Warum kann mein NLOPT-Optimierungsfehler nicht gelöst werden?

Question

Ich bin ratlos. Ich habe ein Problem für NLOPT in R formuliert. Das aktuelle Problem löst für 180 Variablen mit 28 Gleichheitsbedingungen

Der Code wird von einer einfacheren Version des Problems, früher in meinem Skript, mit 36 ​​Variablen und 20 Gleichheit wiederverwendet Constraints, die sofort mit NLOPT_LD_SLSQP als Algorithmus löst.

Die größere Version des Problems mit 180 Variablen erzeugt die folgende, sofort, wenn NLOPT_LD_SLSQP mit:

NLopt solver status: -4 (NLOPT_ROUNDOFF_LIMITED: Roundoff errors led 
to a breakdown of the optimization algorithm. In this case, the 
returned minimum may 
still be useful. (e.g. this error occurs in NEWUOA if one tries to 
achieve a tolerance too close to machine precision.)) 

Das verwirrte mich, da es auf der kleineren Version des Problems gearbeitet. Außerdem gibt es die Startwerte zurück und führt keine Iterationen durch. Also implementierte ich NLOPT_AUGLAG_LD_EQ als Hauptalgorithmus und NLOPT_LD_SLSQP als lokalen Algorithmus. Nun schlägt das Problem zu lösen und produziert diese:

NLopt solver status: -1 (NLOPT_FAILURE: Generic failure code.) 

Wenn ich die Toleranz reduzieren sie nicht nur schneller ... ich das Spielzeug Problem in Excel nahm, um zu sehen, ob ich richtig zu formulieren, nicht gelungen war oder ob es irgendwie undurchführbar, aber es hat sofort gelöst. Ich kann dir diese Datei geben, wenn du willst. Ich nahm die Werte der Excel-Lösung und füllte die Funktionen in R, und tatsächlich scheinen meine Einschränkungen und Zielfunktion gut zu sein.

Ich frage mich, ob mir jemand mit diesem Problem helfen kann. Hier ist ein Code, der das Problem (getestet und bestätigt) für jedermann in R erzeugt:

library(pracma) 
library(nloptr) 

#My constraint function uses the following: 
#RHS of the equality constraints 
f.rhs <- c(590.0000,4781.0000,4414.0000,120.0000,224.0000, 
849.0000,4693.0000,4374.0000,85.0000,697.0000,0.0000, 
0.0000,0.0000,1092.0000,1434.0000,2251.0133,3482.9867, 
2316.1813,1873.8187,1622.1450,1206.8550,1240.0000,1233.0000, 
933.8532,733.1468,486.7907,395.2093,526.0000) 

#matrix of constraints 
#the first 10 rows are row total constraints 
#the remaining 18 constraints are column total constraints 
#this is a 28x180 matrix. It's sort of big to have it in this 
#code window, but this code should produce the matrix for you 
conmat <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.101385354477184,0,0,0,0,0,0,0,0,0,0,0,0,0.101385354477184,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.879602416076347,0,0,0,0,0,0,0,0,0,0,0,0,0,0.879602416076347,0,0,0,0,0,0,0,0,0,0,0,0,0,9.72187831641552,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9.72187831641552,0,0,0,0,0,0,0,0,0,0,0,0,50.825662951981,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,50.825662951981,0,0,0,0,0,0,0,0,0,0,0,61.4161196898944,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,61.4161196898944,0,0,0,0,0,0,0,0,0,0,76.5722969856399,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,76.5722969856399,0,0,0,0,0,0,0,0,0,85.0874667314792,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,85.0874667314792,0,0,0,0,0,0,0,0,70.1228611430807,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,70.1228611430807,0,0,0,0,0,0,0,72.2969630445657,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72.2969630445657,0,0,0,0,0,0,70.9315070452785,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,70.9315070452785,0,0,0,0,0,54.6520210670868,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,54.6520210670868,0,0,0,0,44.0086494626126,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,44.0086494626126,0,0,0,20.019587567467,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20.019587567467,0,0,14.1724093345295,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14.1724093345295,0,10.4922206705268,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10.4922206705268,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3.56379085643383,0,0,0,0,0,0,0,0,0,0,0,3.56379085643383,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36.3843642437252,0,0,0,0,0,0,0,0,0,0,0,0,36.3843642437252,0,0,0,0,0,0,0,0,0,0,0,0,0,0,208.581934690648,0,0,0,0,0,0,0,0,0,0,0,0,0,208.581934690648,0,0,0,0,0,0,0,0,0,0,0,0,0,649.993541449925,0,0,0,0,0,0,0,0,0,0,0,0,0,0,649.993541449925,0,0,0,0,0,0,0,0,0,0,0,0,620.425879840303,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,620.425879840303,0,0,0,0,0,0,0,0,0,0,0,532.113517313307,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,532.113517313307,0,0,0,0,0,0,0,0,0,0,487.289086271457,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,487.289086271457,0,0,0,0,0,0,0,0,0,355.571461649492,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,355.571461649492,0,0,0,0,0,0,0,0,370.187737611463,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,370.187737611463,0,0,0,0,0,0,0,377.604110342457,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,377.604110342457,0,0,0,0,0,0,286.391885309974,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,286.391885309974,0,0,0,0,0,230.617639447808,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,230.617639447808,0,0,0,0,150.901768695456,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,150.901768695456,0,0,0,106.827457261503,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,106.827457261503,0,0,102.516716725934,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 
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    0,0,0,0,0,0,103.53378703525,0,0,0,0,0,0,0,0,0,0,216.919314868552,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,216.919314868552) 

#Create the matrix from my list of values 
conmat <- matrix(conmat,nrow=28,ncol=180) 

#Create the constraint function so that it produces 0s 
#for the equality constraints 

eqn <- function(x){ 
z=c() 
for (i in 1:length(f.rhs)){ 
    z[i]=x%*%conmat[i,]-f.rhs[i] 
} 
return(z) 
} 

#Function for the Jacobian of the constraint function 
eqn_grad <- function(x){ 
jacobian(eqn,x) 
} 

#Create my objective function: sum of squared error 
#Data for the function is in f.obj 
f.obj<- c(0,0,0,0.101385354477184,0.879602416076347,9.72187831641552,50.825662951981,61.4161196898944,76.5722969856399,85.0874667314792,70.1228611430807,72.2969630445657,70.9315070452785,54.6520210670868,44.0086494626126,20.019587567467,14.1724093345295,10.4922206705268,0,0,0,3.56379085643383,36.3843642437252,208.581934690648,649.993541449925,620.425879840303,532.113517313307,487.289086271457,355.571461649492,370.187737611463,377.604110342457,286.391885309974,230.617639447808,150.901768695456,106.827457261503,102.516716725934,0,0,0,478.206023100627,656.900445375174,835.409246572897,913.757767618374,483.249080637494,317.775206732195,237.647569920133,144.341667617821,124.724396017039,98.7489023828154,55.1727766234815,44.4279889177619,25.9420568421214,18.3650860591977,18.8065580202466,0,0,0,0.625528507812058,1.14682340323878,8.19022941749738,25.5271393599384,30.1338658434231,33.325484734614,34.6159582253122,25.9425263509155,28.6721543150432,24.6001930717219,19.4063381820094,15.6269927027328,7.86354283325046,5.56681537403581,4.09245697782775,0,0,0,0.0478232804137659,0.4097388469824,1.18088960908956,2.13050439017778,2.88537630873373,4.47104432861129,6.7066648766379,7.21292740479352,13.2288618280058,17.7096180695658,32.1794159026695,25.9125391288607,39.9666473126856,28.293474255415,71.410473393917,0,0,0,0.182608374202251,2.83538237187642,19.2609656330223,80.7233037904926,85.6477154102395,99.9589781355406,111.246861310206,95.4680594823716,110.647972395614,121.020177240488,82.2865998682753,63.071831445587,28.5766379506572,26.2025287609293,24.3533348857096,0,0,0,6.77378361061052,66.8658893692034,301.030835688858,790.527770127371,571.353574962507,465.451417348464,385.48085698754,298.799798756715,305.715022425598,296.891932828816,189.45276364351,145.213592426377,62.942694262135,57.7134986817454,35.5644312539887,0,0,0,600.919741246423,664.169162623053,847.398221128337,903.249824396504,402.366866953328,280.141092267195,201.892549678482,139.225117820461,123.454735574923,113.363999137925,72.6983780508768,55.7225581581023,31.2411980593257,28.6457207488449,38.4996441218375,0,0,0,1.10058560667843,3.80905898924575,18.1504505832434,61.5743643896698,52.0421527000771,53.6771028451664,54.7413143789797,46.2299586575942,43.5728165091606,36.8659311062071,21.583566050924,16.5435811193775,6.42202062218569,5.88848766417714,3.34484908130525,0,0,0,0.478730062097792,0.599532361620929,2.08861096083086,4.67685892604545,6.66062843004317,10.3325985324682,17.4366302473174,23.9406624891688,47.4993402782371,75.2636287745429,120.029503824241,92.0013786669861,112.914580678886,103.53378703525,216.919314868552) 

#Objective Function is SSE 
fn <- function(x){ 
sum((f.obj*x-f.obj)^2) 
} 

#Objective Gradient 
fn_grad <- function(x){ 
grad(fn,x) 
} 

#Optimization options: 
#Starting Values 
x0 <- c(matrix(1,1,ncol=length(f.obj))) 
#Lower Bound 
lb_x <- c(matrix(0,1,ncol=length(f.obj))) 
#Upper Bound 
ub_x <- c(matrix(2,1,ncol=length(f.obj))) 

Hier ist der Code mit SLSQP zu lösen:

opts_list <- list("algorithm"="NLOPT_LD_SLSQP", "xtol_rel"=1.0e-8,"maxeval"=1000) 

SOLUTION <- nloptr(x0, 
       eval_f=fn, 
       eval_grad_f=fn_grad, 
       lb=lb_x, 
       ub=ub_x, 
       eval_g_eq = eqn, 
       eval_jac_g_eq = eqn_grad, 
       opts=opts_list) 

SOLUTION 

Und hier ist der Code zu lösen mit AUGLAG:

local_opts_list <- list("algorithm" = "NLOPT_LD_SLSQP","xtol_rel"=1.0e-8) 
opts_list <- list("algorithm"="NLOPT_LD_AUGLAG_EQ", "xtol_rel"=1.0e-8,"maxeval"=1000, "local_opts"=local_opts_list) 

SOLUTION <- nloptr(x0, 
       eval_f=fn, 
       eval_grad_f=fn_grad, 
       lb=lb_x, 
       ub=ub_x, 
       eval_g_eq = eqn, 
       eval_jac_g_eq = eqn_grad, 
       opts=opts_list) 

SOLUTION 

Last, but not least, hier ist eine realisierbare Lösung von Excel erzeugt:

XL_Sol <- c(1,1,1,0.999840758,0.998619343,0.984736297,0.917305327,0.895766204,0.902054225,0.930449654,0.908095562,0.90555472,0.907109309,0.92775884,0.941019433,0.973119106,0.980882453,1.00818501,1,1,1,0.999277369,0.99265595,0.957137798,1.068940265,0.971466259,0.916751383,1.031659407,1.088943875,1.106986398,1.09298138,1.169120549,1.136193658,1.242584994,1.260102262,1.218743178,1,1,1,1.036812057,1.051986714,1.066969218,0.764543983,0.983061755,1.117743946,1.086522598,1.048114102,1.042021839,1.032920906,1.017781177,1.013537363,1.007869529,1.005459785,1.04562595,1,1,1,0.986367866,0.975008481,0.821522583,0.442429864,0.339768188,0.28380776,0.279905133,0.44151851,0.382930646,0.470415572,0.581960382,0.663070622,0.830426413,0.879918273,0.920434632,1,1,1,0.99976331,0.997972512,0.994156123,0.989331379,0.985345491,0.979153535,0.96871128,0.966148601,0.937972897,0.916916787,0.848764035,0.877734945,0.811485291,0.866305968,0.895221739,1,1,1,0.999728838,0.995792558,0.971414471,0.875797499,0.862210736,0.88101002,0.867466136,0.883358551,0.865529128,0.894845599,0.898520124,0.920974341,0.964087977,0.966910775,1.02107896,1,1,1,1.000481883,1.004812176,0.752446473,1.516261232,1.234618416,1.169698414,1.138623218,1.098571557,1.102118823,1.098211297,1.060360506,1.043658294,1.01881129,1.016899233,1.086075848,1,1,1,0.970757235,0.948975599,1.042336122,0.836515328,0.953263454,1.082324289,1.058181611,1.036019128,1.032395622,1.02937951,1.018031769,1.012837354,1.007153339,1.006385936,1.090482373,1,1,1,0.979339786,0.92850077,0.659360317,0,0.01800471,0.038870008,0,0.145186371,0.194368327,0.318022546,0.600269991,0.693283542,0.880910409,0.890768254,0.94507392,1,1,1,0.998868123,0.998583119,0.995062994,0.988669793,0.983388904,0.978533683,0.963729915,0.94954262,0.90018972,0.84196737,0.764585537,0.80459333,0.760693136,0.77966356,0.896573068) 
#eqn(XL_Sol) should give you a vector of numbers that are pretty close to zero. 

Meine Frage ist: Warum erzeugt dieses Modell die Fehler, die ich hier für jeden der SLSQP gemeldet habe, und AUGLAG Algorithmen im Beispielcode implementiert?

Würde ich einige Eingabe hier lieben. Bitte lassen Sie mich wissen, wenn Sie zusätzliche Informationen benötigen!

Answer 1

Eine interessante Sache über Ihr System ist, dass conmat ist Rang mangelhaft:

qr(conmat)$rank # 24 < 28 (number of rows in the system) 

Aber das System conmat%*%x = f.rhs tatsächlich konsistent ist und Sie können eine ihrer Lösungen, die Antworten von diesen post mit finden.

Das bedeutet, dass Ihr Einschränkungssystem 4 redundante Beziehungen hat. Manchmal können solche Probleme zu numerischer Instabilität führen. Sehen wir uns an, was passiert, wenn wir die redundanten Gleichungen entfernen. Um dies zu tun, können wir die Antwort in diesen post anpassen:

# let's relabel 
conmat_raw = conmat 
f.rhs_raw = f.rhs 

# append rhs as a column to conmat 
augmented_matrix <- cbind(conmat, f.rhs) 

# qr transform of the transpose 
q <- qr(t(augmented_matrix)) 

# extract a set of linearly dependent rows for the original augmented matrix 
reduced <- t(t(augmented_matrix)[,q$pivot[seq(q$rank)]]) 

# decompose augmented matrix 
conmat = reduced[,1:180] 
f.rhs = reduced[,181] 

Jetzt können wir die NLOPT_LD_SLSQP Solver laufen. Nach 1350 Iterationen bekomme ich eine Lösung. Mal sehen, wie es mit der XL_Sol vergleicht.

> sol = SOLUTION$solution 
> fn(sol) 
[1] 58404.02 
> fn(XL_Sol) 
[1] 317837 

So haben wir gegen das Ziel verbessert. Lassen Sie uns die Lösungen überprüfen Sie die ursprünglichen Zwänge mit:

> sum((conmat_raw%*%sol - f.rhs_raw)^2) 
[1] 6.071796e-08 
> sum((conmat_raw%*%XL_Sol - f.rhs_raw)^2) 
[1] 2.771957e-08