使用约束进行优化

问题描述 投票：0回答：1

我有以下数据

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))

并且，下面是我的函数（为了简单起见，我只考虑三次样条拟合）

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

我想知道如何设置 

optim()
功能以便我可以估计 
a,b,c,d

这里是我的数据和函数，只是举例，所以我正在寻找一些通用方法来最佳估计参数。

r optimization
1个回答
0
投票

您正在解决约束优化问题，其中存在非线性约束。我想你可以使用 

fmincon
包中的 
pracma
，例如，

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

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

# define constraints
A <- X[100, , drop = FALSE]
b <- 2 - eps
hin <- function(theta) {
  abs(sum(X %*% theta) - sum(y)) - 1e-3 + eps
}

# use fmincon
p <- fmincon(rep(0, 4), f, A = A, b = b, hin = hin)
sol <- p$par
res <- transform(
  Rand_Data,
  Val_est = X %*% sol
)

# verify the solution
with(
  res,
  abs(sum(Val) - sum(Val_est)) < 1e-3 & Val_est[100] < 2
)

您将获得解决方案

(d,c,b,a)
，如下所示

> sol
[1] -1.281027e+03  5.661665e+01 -6.601774e-01  2.214139e-03

和估计的

Val
，即
Val_est
看起来像

> res
      x           Val      Val_est
1     0  0.0000000000 -1281.027073
2     1  0.0775383362 -1225.068384
3     2 -1.3427550500 -1170.416765
4     3 -1.6549720500 -1117.058931
5     4 -1.1856417100 -1064.981597
6     5  1.7766087800 -1014.171479
7     6  1.8459308800  -964.615291
8     7  1.7196369400  -916.299749
9     8  0.8219643400  -869.211568
10    9  1.7404113700  -823.337462
11   10  2.6463786100  -778.664148
12   11  2.0696575000  -735.178341
13   12  1.0282833600  -692.866755
14   13 -0.5003956840  -651.716105
15   14 -0.0693668766  -611.713108
16   15  0.4220301960  -572.844477
17   16  0.1161984220  -535.096929
18   17 -0.0005117223  -498.457178
19   18 -0.6579449610  -462.911940
20   19 -0.3744341420  -428.447929
21   20 -1.1582721300  -395.051861
22   21 -1.4583076600  -362.710452
23   22 -0.4904586250  -331.410415
24   23  0.0247042268  -301.138467
25   24  0.6933279730  -271.881322
26   25  1.3093847800  -243.625697
27   26 -0.4061428080  -216.358305
28   27 -0.4611682890  -190.065862
29   28 -0.6721377690  -164.735083
30   29 -0.4848999220  -140.352684
31   30 -0.8299153650  -116.905379
32   31 -2.4182894200   -94.379884
33   32 -1.2076593900   -72.762914
34   33  0.5763534060   -52.041184
35   34  2.6195578800   -32.201409
36   35  2.9879581000   -13.230305
37   36  4.1690419500     4.885414
38   37  2.9385801600    22.159033
39   38  5.3552796100    38.603835
40   39  5.1184096200    54.233106
41   40  5.5123663700    69.060131
42   41  5.8867108100    83.098195
43   42  5.3841550600    96.360582
44   43  6.3240526400   108.860577
45   44  5.6583801600   120.611465
46   45  5.0796049800   131.626531
47   46  5.7040721900   141.919061
48   47  7.2997826000   151.502337
49   48  6.0845629000   160.389646
50   49  7.4305108900   168.594273
51   50  8.5580719100   176.129501
52   51  8.5298150500   183.008616
53   52  8.2332918000   189.244904
54   53  9.1045057700   194.851647
55   54  9.8528828100   199.842133
56   55  9.6298924900   204.229644
57   56  8.6906959900   208.027466
58   57  8.3405223800   211.248885
59   58  7.1964773300   213.907184
60   59  7.2491794900   216.015649
61   60  6.6342291500   217.587564
62   61  4.9591171100   218.636214
63   62  3.4965470400   219.174885
64   63  1.8598040100   219.216860
65   64  3.0895706500   218.775426
66   65  4.2426726200   217.863865
67   66  2.3850828500   216.495465
68   67  1.0208983800   214.683508
69   68  0.2196703080   212.441281
70   69  1.5385834000   209.782067
71   70  2.4165318200   206.719152
72   71  3.6986141100   203.265821
73   72  2.8760787000   199.435358
74   73  3.2181964300   195.241049
75   74  2.0735907200   190.696177
76   75  1.1300032300   185.814029
77   76  1.1704777200   180.607888
78   77  1.5591304500   175.091040
79   78  0.8415140760   169.276769
80   79  2.2361863400   163.178360
81   80  2.9990507300   156.809098
82   81  3.2125967200   150.182269
83   82  4.3927481900   143.311156
84   83  1.9951500000   136.209045
85   84  1.1617560600   128.889220
86   85  1.7075889000   121.364966
87   86  0.9741604440   113.649568
88   87 -0.0303140114   105.756312
89   88  1.1765921300    97.698481
90   89  0.3072323230    89.489360
91   90  1.0803438100    81.142235
92   91  1.2919956800    72.670390
93   92  0.2456073870    64.087111
94   93 -0.5210215660    55.405681
95   94 -0.4346394020    46.639386
96   95 -0.9601274690    37.801510
97   96 -1.0325848200    28.905339
98   97 -1.4332517200    19.964157
99   98 -2.2821782300    10.991249
100  99 -2.0992371200     1.999900
101 100 -2.0605113400    -6.996605
102 101 -2.1976763700   -15.984982
103 102 -2.1661785600   -24.951945
104 103 -2.4581108300   -33.884210
105 104 -2.7986530900   -42.768492
106 105 -3.7052945900   -51.591507
107 106 -2.7746150200   -60.339968
108 107 -3.2721697200   -69.000592
109 108 -4.7984424400   -77.560094
110 109 -3.4255700100   -86.005188
111 110 -4.1431435800   -94.322590
112 111 -5.2573845700  -102.499015
113 112 -3.4982710700  -110.521179
114 113 -3.5564346800  -118.375796
115 114 -5.4562663300  -126.049582
116 115 -5.3310630500  -133.529251
117 116 -5.2533232200  -140.801520
118 117 -4.9484594600  -147.853103
119 118 -5.0479570000  -154.670714
120 119 -4.7505896100  -161.241071
121 120 -3.3781539200  -167.550887
122 121 -3.2409997400  -173.586878
123 122 -4.2677426200  -179.335758
124 123 -4.3529317800  -184.784244
125 124 -3.7040913000  -189.919050
126 125 -4.8506259900  -194.726892
127 126 -4.4603942800  -199.194484
128 127 -4.7371994400  -203.308542
129 128 -5.5339222800  -207.055781
130 129 -6.1504913300  -210.422916
131 130 -6.1097728600  -213.396663
132 131 -4.8741113100  -215.963736
133 132 -4.4953089800  -218.110850
134 133 -4.6680220500  -219.824722
135 134 -6.9810563800  -221.092065
136 135 -5.7700606100  -221.899595
137 136 -6.2715214900  -222.234028
138 137 -6.0787787900  -222.082078
139 138 -5.0985027600  -221.430461
140 139 -5.6877227600  -220.265892
141 140 -5.1644660200  -218.575086
142 141 -3.5811549700  -216.344757
143 142 -3.2120907200  -213.561623
144 143 -3.6117908900  -210.212396
145 144 -4.4016356100  -206.283793
146 145 -3.1432813700  -201.762529
147 146 -3.8044424700  -196.635319
148 147 -3.9133892900  -190.888879
149 148 -3.9839541200  -184.509922
150 149 -4.7108615000  -177.485165
151 150 -2.9830829400  -169.801323
152 151 -2.5160844300  -161.445110
153 152 -2.8684858600  -152.403242
154 153 -3.2765328800  -142.662435
155 154 -3.1768430200  -132.209403
156 155 -2.8810320800  -121.030861
157 156 -4.9073773600  -109.113525
158 157 -4.2754196700   -96.444110
159 158 -2.9269137700   -83.009331
160 159 -2.1914266500   -68.795903
161 160 -3.0818525800   -53.790541
162 161 -2.5439370400   -37.979961
163 162 -2.4248514200   -21.350878
164 163 -1.9017676200    -3.890006
165 164 -0.6523724760    14.415939
166 165 -1.0416943000    33.580241
167 166 -0.5797177970    53.616187
168 167 -0.4036966960    74.537059
169 168  0.5924958810    96.356145
170 169 -2.1469494200   119.086727
171 170 -1.4267929900   142.742091
172 171 -0.9816657510   167.335522
173 172 -3.5183029300   192.880305
174 173 -2.2467204400   219.389724
175 174 -2.2357891800   246.877065
176 175 -0.9446893260   275.355611
177 176 -1.5591324800   304.838649
178 177 -0.6176180420   335.339463
179 178 -0.9560827940   366.871337
180 179 -1.3832677100   399.447557
181 180 -1.5510867000   433.081407
182 181 -0.5740618870   467.786172
183 182  0.1891419950   503.575137
184 183  0.8114323270   540.461587
185 184  1.9957411500   578.458807
186 185  1.7268617600   617.580081
187 186  0.7178595580   657.838695
188 187  0.5035188050   699.247932
189 188  1.9125641700   741.821079
190 189  0.9594114840   785.571419
191 190  0.5286134880   830.512238
192 191  1.1708664900   876.656821
193 192  1.8807594300   924.018452
194 193  2.4199951400   972.610416
195 194  2.5111568000  1022.445998
196 195  4.7797643200  1073.538483
197 196  6.1432360700  1125.901156
198 197  6.4256431400  1179.547301
199 198  4.8200699600  1234.490203
200 199  4.6338606700  1290.743147
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