为什么非线性规划（NLP）解算器Rsolnp中的目标函数不符合？

Question

案例：

我有3个地区（巴西，新西兰，美国）的宇宙（我的实际问题是更大 - 31个区）。这三个地区通过迁移相连。例如，如果有10人从巴西转移到美国（BRZ-USA），我们就有移民到美国（人流入）和从巴西移民（人流出）。我有一个关于给定宇宙中所有可能的迁移流的迁移率数据集（3 * 2 = 6）。另外，我还有每个地区的人口数据集。当我将迁移率乘以人口时，我获得了移民数量。然后我可以计算每个地区的移民人数和移民人数。从移民中扣除移民导致净移民数量（可以是正数或负数）。然而，由于我们有一个平衡的制度（每个地区的流入流量相等），所有地区的净流动人口总和应为零。除了净移民率和人口外，我还会从每个地区假设的未来情景中获得净移民数量。但情景净移民数量与我可以从我的数据计算的数量不同。因此，我想通过增加或减少一个固定数字来扩大和缩小6个迁移率，以使得到的净迁移数量符合方案值。我使用非线性规划（NLP）解算器Rsolnp来完成此任务（请参阅下面的示例脚本）。

问题：

我已指定在最小二乘方程的形式的目标函数，因为它是迫使6个定标器以尽可能接近零越好目标。另外，我正在使用等式约束函数来满足场景值。这一切都正常工作，解决方案提供了可以添加到迁移率的标量，从而导致迁移计数与场景值完美匹配（请参阅脚本部分“测试目标是否已达到”）。但是，我还想将权重（变量：w）应用于目标函数，以便某些标量的较高值受到较大的惩罚。但是，无论我如何指定权重，我总是会获得相同的解决方案（请参阅“不同权重的示例结果”）。所以看起来求解器不遵守目标函数。有没有人有一个想法，为什么是这种情况，以及如何改变目标函数，以便使用权重是可能的？非常感谢您的帮助！

library(Rsolnp) 

# Regions 
regUAll=c("BRZ","NZL","USA") # "BRZ"=Brazil; "NZL"=New Zealand; "USA"=United States 

#* Generate unique combinations of regions 
uCombi=expand.grid(regUAll,regUAll,stringsAsFactors=F) 
uCombi=uCombi[uCombi$Var1!=uCombi$Var2,] # remove same region combination (e.g., BRZ-BRZ) 
uCombi=paste(uCombi$Var2,uCombi$Var1,sep="-") 

#* Generate data frames 
# Migration rates - rows represent major age groups (row1=0-25 years, row2=26-50 years, row3=51-75 years) 
dfnm=data.frame(matrix(rep(c(0.01,0.04,0.02),length(uCombi)),ncol=length(uCombi),nrow=3),stringsAsFactors=F) # generate empty df 
names(dfnm)=uCombi # assign variable names 

# Population (number of people) in region of origin 
pop=c(rep(c(20,40,10),2),rep(c(4,7,2),2),rep(c(30,70,50),2)) 
dfpop=data.frame(matrix(pop,ncol=length(uCombi),nrow=3),stringsAsFactors=F) # generate empty df 
names(dfpop)=uCombi # assign variable names 

#* Objective function for optimization 
# Note: Least squares method to keep the additive scalers as close to 0 as possible 
#  The sum expression allows for flexible numbers of scalars to be included but is identical to: w[1](scal[1]-0)^2+w[2](scal[2]-0)^2+w[3](scal[3]-0)^2+w[4](scal[4]-0)^2+w[5](scal[5]-0)^2+w[6](scal[6]-0)^2 
f.main=function(scal,nScal,w,dfnm,dfpop,regUAll){ 
    sum(w*(scal[1:nScal]-0)^2) 
} 

#* Equality contraint function 
f.equal=function(scal,nScal,w,dfnm,dfpop,regUAll){ 

    #* Adjust net migration rates by scalar 
    for(s in 1:nScal){ 
    dfnm[,s]=dfnm[,s]+scal[s] 
    } 

    #* Compute migration population from data 
    nmp=sapply(dfpop*dfnm,sum) # sums migration population across age groups 

    nmd=numeric(length(regUAll)); names(nmd)=regUAll      # generate named vector to be filled with values 
    for(i in 1:length(regUAll)){ 
    colnEm=names(nmp)[grep(paste0("^",regUAll[i],"-.*"),names(nmp))]  # emigration columns 
    colnIm=names(nmp)[grep(paste0("^.*","-",regUAll[i],"$"),names(nmp))] # immigration columns 
    nmd[regUAll[i]]=sum(nmp[colnIm])-sum(nmp[colnEm])      # compute net migration population = immigration - emigration 
    } 
    nmd=nmd[1:(length(nmd)-1)] # remove the last equality constraint value - not needed because we have a closed system in which global net migration=0 

    return(nmd) 
} 

#* Set optimization parameters 
cpar2=list(delta=1,tol=1,outer.iter=10,trace=1) # optimizer settings 
nScal=ncol(dfnm)     # number of scalars to be used 
initScal=rep(0,nScal)    # initial values of additive scalars 
lowScal=rep(-1,nScal)    # lower bounds on scalars 
highScal=rep(1,nScal)    # upper bounds on scalars 
nms=c(-50,10)      # target values: BRZ=-50, NZL=10, USA=40; last target value does not need to be included since we deal with a closed system in which global net migration sums to 0 
w=c(1,1,1,1,1,1) # unity weights 
#w=c(1,1,2,2,1,1) # double weight on NZL 
#w=c(5,1,2,7,1,0.5) # mixed weights 


#* Perform optimization using solnp 
solRes=solnp(initScal,fun=f.main,eqfun=f.equal,eqB=nms,LB=lowScal,UB=highScal,control=cpar2, 
      nScal=nScal,w=w,dfnm=dfnm,dfpop=dfpop,regUAll=regUAll) 

scalSol=solRes$pars # return optimized values of scalars 

# Example results for different weights 
#[1] 0.101645349 0.110108019 -0.018876993 0.001571639 -0.235945755 -0.018134294 # w=c(1,1,1,1,1,1) 
#[1] 0.101645349 0.110108019 -0.018876993 0.001571639 -0.235945755 -0.018134294 # w=c(1,1,2,2,1,1) 
#[1] 0.101645349 0.110108019 -0.018876993 0.001571639 -0.235945755 -0.018134294 # w=c(5,1,2,7,1,0.5) 

#*** Test if target was reached 
# Adjust net migration rates using the optimized scalars 
for(s in 1:nScal){ 
    dfnm[,s]=dfnm[,s]+scalSol[s] 
} 

# Compute new migration population 
nmp=sapply(dfpop*dfnm,sum) # sums migration population across age groups 

nmd=numeric(length(regUAll)); names(nmd)=regUAll      # generate named vector to be filled with values 
for(i in 1:length(regUAll)){ 
    colnEm=names(nmp)[grep(paste0("^",regUAll[i],"-.*"),names(nmp))]  # emigration columns 
    colnIm=names(nmp)[grep(paste0("^.*","-",regUAll[i],"$"),names(nmp))] # immigration columns 
    nmd[regUAll[i]]=sum(nmp[colnIm])-sum(nmp[colnEm])      # compute net migration population = immigration - emigration 
} 

nmd # should be -50,10,40 if scalars work correctly 

Answer 1

尝试不同的起始值（initScal）大于零;所有w的sum(w * 0^2) = 0。