使用约束进行优化，不同的包给出截然不同的结果

问题描述 投票：0回答：1

这是我在使用约束进行优化

中的原始帖子的后续内容

数据

Rand_Data = data.frame(x = c(  0.,   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.,  46.,  47.,  48.,  49.,  50.,  51.,  52.,  53.,  54.,
        55.,  56.,  57.,  58.,  59.,  60.,  61.,  62.,  63.,  64.,  65.,
        66.,  67.,  68.,  69.,  70.,  71.,  72.,  73.,  74.,  75.,  76.,
        77.,  78.,  79.,  80.,  81.,  82.,  83.,  84.,  85.,  86.,  87.,
        88.,  89.,  90.,  91.,  92.,  93.,  94.,  95.,  96.,  97.,  98.,
        99., 100., 101., 102., 103., 104., 105., 106., 107., 108., 109.,
       110., 111., 112., 113., 114., 115., 116., 117., 118., 119., 120.,
       121., 122., 123., 124., 125., 126., 127., 128., 129., 130., 131.,
       132., 133., 134., 135., 136., 137., 138., 139., 140., 141., 142.,
       143., 144., 145., 146., 147., 148., 149., 150., 151., 152., 153.,
       154., 155., 156., 157., 158., 159., 160., 161., 162., 163., 164.,
       165., 166., 167., 168., 169., 170., 171., 172., 173., 174., 175.,
       176., 177., 178., 179., 180., 181., 182., 183., 184., 185., 186.,
       187., 188., 189., 190., 191., 192., 193., 194., 195., 196., 197.,
       198., 199), Val = c( 0.00000000e+00,  7.75383362e-02, -1.34275505e+00, -1.65497205e+00,
       -1.18564171e+00,  1.77660878e+00,  1.84593088e+00,  1.71963694e+00,
        8.21964340e-01,  1.74041137e+00,  2.64637861e+00,  2.06965750e+00,
        1.02828336e+00, -5.00395684e-01, -6.93668766e-02,  4.22030196e-01,
        1.16198422e-01, -5.11722287e-04, -6.57944961e-01, -3.74434142e-01,
       -1.15827213e+00, -1.45830766e+00, -4.90458625e-01,  2.47042268e-02,
        6.93327973e-01,  1.30938478e+00, -4.06142808e-01, -4.61168289e-01,
       -6.72137769e-01, -4.84899922e-01, -8.29915365e-01, -2.41828942e+00,
       -1.20765939e+00,  5.76353406e-01,  2.61955788e+00,  2.98795810e+00,
        4.16904195e+00,  2.93858016e+00,  5.35527961e+00,  5.11840962e+00,
        5.51236637e+00,  5.88671081e+00,  5.38415506e+00,  6.32405264e+00,
        5.65838016e+00,  5.07960498e+00,  5.70407219e+00,  7.29978260e+00,
        6.08456290e+00,  7.43051089e+00,  8.55807191e+00,  8.52981505e+00,
        8.23329180e+00,  9.10450577e+00,  9.85288281e+00,  9.62989249e+00,
        8.69069599e+00,  8.34052238e+00,  7.19647733e+00,  7.24917949e+00,
        6.63422915e+00,  4.95911711e+00,  3.49654704e+00,  1.85980401e+00,
        3.08957065e+00,  4.24267262e+00,  2.38508285e+00,  1.02089838e+00,
        2.19670308e-01,  1.53858340e+00,  2.41653182e+00,  3.69861411e+00,
        2.87607870e+00,  3.21819643e+00,  2.07359072e+00,  1.13000323e+00,
        1.17047772e+00,  1.55913045e+00,  8.41514076e-01,  2.23618634e+00,
        2.99905073e+00,  3.21259672e+00,  4.39274819e+00,  1.99515000e+00,
        1.16175606e+00,  1.70758890e+00,  9.74160444e-01, -3.03140114e-02,
        1.17659213e+00,  3.07232323e-01,  1.08034381e+00,  1.29199568e+00,
        2.45607387e-01, -5.21021566e-01, -4.34639402e-01, -9.60127469e-01,
       -1.03258482e+00, -1.43325172e+00, -2.28217823e+00, -2.09923712e+00,
       -2.06051134e+00, -2.19767637e+00, -2.16617856e+00, -2.45811083e+00,
       -2.79865309e+00, -3.70529459e+00, -2.77461502e+00, -3.27216972e+00,
       -4.79844244e+00, -3.42557001e+00, -4.14314358e+00, -5.25738457e+00,
       -3.49827107e+00, -3.55643468e+00, -5.45626633e+00, -5.33106305e+00,
       -5.25332322e+00, -4.94845946e+00, -5.04795700e+00, -4.75058961e+00,
       -3.37815392e+00, -3.24099974e+00, -4.26774262e+00, -4.35293178e+00,
       -3.70409130e+00, -4.85062599e+00, -4.46039428e+00, -4.73719944e+00,
       -5.53392228e+00, -6.15049133e+00, -6.10977286e+00, -4.87411131e+00,
       -4.49530898e+00, -4.66802205e+00, -6.98105638e+00, -5.77006061e+00,
       -6.27152149e+00, -6.07877879e+00, -5.09850276e+00, -5.68772276e+00,
       -5.16446602e+00, -3.58115497e+00, -3.21209072e+00, -3.61179089e+00,
       -4.40163561e+00, -3.14328137e+00, -3.80444247e+00, -3.91338929e+00,
       -3.98395412e+00, -4.71086150e+00, -2.98308294e+00, -2.51608443e+00,
       -2.86848586e+00, -3.27653288e+00, -3.17684302e+00, -2.88103208e+00,
       -4.90737736e+00, -4.27541967e+00, -2.92691377e+00, -2.19142665e+00,
       -3.08185258e+00, -2.54393704e+00, -2.42485142e+00, -1.90176762e+00,
       -6.52372476e-01, -1.04169430e+00, -5.79717797e-01, -4.03696696e-01,
        5.92495881e-01, -2.14694942e+00, -1.42679299e+00, -9.81665751e-01,
       -3.51830293e+00, -2.24672044e+00, -2.23578918e+00, -9.44689326e-01,
       -1.55913248e+00, -6.17618042e-01, -9.56082794e-01, -1.38326771e+00,
       -1.55108670e+00, -5.74061887e-01,  1.89141995e-01,  8.11432327e-01,
        1.99574115e+00,  1.72686176e+00,  7.17859558e-01,  5.03518805e-01,
        1.91256417e+00,  9.59411484e-01,  5.28613488e-01,  1.17086649e+00,
        1.88075943e+00,  2.41999514e+00,  2.51115680e+00,  4.77976432e+00,
        6.14323607e+00,  6.42564314e+00,  4.82006996e+00,  4.63386067e+00))

优化问题

MyFN = function(a, b, c, d) return(a * Rand_Data$x^3 + b * Rand_Data$x^2  + c * Rand_Data$x + d)

现在我想估计参数

a, b, c, d
，以最小化 Rand_Data$Val 与具有以下约束的拟合值之间的绝对差之和

  1. MyFN
    第 100 个位置的拟合值应小于 2，即 
    MyFN[100] < 2
  2. 拟合值之和与观测值之和
    MyFN$Val
    之差应小于0.001

最优解

使用

pracma
包进行优化（由@ThomasIsCoding作为解决方案提供）

library(pracma)
x <- Rand_Data$x
y <- Rand_Data$Val
eps <- 1e-4
X <- outer(x, 0:3, `^`)


f <- function(theta) {
  sum(abs(X %*% theta - y))
}

A <- X[100, , drop = FALSE]
b <- 2 - eps
hin <- function(theta) {
  abs(sum(X %*% theta) - sum(y)) - 1e-3 + eps
}
fmincon(rep(0, 4), f, A = A, b = b, hin = hin)$par
#[1] -1.111242e+03  4.710255e+01 -5.408485e-01  1.804586e-03

使用

nloptr
包进行优化

library(nloptr)
Hx <- function(theta) {
    X[100, , drop = FALSE] %*% theta - (2 - eps)
}
print(nloptr(rep(0, 4), f, eval_g_ineq = hin, eval_g_eq = Hx, opts = list("algorithm"="NLOPT_LN_COBYLA")))
#-0.1000802 2.229485e-05 -3.691354e-05 2.53485e-06

可以看出，这两个例程给出了非常不同的最佳解决方案。

您能帮忙找出为什么最佳解决方案如此不同吗？我该如何解决这个问题？

场次信息

> sessionInfo()
R version 4.4.0 (2024-04-24)
Platform: aarch64-apple-darwin20
Running under: macOS 15.1.1

...

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] nloptr_2.1.1 pracma_2.4.4

loaded via a namespace (and not attached):
[1] MASS_7.3-60.2  compiler_4.4.0 quadprog_1.5-8 NlcOptim_0.6
r optimization
1个回答
0
投票

看起来您正在尝试拟合三次方程。 我建议首先使用更简单的最小二乘法 

lm()
和 
poly()
函数。

model <- lm(Val ~ poly(x, degree=3, raw=TRUE), data=Rand_Data)

out <- MyFN(model$coefficients[4], model$coefficients[3], model$coefficients[2], model$coefficients[1])
sum(abs(Rand_Data$Val-out))

plot(Rand_Data$x, Rand_Data$Val)
lines(0:199, predict(model, data.frame(x=0:199)), col="blue")

这应该接近你的第一个条件。 这不符合你的第二个条件，但从数据来看，我认为任何三次方程都不会通过该标准。

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