r - 为什么我的 NLOPT 优化错误/无法解决？

Question

我难住了。我在 R 中为 NLOPT 制定了一个问题。当前问题解决了 180 个具有 28 个等式约束的变量

该代码是从一个更简单的问题版本中重新使用的，在我的脚本的前面，有 36 个变量和 20 个等式约束，使用 NLOPT_LD_SLSQP 作为算法立即解决。

使用 NLOPT_LD_SLSQP 时，带有 180 个变量的问题的较大版本会立即产生以下结果:

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.) )

这让我感到困惑，因为它可以解决问题的较小版本。同样，它返回起始值并且实际上并不完成任何迭代。所以我实现了， NLOPT_AUGLAG_LD_EQ为主算法， NLOPT_LD_SLSQP为局部算法。现在问题无法解决并产生:

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

如果我降低容忍度，它只会更快地失败......我将玩具问题放入 Excel 中，看看我是否未能正确制定或者是否以某种方式不可行，但它立即解决了。如果你愿意，我可以给你这个文件。我采用 Excel 解决方案值并在 R 中填充函数，果然我的约束和目标函数似乎很好。

我想知道是否有人可以帮助我解决这个问题。下面是一些代码，它会为 R 中的任何人产生问题(经过测试和确认):

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)))   

现在这里是用 SLSQP 解决的代码:

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

这是用 AUGLAG 解决的代码:

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

最后但并非最不重要的是，这是 Excel 生成的一个可行的解决方案:

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.

我的问题是:为什么这个模型会为示例代码中实现的每个 SLSQP 和 AUGLAG 算法产生我在此处报告的错误？

会喜欢这里的一些输入。如果您需要任何其他信息，请告诉我!

Answer 1

关于您的系统的一个有趣的事情是 conmat等级不足:

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

但是系统 conmat%*%x = f.rhs实际上是一致的，您可以使用此 post 中的答案找到其解决方案之一。 .

这意味着您的约束系统有 4 个冗余关系。有时这样的问题会导致数值不稳定，所以让我们看看如果我们删除冗余方程会发生什么。为此，我们可以修改此 post 中的答案。 :

# 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]  

现在，我们可以运行 NLOPT_LD_SLSQP求解器。经过 1350 次迭代，我得到了一个解决方案。让我们看看它与 XL_Sol 相比如何.

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

所以我们已经针对目标进行了改进。让我们使用原始约束检查解决方案:

> 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