.9998295 0.9999147 0.9998291 1.15442128
[9,] 0.9999686 0.9999371 0.9999700 0.9999400 -0.23069300
[10,] 0.9997891 0.9995774 0.9997890 0.9995773 -0.36508877
[11,] 0.9999589 0.9999176 0.9999588 0.9999176 0.15751793
[12,] 0.9996967 0.9993924 0.9998152 0.9996297 0.76593222
[13,] 0.9999699 0.9999398 0.9999699 0.9999398 0.42608697
[14,] 0.9999703 0.9999403 0.9999706 0.9999410 0.92726540
[15,] 0.9999019 0.9998035 0.9999019 0.9998036 0.04613459
[16,] 0.9999665 0.9999330 0.9999665 0.9999330 -0.50428564
[17,] 0.9999615 0.9999228 0.9999615 0.9999228 -0.30044941
[18,] 0.9999697 0.9999394 0.9999697 0.9999394 0.82291598
[19,] 0.9998285 0.9996564 0.9998284 0.9996563 -0.84926880
[20,] 0.9999693 0.9999385 0.9999692 0.9999384 -0.95361010
$fvalue
[1] 8.638806e-10 1.901873e-09 2.423288e-09 4.216728e-09 1.806844e-09
[6] 6.911578e-09 1.809587e-09 1.454471e-08 1.886645e-09 8.911840e-08
[11] 3.387040e-09 1.263133e-07 1.808496e-09 1.761743e-09 1.926060e-08
[16] 2.239977e-09 2.971137e-09 1.832471e-09 5.892285e-08 1.892542e-09
$converged
[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[16] TRUE TRUE TRUE TRUE TRUE
NULL
[,1] [,2] [,3] [,4] [,5]
[1,] 1 1.000 1 1 -0.962
[2,] 1 1.000 1 1 0.475
[3,] 1 1.000 1 1 0.352
[4,] 1 1.000 1 1 -0.107
[5,] 1 1.000 1 1 0.538
[6,] 1 1.000 1 1 -0.602
[7,] 1 1.000 1 1 -2.023
[8,] 1 1.000 1 1 1.154
[9,] 1 1.000 1 1 -0.231
[10,] 1 1.000 1 1 -0.365
[11,] 1 1.000 1 1 0.158
[12,] 1 0.999 1 1 0.766
[13,] 1 1.000 1 1 0.426
[14,] 1 1.000 1 1 0.927
[15,] 1 1.000 1 1 0.046
[16,] 1 1.000 1 1 -0.504
[17,] 1 1.000 1 1 -0.300
[18,] 1 1.000 1 1 0.823
[19,] 1 1.000 1 1 -0.849
[20,] 1 1.000 1 1 -0.954
BB documentation built on Oct. 30, 2019, 11:41 a.m.
R Package Documentation
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Description Usage Arguments Details Value References See Also Examples
Description
Start BBsolve or BBoptim from multiple starting
points to obtain multiple solutions and to test sensitivity to starting values.
Usage
1 2 3 4 5 |
Arguments
par |
A real matrix, each row of which is an argument to |
fn |
see |
gr |
Only required for optimization. See |
action |
A character string indicating whether to solve a nonlinear system or to optimize. Default is “solve”. |
method |
see |
upper |
An upper bound for box constraints. See |
lower |
An lower bound for box constraints. See |
project |
A projection
function or character string indicating its name. 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 |
A list with arguments to the |
control |
See |
quiet |
A logical variable (TRUE/FALSE). If |
details |
Logical indicating if the result should include the full
result from |
... |
arguments passed fn (via the optimization algorithm). |
Details
The optimization or root-finder is run with each row of par indicating
initial guesses.
Value
list with elements par, values, and converged.
It optionally returns an attribute called “details”, which is a list as long as
the number of starting values, which contains the complete object returned
by dfsane or spg for each starting value.
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
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | # Use a preset seed so the example is reproducable.
require("setRNG")
old.seed <- setRNG(list(kind="Mersenne-Twister", normal.kind="Inversion",
seed=1234))
# Finding multiple roots of a nonlinear system
brownlin <- function(x) {
# Brown's almost linear system(A.P. Morgan, ACM 1983)
# two distinct solutions if n is even
# three distinct solutions if n is odd
n <- length(x)
f <- rep(NA, n)
nm1 <- 1:(n-1)
f[nm1] <- x[nm1] + sum(x) - (n+1)
f[n] <- prod(x) - 1
f
}
p <- 9
n <- 50
p0 <- matrix(rnorm(n*p), n, p) # n starting values, each of length p
ans <- multiStart(par=p0, fn=brownlin)
pmat <- ans$par[ans$conv, 1:p] # selecting only converged solutions
ord1 <- order(abs(pmat[,1]))
round(pmat[ord1, ], 3) # all 3 roots can be seen
# An optimization example
rosbkext <- function(x){
n <- length(x)
j <- 2 * (1:(n/2))
jm1 <- j - 1
sum(100 * (x[j] - x[jm1]^2)^2 + (1 - x[jm1])^2)
}
p0 <- rnorm(50)
spg(par=p0, fn=rosbkext)
BBoptim(par=p0, fn=rosbkext)
pmat <- matrix(rnorm(100), 20, 5) # 20 starting values each of length 5
ans <- multiStart(par=pmat, fn=rosbkext, action="optimize")
ans
attr(ans, "details")[[1]] #
pmat <- ans$par[ans$conv, 1:5] # selecting only converged solutions
round(pmat, 3)
|
Example output
Loading required package: setRNG
Parameter set : 1 ...
Successful convergence.
Parameter set : 2 ...
Successful convergence.
Parameter set : 3 ...
Successful convergence.
Parameter set : 4 ...
Successful convergence.
Parameter set : 5 ...
Successful convergence.
Parameter set : 6 ...
Successful convergence.
Parameter set : 7 ...
Successful convergence.
Parameter set : 8 ...
Successful convergence.
Parameter set : 9 ...
Successful convergence.
Parameter set : 10 ...
Successful convergence.
Parameter set : 11 ...
Successful convergence.
Parameter set : 12 ...
Successful convergence.
Parameter set : 13 ...
Successful convergence.
Parameter set : 14 ...
Successful convergence.
Parameter set : 15 ...
Successful convergence.
Parameter set : 16 ...
Successful convergence.
Parameter set : 17 ...
Successful convergence.
Parameter set : 18 ...
Successful convergence.
Parameter set : 19 ...
Successful convergence.
Parameter set : 20 ...
Successful convergence.
Parameter set : 21 ...
Successful convergence.
Parameter set : 22 ...
Successful convergence.
Parameter set : 23 ...
Successful convergence.
Parameter set : 24 ...
Successful convergence.
Parameter set : 25 ...
Successful convergence.
Parameter set : 26 ...
Successful convergence.
Parameter set : 27 ...
Successful convergence.
Parameter set : 28 ...
Successful convergence.
Parameter set : 29 ...
Successful convergence.
Parameter set : 30 ...
Successful convergence.
Parameter set : 31 ...
Successful convergence.
Parameter set : 32 ...
Successful convergence.
Parameter set : 33 ...
Successful convergence.
Parameter set : 34 ...
Successful convergence.
Parameter set : 35 ...
Successful convergence.
Parameter set : 36 ...
Successful convergence.
Parameter set : 37 ...
Successful convergence.
Parameter set : 38 ...
Successful convergence.
Parameter set : 39 ...
Successful convergence.
Parameter set : 40 ...
Successful convergence.
Parameter set : 41 ...
Successful convergence.
Parameter set : 42 ...
Successful convergence.
Parameter set : 43 ...
Successful convergence.
Parameter set : 44 ...
Successful convergence.
Parameter set : 45 ...
Successful convergence.
Parameter set : 46 ...
Successful convergence.
Parameter set : 47 ...
Successful convergence.
Parameter set : 48 ...
Successful convergence.
Parameter set : 49 ...
Successful convergence.
Parameter set : 50 ...
Successful convergence.
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 -0.705 16.347
[2,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[3,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[4,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[5,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[6,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[7,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[8,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[9,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[10,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[11,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[12,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[13,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[14,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[15,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[16,] 0.975 0.975 0.975 0.975 0.975 0.975 0.975 0.975 1.229
[17,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[18,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[19,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[20,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[21,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[22,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[23,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[24,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[25,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[26,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[27,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[28,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[29,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[30,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[31,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[32,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[33,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[34,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[35,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[36,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[37,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[38,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[39,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[40,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[41,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[42,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[43,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[44,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[45,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[46,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[47,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[48,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[49,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
[50,] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
iter: 0 f-value: 22648.33 pgrad: 9061.209
iter: 10 f-value: 44.83298 pgrad: 168.1783
iter: 20 f-value: 29.42533 pgrad: 2.073708
iter: 30 f-value: 26.56617 pgrad: 52.06237
iter: 40 f-value: 23.01367 pgrad: 6.263308
iter: 50 f-value: 21.83825 pgrad: 2.040631
iter: 60 f-value: 18.25856 pgrad: 2.166934
iter: 70 f-value: 16.78601 pgrad: 2.00985
iter: 80 f-value: 14.67511 pgrad: 1.701071
iter: 90 f-value: 11.62743 pgrad: 2.440629
iter: 100 f-value: 9.914879 pgrad: 3.168448
iter: 110 f-value: 8.882793 pgrad: 31.73547
iter: 120 f-value: 7.576571 pgrad: 1.906156
iter: 130 f-value: 6.864912 pgrad: 1.99973
iter: 140 f-value: 6.328198 pgrad: 20.60001
iter: 150 f-value: 5.582651 pgrad: 2.100997
iter: 160 f-value: 4.727286 pgrad: 1.871804
iter: 170 f-value: 4.134273 pgrad: 1.568214
iter: 180 f-value: 2.987975 pgrad: 20.52218
iter: 190 f-value: 2.623606 pgrad: 0.689513
iter: 200 f-value: 2.461139 pgrad: 5.85653
iter: 210 f-value: 2.211638 pgrad: 1.01064
iter: 220 f-value: 2.13492 pgrad: 0.513771
iter: 230 f-value: 1.799606 pgrad: 6.160031
iter: 240 f-value: 1.699531 pgrad: 1.723531
iter: 250 f-value: 1.638155 pgrad: 0.421163
iter: 260 f-value: 1.346872 pgrad: 4.516073
iter: 270 f-value: 1.295948 pgrad: 0.3287437
iter: 280 f-value: 1.241569 pgrad: 0.3423485
iter: 290 f-value: 1.003999 pgrad: 0.4740795
iter: 300 f-value: 0.9668864 pgrad: 0.3245561
iter: 310 f-value: 0.7856029 pgrad: 0.2984773
iter: 320 f-value: 0.7358545 pgrad: 0.2307154
iter: 330 f-value: 0.7062176 pgrad: 6.145852
iter: 340 f-value: 0.7024254 pgrad: 11.1443
iter: 350 f-value: 0.6192393 pgrad: 0.2122829
iter: 360 f-value: 0.590852 pgrad: 0.2078623
iter: 370 f-value: 0.1914484 pgrad: 0.3003456
iter: 380 f-value: 0.1832165 pgrad: 0.1156638
iter: 390 f-value: 0.1676437 pgrad: 0.114445
iter: 400 f-value: 0.1374987 pgrad: 0.106399
iter: 410 f-value: 0.1414682 pgrad: 4.007048
iter: 420 f-value: 0.1207254 pgrad: 0.3420549
iter: 430 f-value: 0.1175114 pgrad: 0.10141
iter: 440 f-value: 0.08686819 pgrad: 0.1062418
iter: 450 f-value: 0.08254599 pgrad: 0.3201123
iter: 460 f-value: 0.07800522 pgrad: 1.575407
iter: 470 f-value: 0.07062682 pgrad: 2.791684
iter: 480 f-value: 0.0628413 pgrad: 0.1299277
iter: 490 f-value: 0.05238513 pgrad: 7.42971
iter: 500 f-value: 0.01946227 pgrad: 0.1119249
iter: 510 f-value: 0.01749382 pgrad: 0.04822032
iter: 520 f-value: 0.01299227 pgrad: 0.04074298
iter: 530 f-value: 0.01173887 pgrad: 0.09360855
iter: 540 f-value: 0.01129308 pgrad: 0.9992779
iter: 550 f-value: 0.0102092 pgrad: 0.5334276
iter: 560 f-value: 0.006627701 pgrad: 0.07300872
iter: 570 f-value: 0.004353487 pgrad: 0.2634308
iter: 580 f-value: 0.004004913 pgrad: 0.02492965
iter: 590 f-value: 0.003009892 pgrad: 0.02181283
iter: 600 f-value: 0.0017318 pgrad: 0.01683172
iter: 610 f-value: 0.001331618 pgrad: 0.01480588
iter: 620 f-value: 0.0002272057 pgrad: 0.00631634
iter: 630 f-value: 2.340518e-05 pgrad: 0.002222858
iter: 640 f-value: 3.057962e-07 pgrad: 0.01303558
$par
[1] 0.9999672 0.9999343 0.9999576 0.9999151 0.9999566 0.9999131 0.9999560
[8] 0.9999119 0.9999532 0.9999063 0.9999562 0.9999123 0.9999585 0.9999169
[15] 0.9999559 0.9999117 0.9999560 0.9999118 0.9999553 0.9999105 0.9999711
[22] 0.9999421 0.9999564 0.9999126 0.9999559 0.9999118 0.9999531 0.9999062
[29] 0.9999563 0.9999125 0.9999522 0.9999043 0.9999532 0.9999064 1.0000130
[36] 1.0000261 0.9999555 0.9999108 0.9999817 0.9999635 0.9999562 0.9999123
[43] 0.9999562 0.9999123 0.9999547 0.9999093 0.9999675 0.9999349 0.9999852
[50] 0.9999704
$value
[1] 4.133372e-08
$gradient
[1] 3.443613e-05
$fn.reduction
[1] 22648.33
$iter
[1] 642
$feval
[1] 778
$convergence
[1] 0
$message
[1] "Successful convergence"
iter: 0 f-value: 22648.33 pgrad: 9061.209
iter: 10 f-value: 40.13237 pgrad: 29.92377
iter: 20 f-value: 29.65503 pgrad: 2.084342
iter: 30 f-value: 24.15897 pgrad: 2.028767
iter: 40 f-value: 22.05032 pgrad: 2.059338
iter: 50 f-value: 19.63944 pgrad: 2.044971
iter: 60 f-value: 17.54232 pgrad: 2.083767
iter: 70 f-value: 16.73025 pgrad: 2.004452
iter: 80 f-value: 14.23287 pgrad: 1.897868
iter: 90 f-value: 13.45627 pgrad: 1.792503
iter: 100 f-value: 10.14956 pgrad: 2.049154
iter: 110 f-value: 11.60104 pgrad: 25.25272
iter: 120 f-value: 5.646377 pgrad: 2.106924
iter: 130 f-value: 4.698865 pgrad: 1.920606
iter: 140 f-value: 3.825852 pgrad: 1.852113
iter: 150 f-value: 3.586715 pgrad: 1.232265
iter: 160 f-value: 1.74063 pgrad: 0.6242387
iter: 170 f-value: 1.686722 pgrad: 0.4283466
iter: 180 f-value: 1.481776 pgrad: 0.3898767
iter: 190 f-value: 1.280082 pgrad: 0.3417461
iter: 200 f-value: 0.6583132 pgrad: 0.2120633
iter: 210 f-value: 0.5228338 pgrad: 1.904417
iter: 220 f-value: 0.4583797 pgrad: 0.165238
iter: 230 f-value: 0.3408936 pgrad: 2.653384
iter: 240 f-value: 0.3160754 pgrad: 0.132524
iter: 250 f-value: 0.213777 pgrad: 0.132754
iter: 260 f-value: 0.1566734 pgrad: 1.005709
iter: 270 f-value: 0.1375318 pgrad: 0.1962938
iter: 280 f-value: 0.1276687 pgrad: 0.115445
iter: 290 f-value: 0.06870504 pgrad: 0.08940269
iter: 300 f-value: 0.05673515 pgrad: 0.08451464
iter: 310 f-value: 0.009130962 pgrad: 0.2581156
iter: 320 f-value: 0.005917028 pgrad: 0.03364445
iter: 330 f-value: 8.363714e-06 pgrad: 0.003680363
Successful convergence.
$par
[1] 0.9999700 0.9999400 0.9999703 0.9999405 0.9999703 0.9999406 0.9999703
[8] 0.9999406 0.9999704 0.9999408 0.9999703 0.9999406 0.9999703 0.9999405
[15] 0.9999703 0.9999406 0.9999703 0.9999406 0.9999703 0.9999407 0.9999699
[22] 0.9999398 0.9999703 0.9999406 0.9999703 0.9999406 0.9999704 0.9999408
[29] 0.9999703 0.9999406 0.9999704 0.9999408 0.9999704 0.9999408 0.9999687
[36] 0.9999374 0.9999703 0.9999406 0.9999697 0.9999394 0.9999703 0.9999406
[43] 0.9999703 0.9999406 0.9999704 0.9999407 0.9999700 0.9999400 0.9999697
[50] 0.9999393
$value
[1] 2.223557e-08
$gradient
[1] 1.429335e-06
$fn.reduction
[1] 22648.33
$iter
[1] 337
$feval
[1] 338
$convergence
[1] 0
$message
[1] "Successful convergence"
$cpar
method M
2 50
Parameter set : 1 ...
iter: 0 f-value: 104.1477 pgrad: 195.7775
iter: 10 f-value: 0.8165461 pgrad: 1.691339
iter: 20 f-value: 0.3660929 pgrad: 0.6701369
iter: 30 f-value: 0.1948056 pgrad: 3.295041
iter: 40 f-value: 0.03397907 pgrad: 0.2969536
iter: 50 f-value: 0.03081283 pgrad: 0.1332274
iter: 60 f-value: 0.006901353 pgrad: 0.05083775
iter: 70 f-value: 0.0005085973 pgrad: 0.02791898
iter: 80 f-value: 3.46663e-05 pgrad: 0.003696932
Successful convergence.
Parameter set : 2 ...
iter: 0 f-value: 1043.794 pgrad: 1446.276
iter: 10 f-value: 2.737732 pgrad: 1.673578
iter: 20 f-value: 1.707397 pgrad: 2.12588
iter: 30 f-value: 0.7376359 pgrad: 1.054303
iter: 40 f-value: 0.6052184 pgrad: 14.46225
iter: 50 f-value: 0.1062305 pgrad: 0.2269856
iter: 60 f-value: 0.129 pgrad: 10.58344
iter: 70 f-value: 3.347751e-06 pgrad: 0.00101306
Successful convergence.
Parameter set : 3 ...
iter: 0 f-value: 118.5023 pgrad: 226.2174
iter: 10 f-value: 1.544505 pgrad: 2.058662
iter: 20 f-value: 1.042624 pgrad: 1.982083
iter: 30 f-value: 0.5748645 pgrad: 6.803917
iter: 40 f-value: 0.2663864 pgrad: 0.5238955
iter: 50 f-value: 0.09414201 pgrad: 0.2876652
iter: 60 f-value: 0.01393715 pgrad: 0.1003782
iter: 70 f-value: 0.005687909 pgrad: 0.06244017
iter: 80 f-value: 0.01253125 pgrad: 4.223294
iter: 90 f-value: 8.144976e-06 pgrad: 0.002220145
Successful convergence.
Parameter set : 4 ...
iter: 0 f-value: 3.438234 pgrad: 21.56433
iter: 10 f-value: 1.160692 pgrad: 4.846107
iter: 20 f-value: 0.6238095 pgrad: 4.638383
iter: 30 f-value: 0.3261322 pgrad: 5.303616
iter: 40 f-value: 0.1370613 pgrad: 6.31439
iter: 50 f-value: 0.05995255 pgrad: 5.840455
iter: 60 f-value: 0.001563663 pgrad: 0.7089548
iter: 70 f-value: 0.0002575129 pgrad: 0.009598437
Successful convergence.
Parameter set : 5 ...
iter: 0 f-value: 1504.554 pgrad: 2749.706
iter: 10 f-value: 3.520653 pgrad: 1.61283
iter: 20 f-value: 2.877953 pgrad: 1.580511
iter: 30 f-value: 2.164019 pgrad: 1.611121
iter: 40 f-value: 4.529342 pgrad: 32.99485
iter: 50 f-value: 1.224356 pgrad: 2.11142
iter: 60 f-value: 0.8293356 pgrad: 1.754804
iter: 70 f-value: 0.4338205 pgrad: 1.227703
iter: 80 f-value: 0.3383105 pgrad: 0.6689532
iter: 90 f-value: 0.2322107 pgrad: 9.774733
iter: 100 f-value: 0.01032914 pgrad: 0.08242389
iter: 110 f-value: 0.01756212 pgrad: 4.194925
iter: 120 f-value: 0.003351996 pgrad: 0.04558846
iter: 130 f-value: 0.0005658241 pgrad: 0.0183096
iter: 140 f-value: 0.01210931 pgrad: 4.355415
Successful convergence.
Parameter set : 6 ...
iter: 0 f-value: 89.85157 pgrad: 181.4924
iter: 10 f-value: 4.993377 pgrad: 1.613387
iter: 20 f-value: 3.852569 pgrad: 1.620627
iter: 30 f-value: 0.6696012 pgrad: 1.338925
iter: 40 f-value: 0.3486178 pgrad: 0.6238029
iter: 50 f-value: 0.253394 pgrad: 0.471531
iter: 60 f-value: 0.1569684 pgrad: 0.3468646
iter: 70 f-value: 0.05717632 pgrad: 0.194028
iter: 80 f-value: 0.002223493 pgrad: 0.03174228
iter: 90 f-value: 0.01307329 pgrad: 3.935886
Successful convergence.
Parameter set : 7 ...
iter: 0 f-value: 2553.425 pgrad: 3161.945
iter: 10 f-value: 3.287716 pgrad: 2.100978
iter: 20 f-value: 2.323605 pgrad: 2.061737
iter: 30 f-value: 1.22322 pgrad: 2.440811
iter: 40 f-value: 0.7073892 pgrad: 0.8351592
iter: 50 f-value: 0.1609366 pgrad: 0.7610845
iter: 60 f-value: 0.007220043 pgrad: 2.257647
iter: 70 f-value: 5.743478e-07 pgrad: 0.0004048528
Successful convergence.
Parameter set : 8 ...
iter: 0 f-value: 7.301386 pgrad: 50.49826
iter: 10 f-value: 0.4876734 pgrad: 0.6070551
iter: 20 f-value: 0.2424819 pgrad: 3.883128
iter: 30 f-value: 0.05690639 pgrad: 0.1565022
iter: 40 f-value: 9.563579e-05 pgrad: 0.005619649
Successful convergence.
Parameter set : 9 ...
iter: 0 f-value: 97.45441 pgrad: 234.7565
iter: 10 f-value: 0.4877573 pgrad: 1.020726
iter: 20 f-value: 0.1865057 pgrad: 0.5415518
iter: 30 f-value: 0.1315637 pgrad: 0.3539517
iter: 40 f-value: 0.07136122 pgrad: 0.2486395
iter: 50 f-value: 0.0427114 pgrad: 0.1848895
iter: 60 f-value: 0.03879811 pgrad: 0.1765162
iter: 70 f-value: 0.01747809 pgrad: 0.2651659
iter: 80 f-value: 0.01392243 pgrad: 0.1011329
iter: 90 f-value: 0.01635814 pgrad: 2.888483
iter: 100 f-value: 0.003261127 pgrad: 0.04737148
iter: 110 f-value: 0.01097115 pgrad: 4.169896
Successful convergence.
Parameter set : 10 ...
iter: 0 f-value: 176.7632 pgrad: 334.1628
iter: 10 f-value: 1.817845 pgrad: 2.074538
iter: 20 f-value: 1.041073 pgrad: 1.298292
iter: 30 f-value: 0.3539918 pgrad: 0.7796873
iter: 40 f-value: 0.0745093 pgrad: 6.304562
iter: 50 f-value: 0.0003067417 pgrad: 0.009971132
Successful convergence.
Parameter set : 11 ...
iter: 0 f-value: 558.4307 pgrad: 1202.867
iter: 10 f-value: 10.22346 pgrad: 53.95344
iter: 20 f-value: 1.957312 pgrad: 2.139882
iter: 30 f-value: 1.132162 pgrad: 1.63763
iter: 40 f-value: 0.6903289 pgrad: 1.115213
iter: 50 f-value: 0.3360284 pgrad: 0.4902542
iter: 60 f-value: 0.06463214 pgrad: 0.1659278
iter: 70 f-value: 2.2056e-05 pgrad: 0.007548476
Successful convergence.
Parameter set : 12 ...
iter: 0 f-value: 936.7399 pgrad: 1381.4
iter: 10 f-value: 3.791033 pgrad: 10.22923
iter: 20 f-value: 2.481176 pgrad: 1.565749
iter: 30 f-value: 1.848268 pgrad: 1.817987
iter: 40 f-value: 1.539006 pgrad: 2.018057
iter: 50 f-value: 1.147663 pgrad: 2.088939
iter: 60 f-value: 0.8899576 pgrad: 1.848375
iter: 70 f-value: 0.3315418 pgrad: 4.199062
iter: 80 f-value: 0.2255326 pgrad: 1.4914
iter: 90 f-value: 0.2123317 pgrad: 0.4528963
iter: 100 f-value: 0.04973951 pgrad: 0.1968087
iter: 110 f-value: 0.0124454 pgrad: 0.09117443
iter: 120 f-value: 0.0001859101 pgrad: 0.01256614
Successful convergence.
Parameter set : 13 ...
iter: 0 f-value: 116.0656 pgrad: 467.5044
iter: 10 f-value: 1.945429 pgrad: 2.08734
iter: 20 f-value: 0.7791629 pgrad: 1.02835
iter: 30 f-value: 0.4993475 pgrad: 0.6020762
iter: 40 f-value: 0.1458184 pgrad: 0.3209832
iter: 50 f-value: 0.0305428 pgrad: 5.139955
iter: 60 f-value: 4.704206e-07 pgrad: 0.0003640162
Successful convergence.
Parameter set : 14 ...
iter: 0 f-value: 158.3783 pgrad: 310.6516
iter: 10 f-value: 1.233519 pgrad: 2.103569
iter: 20 f-value: 0.8147884 pgrad: 17.53057
iter: 30 f-value: 0.5689345 pgrad: 1.090678
iter: 40 f-value: 0.2180282 pgrad: 0.3935711
iter: 50 f-value: 0.1540431 pgrad: 4.261513
iter: 60 f-value: 0.1201162 pgrad: 0.2838095
iter: 70 f-value: 0.07007922 pgrad: 0.8969035
iter: 80 f-value: 0.06464419 pgrad: 0.1881796
iter: 90 f-value: 0.04395621 pgrad: 0.1466053
iter: 100 f-value: 0.03089696 pgrad: 0.457613
iter: 110 f-value: 0.02483754 pgrad: 0.102058
iter: 120 f-value: 0.01280933 pgrad: 0.3240663
iter: 130 f-value: 0.008748824 pgrad: 0.0543076
iter: 140 f-value: 0.0005616496 pgrad: 0.6241075
iter: 150 f-value: 4.384569e-05 pgrad: 0.004445839
iter: 160 f-value: 1.805938e-09 pgrad: 1.906689e-05
Successful convergence.
Parameter set : 15 ...
iter: 0 f-value: 102.838 pgrad: 190.781
iter: 10 f-value: 2.725574 pgrad: 2.147047
iter: 20 f-value: 1.684569 pgrad: 2.149739
iter: 30 f-value: 0.8545071 pgrad: 4.062114
iter: 40 f-value: 0.5186857 pgrad: 0.5901075
iter: 50 f-value: 0.08392794 pgrad: 0.370899
iter: 60 f-value: 0.005359188 pgrad: 2.108803
Successful convergence.
Parameter set : 16 ...
iter: 0 f-value: 513.5289 pgrad: 684.5052
iter: 10 f-value: 1.109876 pgrad: 1.229066
iter: 20 f-value: 0.2733764 pgrad: 0.3807037
iter: 30 f-value: 0.1347278 pgrad: 9.811595
iter: 40 f-value: 8.692025e-06 pgrad: 0.001645287
Successful convergence.
Parameter set : 17 ...
iter: 0 f-value: 5781.69 pgrad: 8216.68
iter: 10 f-value: 2958.348 pgrad: 2595.746
iter: 20 f-value: 2.205485 pgrad: 16.35224
iter: 30 f-value: 0.3015938 pgrad: 0.3812378
iter: 40 f-value: 0.004954546 pgrad: 0.04162067
iter: 50 f-value: 0.009318093 pgrad: 2.72599
Successful convergence.
Parameter set : 18 ...
iter: 0 f-value: 10769.93 pgrad: 11885.55
iter: 10 f-value: 1.086693 pgrad: 1.944855
iter: 20 f-value: 0.5642495 pgrad: 3.709624
iter: 30 f-value: 0.3503109 pgrad: 0.5451841
iter: 40 f-value: 0.1757639 pgrad: 0.3478917
iter: 50 f-value: 0.06314237 pgrad: 3.063494
iter: 60 f-value: 0.01987791 pgrad: 0.09774254
iter: 70 f-value: 1.594943e-06 pgrad: 0.003916945
Successful convergence.
Parameter set : 19 ...
iter: 0 f-value: 213.8751 pgrad: 509.3601
iter: 10 f-value: 1.233618 pgrad: 1.574912
iter: 20 f-value: 0.4427763 pgrad: 1.342841
iter: 30 f-value: 0.1468199 pgrad: 0.7130403
iter: 40 f-value: 0.009736555 pgrad: 0.05829691
iter: 50 f-value: 5.906892e-08 pgrad: 0.0003150453
Successful convergence.
Parameter set : 20 ...
iter: 0 f-value: 381.5038 pgrad: 717.5239
iter: 10 f-value: 1.041809 pgrad: 1.821658
iter: 20 f-value: 0.7302731 pgrad: 4.555018
iter: 30 f-value: 0.5153623 pgrad: 0.8002466
iter: 40 f-value: 0.1290155 pgrad: 0.4946614
iter: 50 f-value: 0.04967761 pgrad: 0.156249
iter: 60 f-value: 0.001800669 pgrad: 0.02640828
iter: 70 f-value: 0.001448272 pgrad: 1.101689
Successful convergence.
$par
[,1] [,2] [,3] [,4] [,5]
[1,] 0.9999871 0.9999743 0.9999736 0.9999471 -0.96211903
[2,] 0.9999692 0.9999383 0.9999692 0.9999383 0.47463463
[3,] 0.9999735 0.9999470 0.9999585 0.9999169 0.35175427
[4,] 0.9999541 0.9999080 0.9999541 0.9999082 -0.10702036
[5,] 0.9999700 0.9999398 0.9999699 0.9999398 0.53751726
[6,] 0.9999431 0.9998860 0.9999395 0.9998788 -0.60210490
[7,] 0.9999699 0.9999398 0.9999699 0.9999398 -2.02329765
[8,] 0.9999149 0.9998295 0.9999147 0.9998291 1.15442128
[9,] 0.9999686 0.9999371 0.9999700 0.9999400 -0.23069300
[10,] 0.9997891 0.9995774 0.9997890 0.9995773 -0.36508877
[11,] 0.9999589 0.9999176 0.9999588 0.9999176 0.15751793
[12,] 0.9996967 0.9993924 0.9998152 0.9996297 0.76593222
[13,] 0.9999699 0.9999398 0.9999699 0.9999398 0.42608697
[14,] 0.9999703 0.9999403 0.9999706 0.9999410 0.92726540
[15,] 0.9999019 0.9998035 0.9999019 0.9998036 0.04613459
[16,] 0.9999665 0.9999330 0.9999665 0.9999330 -0.50428564
[17,] 0.9999615 0.9999228 0.9999615 0.9999228 -0.30044941
[18,] 0.9999697 0.9999394 0.9999697 0.9999394 0.82291598
[19,] 0.9998285 0.9996564 0.9998284 0.9996563 -0.84926880
[20,] 0.9999693 0.9999385 0.9999692 0.9999384 -0.95361010
$fvalue
[1] 8.638806e-10 1.901873e-09 2.423288e-09 4.216728e-09 1.806844e-09
[6] 6.911578e-09 1.809587e-09 1.454471e-08 1.886645e-09 8.911840e-08
[11] 3.387040e-09 1.263133e-07 1.808496e-09 1.761743e-09 1.926060e-08
[16] 2.239977e-09 2.971137e-09 1.832471e-09 5.892285e-08 1.892542e-09
$converged
[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[16] TRUE TRUE TRUE TRUE TRUE
NULL
[,1] [,2] [,3] [,4] [,5]
[1,] 1 1.000 1 1 -0.962
[2,] 1 1.000 1 1 0.475
[3,] 1 1.000 1 1 0.352
[4,] 1 1.000 1 1 -0.107
[5,] 1 1.000 1 1 0.538
[6,] 1 1.000 1 1 -0.602
[7,] 1 1.000 1 1 -2.023
[8,] 1 1.000 1 1 1.154
[9,] 1 1.000 1 1 -0.231
[10,] 1 1.000 1 1 -0.365
[11,] 1 1.000 1 1 0.158
[12,] 1 0.999 1 1 0.766
[13,] 1 1.000 1 1 0.426
[14,] 1 1.000 1 1 0.927
[15,] 1 1.000 1 1 0.046
[16,] 1 1.000 1 1 -0.504
[17,] 1 1.000 1 1 -0.300
[18,] 1 1.000 1 1 0.823
[19,] 1 1.000 1 1 -0.849
[20,] 1 1.000 1 1 -0.954
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