当解决投资组合权重的额外1范数约束的投资组合优化问题时，我遇到了收敛问题。
练习描述：
对于具有T观察值的给定N个资产，它们找到1Norm约束Theta的值，使得最后一个期间投资组合返回（第T个）最大化。也就是说，解决问题：min_w w'COVw $ s.t. w1 + w2 + ... + wN = 1和| w1 | + | w2 | + ... + | wN |＆lt; = Theta和，在Theta的所有值中，选择最大值为w'r_T的那个其中r_T是上一期间资产收益率的向量，COV是资产收益率的方差 - 协方差矩阵。
问题描述：
首先，我尝试了“天真”的方法：对于Theta从1到6的网格为0.001，我将解决投资组合优化问题并计算每个Theta的最后一个期间投资组合回报。我们的想法是选择具有最大相应的最后期间回报的Theta的值。但是，我注意到，对于Theta的相当多的值，solnp函数没有收敛。问题主要发生在Theta的小值上：从1到3.对于较大的值，没有检测到收敛问题。
第二种方法是使用solnp函数两次：首先作为找到Theta的函数，第二个作为目标函数的内部部分。但是，我无法通过这种方式找到可靠的估算值：我得到的值并未提供最佳解决方案。显然，目标函数并不平滑，但是gosolnp函数找不到解决方案。
下面提供了包含数据的代码（6个资产，120个返回观察值）。欢迎任何建议。
> exp_d
[,1] [,2] [,3] [,4] [,5] [,6]
1 1.3724 0.9081 -0.0695 5.7168 1.9642 1.4222
2 0.6095 1.5075 5.3842 2.7154 2.6838 6.3154
3 -2.6779 -0.1359 -0.4374 1.4287 0.0709 -0.7967
4 -3.5365 -4.3572 -2.0112 -3.5898 -2.3460 -4.0970
5 3.1210 3.6608 2.0944 3.1292 2.8965 3.4614
6 2.7364 1.8411 3.2639 2.9678 2.6067 2.3950
7 -1.0001 -0.3782 3.9316 -0.2621 0.0347 4.4635
8 3.9022 6.3784 6.6192 5.0044 3.5568 8.6305
9 -1.6000 -0.9889 -3.1676 1.3025 0.2071 -2.4764
10 -1.3184 0.8741 3.4796 3.0510 -0.7634 -0.5452
11 5.5482 3.3467 13.3256 5.4076 5.1017 7.4921
12 -1.5484 1.3040 -3.9474 -0.9628 -3.0156 -1.6326
13 4.3331 5.2347 3.9846 9.2369 6.7429 7.2603
14 2.3503 -2.2044 0.8600 3.9191 1.2181 -1.9651
15 2.3981 2.2316 0.3990 5.3864 4.3919 5.9674
16 -0.1633 -2.1458 -5.8357 -3.6349 -4.2840 -6.6219
17 10.4346 8.0620 10.2275 7.0560 6.7676 6.6346
18 5.5505 2.6016 2.4506 2.4954 1.8547 3.4755
19 3.2031 2.7804 3.5948 -0.4774 -0.3667 -2.3168
20 -4.7913 -1.7203 -4.1271 -0.6762 -1.1395 -2.7296
21 7.3930 8.6229 9.4570 12.2800 6.1327 7.8254
22 4.2158 10.6845 9.9723 2.9145 6.0000 4.4979
23 7.5326 1.9540 2.5740 2.6065 -0.1128 0.6388
24 -8.5131 -8.3044 -6.8294 -3.6094 -4.1224 -5.4164
25 -0.4048 -0.4017 -0.8867 1.3590 0.1098 0.9017
26 6.1240 5.0517 3.6990 8.7368 5.3867 6.9468
27 5.7317 4.4538 6.1762 3.4108 1.9153 4.4896
28 6.8299 3.1244 1.6621 1.3590 1.4325 2.0067
29 8.6705 11.1936 12.2831 11.1602 13.2781 13.1497
30 0.9055 -0.5953 -0.6462 0.9332 -0.0008 1.2917
31 -0.0340 1.9379 1.4480 6.5262 4.8373 2.6307
32 0.7414 1.1014 0.3820 -0.5791 0.8306 3.1476
33 -5.9533 -3.4602 -4.1597 -1.3835 2.2098 -0.0642
34 -0.0822 2.7549 0.5136 2.2172 1.1145 2.8362
35 -10.2009 -9.3603 -12.8907 -5.7297 -4.1622 -6.1709
36 6.9673 9.4356 6.4064 12.4593 9.3500 7.5052
37 -0.0681 -0.7029 3.8633 3.5471 6.2018 4.9529
38 -0.8885 -0.4641 -0.4544 7.2353 12.5430 6.9497
39 -4.2569 -4.3105 -3.6218 -4.4211 -4.9685 -6.3175
40 -21.7261 -19.3351 -19.8329 -24.1865 -14.0943 -10.3723
41 -16.4096 -9.8425 -12.3366 -14.2535 -10.9314 -7.6677
42 -2.7511 -2.7249 -1.8488 3.3988 0.5515 1.3170
43 6.2347 9.4395 8.4631 7.1735 3.3113 2.9221
44 1.3502 1.2075 4.3779 3.4639 1.8818 1.2221
45 9.0009 11.0661 10.7647 6.8160 8.3348 5.3573
46 -6.0956 -1.7406 -3.4814 -2.3610 -1.7939 -6.6523
47 -3.7611 -2.3132 -2.9578 0.7061 -2.2643 -1.3650
48 -16.9330 -16.6497 -18.9287 -17.8490 -13.1565 -11.8439
49 6.5208 3.7280 2.5406 4.4017 4.2198 5.2644
50 -3.7646 -1.6135 -2.5073 1.1969 -0.8000 -1.6232
51 -13.6823 -14.6721 -19.1619 -12.0325 -11.7394 -17.0858
52 -10.1007 -7.6801 -10.7443 -9.4278 -7.3264 -11.4835
53 0.2234 -2.7838 -2.6023 -2.3616 -2.8339 -6.6002
54 -10.4236 -11.0266 -17.2866 -5.8057 -9.1776 -9.7245
55 9.9121 11.3602 15.4627 4.7170 6.9823 13.5911
56 13.2032 11.6577 17.0900 12.8706 6.7357 12.1756
57 -4.3440 -2.9204 -6.9601 -4.1971 -10.3577 -8.8826
58 -12.8949 -14.0212 -18.2229 -8.8412 -11.1070 -11.3490
59 -9.6709 -8.7006 -13.8440 -11.3932 -15.9343 -20.3630
60 10.1262 7.7295 20.3174 12.8509 16.1956 25.2657
61 -8.5867 -8.9411 -5.7363 -5.1668 -10.1775 -12.2143
62 -1.6114 -1.1797 -3.7504 0.8986 -1.5948 0.0531
63 -26.5722 -30.0521 -33.8286 -28.8671 -28.1493 -35.1131
64 4.2319 9.2071 5.4593 9.5401 3.2588 11.7214
65 -7.7335 -7.2915 -9.3902 -6.1288 -16.6307 -14.5665
66 -13.7987 -16.5088 -22.4192 -11.5667 -18.8221 -20.6686
67 2.4924 3.9264 10.5116 -3.4347 1.9132 6.6353
68 -2.4754 1.9822 1.3833 7.1142 1.7366 0.3507
69 -10.5259 -12.1366 -10.6244 -10.2031 -15.1403 -14.7460
70 -19.3072 -17.1449 -16.0430 -18.1275 -20.1476 -18.5418
71 -17.1387 -24.8400 -17.4741 -19.2184 -25.9531 -25.4360
72 0.9540 6.2481 1.0364 -1.7110 0.0545 8.8476
73 31.9086 36.1125 63.2475 28.1955 48.6209 67.7847
74 48.7266 55.3725 83.5754 31.3211 50.9661 62.5277
75 -2.8054 -4.5841 -12.4737 -1.0366 -6.5149 -5.0087
76 -16.2535 -19.6450 -23.8672 -10.7514 -17.5126 -23.2272
77 -0.2972 -8.2786 -13.2053 -2.7223 -8.9288 -16.3816
78 -5.1581 -4.2529 -11.0631 6.8503 1.0438 -3.5633
79 4.3896 1.3914 7.8950 -0.0120 3.0297 8.7685
80 -17.8468 -18.4137 -20.8881 -14.9493 -16.5816 -17.3738
81 7.0790 6.1577 16.0332 0.9569 9.8608 6.7577
82 45.4453 54.8024 56.5610 33.7390 51.8933 57.4905
83 59.9378 62.1965 73.3394 16.9717 26.7781 41.6260
84 32.9373 23.3035 18.5882 11.3683 15.5089 21.9496
85 -14.1385 -12.7712 -7.2418 -8.8583 -13.0573 -9.2208
86 10.5585 10.1102 8.2144 10.5958 15.7784 18.8615
87 -7.6121 -13.3352 -20.9042 -9.1454 -12.3603 -19.3866
88 -8.4949 -12.4200 -13.8362 -6.1614 -10.5922 -17.7448
89 4.4492 4.6618 6.1086 9.6900 12.2772 12.6924
90 4.7702 4.0567 0.4565 2.1093 1.8286 3.3536
91 21.5203 28.5643 38.3403 9.8669 16.8246 24.1368
92 -0.2935 1.1796 5.3723 -2.3815 -3.1241 -3.8100
93 4.7722 2.0674 -0.0727 -0.0029 -0.1362 -0.6200
94 2.8356 -1.3170 -1.8303 -1.7780 -2.3152 -4.6160
95 -7.3381 -9.8907 -12.0504 -6.4033 -8.5879 -13.5106
96 3.7672 0.0811 -2.3100 2.6280 2.4811 2.8615
97 -18.0226 -21.5042 -24.4760 -8.3110 -11.7051 -23.1790
98 10.9235 9.4125 12.2230 6.0718 4.4464 5.7691
99 -0.9255 -0.5806 -3.5672 -0.2597 -0.1487 -0.1050
100 -0.9250 -1.6380 -4.1937 -0.4115 -2.7080 -7.3690
101 17.4218 15.4433 12.5284 10.2402 5.0137 10.6573
102 4.9409 1.7554 1.5981 0.7507 0.5333 -2.2372
103 -5.5523 -3.3988 -3.0325 -3.0099 -2.8838 -9.3128
104 -3.5225 -4.8589 -5.8780 -0.8581 -1.6838 -12.9875
105 -5.9234 -7.4692 -11.2668 -2.9715 -3.3289 -7.7499
106 6.7174 9.3484 10.2238 8.0311 9.8871 13.0598
107 -2.4801 2.3317 1.7345 3.1370 4.3390 3.9570
108 -1.7502 5.9262 0.8071 6.3821 5.6136 8.0813
109 9.1528 12.9358 12.7537 6.9882 6.3196 17.1148
110 4.2227 8.6796 14.4462 2.5694 2.2001 3.7475
111 5.0917 5.5331 0.5575 4.6757 0.6768 1.1198
112 10.9064 10.8504 6.8068 6.6016 7.9769 6.0328
113 6.6969 10.4475 18.9743 3.0389 5.6087 14.5739
114 5.7713 10.1556 2.2152 2.9835 5.8064 8.6858
115 10.3194 7.6727 22.3771 3.2881 9.5299 12.2406
116 1.9010 6.5601 6.7824 1.9137 2.8060 7.1039
117 0.5096 2.3380 0.8324 2.6378 0.0632 -1.0088
118 -14.3931 -13.9743 -15.4640 -7.1563 -8.3505 -10.2494
119 4.9011 5.5856 8.6767 5.0578 5.1107 6.5508
120 -2.2080 -0.2588 -1.2498 3.6325 2.1402 0.2676
#define equality constraint function
equal <- function(x) c(sum(x))
#define inequality constraint function
in_inequal <- function(x) c(sum(abs(x)))
#define objective function1
obj_f <- function(x) {
int_r <- t(x)%*%V_C_M%*%x
c(as.numeric(int_r))
}
#define objective function2
ex_obj_f <- function(x) {
tteta <- x
port_w <- solnp(rep(1/n,n), fun = obj_f, eqfun=equal, eqB=1,
ineqfun = in_inequal, ineqLB = 0, ineqUB = tteta, control = list(trace=0))
lp_ret <- exp_d[nrow(exp_d),]%*%port_w$pars
-lp_ret
}
#First "naive" attempt
exp_d <- as.matrix(exp_d)
n <- 6
V_C_M <- cov(exp_d)
res <- matrix(0:0, nrow = 5000, ncol = 3)
for (i in 1:5000) {
tteta <- 1 + i*0.001
port_w <- solnp(rep(1/n,n), fun = obj_f, eqfun=equal, eqB=1, ineqfun = in_inequal, ineqLB = 0, ineqUB = tteta, control = list(trace=0))
lp_ret <- exp_d[nrow(exp_d),]%*%port_w$pars
res[i,1] <- tteta
res[i,2] <- lp_ret
res[i,3] <- port_w$convergence
}
#Second Approach (the result really depends on the starting value of the parameter)
tt_op1 <- solnp(pars = 1.5, fun = ex_obj_f, LB = 1, UB = 10, control = list(trace=1))
tt_op2 <- gosolnp(pars = 1.5, fun = ex_obj_f, LB = 1, UB = 10)
P.S。我在这里阅读了类似问题的帖子，但是我找不到解决问题的方法。
答案 0 :(得分：1)
您的模型可以表示为纯QP（二次规划）问题，而不是具有非线性不可微约束的困难非线性问题。
约束
sum(i, |x(i)|) <= Theta
可以以不同方式线性化。一种可能的重新制定是
-y(i) <= x(i) <= y(i)
sum(i, y(i)) <= Theta
non-negative (or free) variable y(i)
现在您可以使用QP求解器而不是通用NLP求解器来解决模型。