我正在尝试解决以下问题:我有很多(~80000)个正在生长的器官表面斑 block 。我随时间(18 个时间点)测量它的每个区域，并希望为其拟合一条增长曲线(双逻辑模型，例如，只是两个逻辑函数 bcs 的总和。有两个“增长突增”发生在观察期)。
我有框约束以确保指数项不会爆炸，还有一个线性约束必须一个接一个地发生。此外，为了在拟合参数中强制执行某种空间连续性，我向目标函数(最小二乘)添加了一个惩罚项，作为相邻补丁(某些)参数之间差异的平方和，因此个人模型拟合不再独立。
我有整个事物的梯度和粗麻布，我在越来越精细的尺度上解决它，从总表面积开始，分割，将全局模型的参数映射到每个补丁，再次运行求解器，分割...直到我得到我想要的分辨率。
因此，使用 IPOPT 进行优化是可行的，但速度非常慢，而且由于我对优化理论的所有事情都不太了解，所以我想知道我是否在设置中做了一些非常愚蠢的事情。我使用 ma86 作为线性求解器，针对在具有 256G RAM 和 56 个内核以及 metis 重新排序的机器上运行的 openBLAS 编译。我使用的其他设置是:
% nlp scaling
solv_options.ipopt.nlp_scaling_method = 'gradient-based';
solv_options.ipopt.nlp_scaling_max_gradient = 1;
solv_options.ipopt.nlp_scaling_min_value = 1e-16;
solv_options.ipopt.bound_mult_init_method='constant';
% Barrier Parameter
solv_options.ipopt.mu_strategy = 'adaptive';
solv_options.ipopt.mu_oracle = 'quality-function';
solv_options.ipopt.fixed_mu_oracle = 'average_compl';
solv_options.ipopt.adaptive_mu_globalization = 'kkt-error';
solv_options.ipopt.corrector_type = 'affine';
% linear solver
solv_options.ipopt.max_soc=0;
solv_options.ipopt.accept_every_trial_step='yes';
solv_options.ipopt.linear_system_scaling = 'none';
solv_options.ipopt.neg_curv_test_tol = 0;
solv_options.ipopt.neg_curv_test_reg = 'yes';
solv_options.ipopt.max_refinement_steps=0;
solv_options.ipopt.min_refinement_steps=0;
% ma86 settings
solv_options.ipopt.linear_solver='ma86';
solv_options.ipopt.ma86_order='auto';
solv_options.ipopt.ma86_scaling='mc64';
solv_options.ipopt.ma86_small=1e-10;
solv_options.ipopt.ma86_static=1;
solv_options.ipopt.recalc_y='yes';
这给了我这样的东西:
This is Ipopt version 3.12, running with linear solver ma86.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2560
Number of nonzeros in Lagrangian Hessian.............: 112280
Total number of variables............................: 8960
variables with only lower bounds: 2560
variables with lower and upper bounds: 3840
variables with only upper bounds: 2560
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1280
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1280
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 9.8260736e-01 0.00e+00 1.19e-02 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 9.6288666e-01 0.00e+00 1.24e-02 -4.5 1.14e-02 0.0 9.91e-01 1.00e+00f 1
2 9.1582880e-01 0.00e+00 1.16e-02 -4.4 2.72e-02 -0.5 1.00e+00 1.00e+00f 1
3 8.2635857e-01 0.00e+00 1.01e-02 -4.6 1.39e-01 -1.0 9.99e-01 1.00e+00f 1
4 7.8943781e-01 0.00e+00 9.40e-03 -4.9 2.70e-02 -0.5 1.00e+00 1.00e+00f 1
5 7.2123624e-01 0.00e+00 8.12e-03 -5.3 8.70e-02 -1.0 1.00e+00 1.00e+00f 1
6 6.9535003e-01 0.00e+00 7.56e-03 -6.1 2.20e-02 -0.6 1.00e+00 9.06e-01f 1
7 6.6635914e-01 0.00e+00 7.00e-03 -6.6 6.21e-02 -1.1 1.00e+00 5.40e-01f 1
8 6.5683787e-01 0.00e+00 6.78e-03 -7.6 2.14e-02 -0.6 1.00e+00 3.81e-01f 1
9 6.4238130e-01 0.00e+00 6.53e-03 -7.7 1.53e-01 -1.1 1.00e+00 2.90e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 6.3440625e-01 0.00e+00 6.33e-03 -8.7 2.20e-02 -0.7 1.00e+00 3.30e-01f 1
11 6.2431009e-01 0.00e+00 6.16e-03 -8.3 3.08e-01 -1.2 1.00e+00 2.04e-01f 1
12 6.1872460e-01 0.00e+00 6.05e-03 -8.8 2.29e-02 -0.7 1.00e+00 2.32e-01f 1
13 6.0753815e-01 0.00e+00 5.86e-03 -8.9 2.73e-01 -1.2 1.00e+00 2.31e-01f 1
14 6.0575477e-01 0.00e+00 5.82e-03 -9.9 2.60e-02 -0.8 1.00e+00 7.44e-02f 1
15 6.0089103e-01 0.00e+00 5.71e-03 -11.0 1.11e-02 -0.4 1.00e+00 4.48e-01f 1
16 5.9426852e-01 0.00e+00 5.58e-03 -11.0 3.14e-02 -0.8 1.00e+00 2.68e-01f 1
17 5.9175418e-01 0.00e+00 5.52e-03 -11.0 1.17e-02 -0.4 1.00e+00 2.26e-01f 1
18 5.8756155e-01 0.00e+00 5.44e-03 -11.0 3.97e-02 -0.9 1.00e+00 1.65e-01f 1
19 5.8651702e-01 0.00e+00 5.42e-03 -9.5 1.26e-02 -0.5 1.00e+00 8.89e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 5.8278151e-01 0.00e+00 5.34e-03 -10.3 5.18e-02 -0.9 1.00e+00 1.40e-01f 1
21 5.8100614e-01 0.00e+00 5.30e-03 -11.0 1.37e-02 -0.5 1.00e+00 1.42e-01f 1
22 5.7827149e-01 0.00e+00 5.25e-03 -11.0 7.12e-02 -1.0 1.00e+00 9.77e-02f 1
23 5.7564914e-01 0.00e+00 5.19e-03 -9.4 1.52e-02 -0.6 1.00e+00 1.97e-01f 1
24 5.7340095e-01 0.00e+00 5.15e-03 -10.0 1.06e-01 -1.0 1.00e+00 7.66e-02f 1
25 5.7134734e-01 0.00e+00 5.10e-03 -11.0 1.78e-02 -0.6 1.00e+00 1.45e-01f 1
26 5.6897098e-01 0.00e+00 5.06e-03 -11.0 1.82e-01 -1.1 1.00e+00 7.65e-02f 1
27 5.6722484e-01 0.00e+00 5.02e-03 -11.0 2.11e-02 -0.7 1.00e+00 1.16e-01f 1
28 5.6402838e-01 0.00e+00 5.25e-03 -11.0 3.29e-01 -1.1 1.00e+00 9.57e-02f 1
29 5.6299118e-01 0.00e+00 5.26e-03 -10.5 2.38e-02 -0.7 1.00e+00 6.53e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 5.5753178e-01 0.00e+00 5.63e-03 -11.0 1.38e-01 -1.2 1.00e+00 1.62e-01f 1
31 5.5730217e-01 0.00e+00 5.63e-03 -11.0 2.37e-02 -0.8 1.00e+00 1.38e-02f 1
32 5.5363398e-01 0.00e+00 5.71e-03 -11.0 8.55e-02 -1.2 1.00e+00 1.06e-01f 1
33 5.5223091e-01 0.00e+00 5.70e-03 -11.0 2.43e-02 -0.8 1.00e+00 7.99e-02f 1
34 5.5163441e-01 0.00e+00 5.70e-03 -9.9 1.30e-01 -1.3 1.00e+00 1.65e-02f 1
35 5.4699694e-01 0.00e+00 5.67e-03 -10.9 2.56e-02 -0.9 1.00e+00 2.48e-01f 1
36 5.4559780e-01 0.00e+00 5.67e-03 -11.0 2.30e-01 -1.3 1.00e+00 3.71e-02f 1
37 5.4248376e-01 0.00e+00 5.63e-03 -11.0 2.58e-02 -0.9 1.00e+00 1.60e-01f 1
38 5.3955718e-01 0.00e+00 5.63e-03 -10.7 6.78e-01 -1.4 1.00e+00 6.97e-02f 1
39 5.3606672e-01 0.00e+00 5.57e-03 -10.9 2.79e-02 -1.0 1.00e+00 1.73e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 5.3481638e-01 0.00e+00 5.55e-03 -11.0 1.32e-02 -0.5 1.00e+00 1.33e-01f 1
41 5.3248177e-01 0.00e+00 5.50e-03 -11.0 3.23e-02 -1.0 1.00e+00 1.10e-01f 1
42 5.3143109e-01 0.00e+00 5.48e-03 -10.4 1.37e-02 -0.6 1.00e+00 1.05e-01f 1
43 5.2975978e-01 0.00e+00 5.45e-03 -11.0 3.92e-02 -1.1 1.00e+00 7.50e-02f 1
44 5.2855042e-01 0.00e+00 5.42e-03 -11.0 1.50e-02 -0.6 1.00e+00 1.13e-01f 1
45 5.2670676e-01 0.00e+00 5.38e-03 -11.0 4.80e-02 -1.1 1.00e+00 7.85e-02f 1
46 5.2428812e-01 0.00e+00 5.31e-03 -11.0 1.64e-02 -0.7 1.00e+00 2.12e-01f 1
47 5.2221881e-01 0.00e+00 5.27e-03 -11.0 5.84e-02 -1.2 1.00e+00 8.45e-02f 1
48 5.2084516e-01 0.00e+00 5.23e-03 -10.8 1.79e-02 -0.8 1.00e+00 1.14e-01f 1
49 5.1928549e-01 0.00e+00 5.19e-03 -11.0 7.06e-02 -1.2 1.00e+00 6.06e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 5.1787195e-01 0.00e+00 5.15e-03 -11.0 1.95e-02 -0.8 1.00e+00 1.10e-01f 1
51 5.1639782e-01 0.00e+00 5.12e-03 -10.9 8.49e-02 -1.3 1.00e+00 5.45e-02f 1
52 5.1606363e-01 0.00e+00 5.11e-03 -11.0 2.13e-02 -0.9 1.00e+00 2.45e-02f 1
53 5.1178863e-01 0.00e+00 5.01e-03 -11.0 1.04e-01 -1.3 1.00e+00 1.50e-01f 1
54 5.1173718e-01 0.00e+00 5.01e-03 -11.0 2.33e-02 -0.9 1.00e+00 3.61e-03f 1
55 5.0944130e-01 0.00e+00 4.96e-03 -11.0 1.40e-01 -1.4 1.00e+00 7.70e-02f 1
56 5.0727659e-01 0.00e+00 4.89e-03 -11.0 2.55e-02 -1.0 1.00e+00 1.43e-01f 1
57 5.0638483e-01 0.00e+00 4.85e-03 -11.0 1.11e-02 -0.5 1.00e+00 1.29e-01f 1
58 5.0426908e-01 0.00e+00 4.78e-03 -11.0 2.79e-02 -1.0 1.00e+00 1.32e-01f 1
59 5.0333858e-01 0.00e+00 4.74e-03 -11.0 1.22e-02 -0.6 1.00e+00 1.26e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 5.0161958e-01 0.00e+00 4.69e-03 -11.0 3.05e-02 -1.1 1.00e+00 1.02e-01f 1
61 5.0037437e-01 0.00e+00 4.64e-03 -11.0 1.34e-02 -0.6 1.00e+00 1.58e-01f 1
62 4.9971029e-01 0.00e+00 4.62e-03 -11.0 3.34e-02 -1.1 1.00e+00 3.72e-02f 1
63 4.9914235e-01 0.00e+00 4.59e-03 -11.0 1.48e-02 -0.7 1.00e+00 6.69e-02f 1
64 4.9664256e-01 0.00e+00 4.51e-03 -11.0 3.70e-02 -1.2 1.00e+00 1.32e-01f 1
65 4.9621567e-01 0.00e+00 4.50e-03 -11.0 1.62e-02 -0.7 1.00e+00 4.72e-02f 1
66 4.9453149e-01 0.00e+00 4.44e-03 -11.0 4.19e-02 -1.2 1.00e+00 8.39e-02f 1
67 4.9304626e-01 0.00e+00 4.38e-03 -11.0 1.79e-02 -0.8 1.00e+00 1.53e-01f 1
68 4.9163087e-01 0.00e+00 4.34e-03 -11.0 6.89e-02 -1.3 1.00e+00 6.69e-02f 1
69 4.9031174e-01 0.00e+00 4.28e-03 -11.0 1.97e-02 -0.8 1.00e+00 1.28e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 4.8795240e-01 0.00e+00 4.22e-03 -8.8 2.01e-01 -1.3 9.38e-01 1.04e-01f 1
71 4.8721723e-01 0.00e+00 4.18e-03 -9.8 2.16e-02 -0.9 1.00e+00 6.73e-02f 1
72 4.8560412e-01 0.00e+00 4.10e-03 -11.0 9.08e-03 -0.5 1.00e+00 3.33e-01f 1
73 4.8395604e-01 0.00e+00 4.03e-03 -11.0 2.38e-02 -0.9 1.00e+00 1.42e-01f 1
74 4.8308417e-01 0.00e+00 3.99e-03 -11.0 9.99e-03 -0.5 1.00e+00 1.69e-01f 1
75 4.8236398e-01 0.00e+00 3.96e-03 -10.8 2.63e-02 -1.0 1.00e+00 5.83e-02f 1
76 4.7919513e-01 0.00e+00 3.79e-03 -10.9 1.11e-02 -0.6 1.00e+00 5.72e-01f 1
77 4.7844811e-01 0.00e+00 3.76e-03 -11.0 3.12e-02 -1.0 1.00e+00 5.79e-02f 1
78 4.7801338e-01 0.00e+00 3.74e-03 -11.0 1.22e-02 -0.6 1.00e+00 7.40e-02f 1
79 4.7631089e-01 0.00e+00 3.67e-03 -11.0 4.30e-02 -1.1 1.00e+00 1.23e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 4.7516941e-01 0.00e+00 3.61e-03 -11.0 1.35e-02 -0.7 1.00e+00 1.81e-01f 1
81 4.7405503e-01 0.00e+00 3.57e-03 -11.0 7.24e-02 -1.1 1.00e+00 7.54e-02f 1
82 4.7394863e-01 0.00e+00 3.57e-03 -11.0 1.49e-02 -0.7 1.00e+00 1.57e-02f 1
83 4.7209146e-01 0.00e+00 3.51e-03 -8.9 1.76e-01 -1.2 9.65e-01 1.15e-01f 1
84 4.7199185e-01 0.00e+00 3.50e-03 -10.1 1.66e-02 -0.8 1.00e+00 1.36e-02f 1
85 4.7051045e-01 0.00e+00 3.41e-03 -11.0 6.70e-03 -0.3 1.00e+00 4.74e-01f 1
86 4.7028360e-01 0.00e+00 3.40e-03 -11.0 1.92e-02 -0.8 1.00e+00 2.87e-02f 1
87 4.6892068e-01 0.00e+00 3.31e-03 -11.0 7.44e-03 -0.4 1.00e+00 4.02e-01f 1
88 4.6754658e-01 0.00e+00 3.25e-03 -11.0 2.37e-02 -0.9 1.00e+00 1.62e-01f 1
89 4.6723856e-01 0.00e+00 3.23e-03 -11.0 8.24e-03 -0.4 1.00e+00 8.47e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 4.6625331e-01 0.00e+00 3.18e-03 -11.0 3.07e-02 -0.9 1.00e+00 1.08e-01f 1
91 4.6621348e-01 0.00e+00 3.18e-03 -11.0 9.16e-03 -0.5 1.00e+00 1.01e-02f 1
92 4.6511327e-01 0.00e+00 3.17e-03 -11.0 4.12e-02 -1.0 1.00e+00 1.11e-01f 1
93 4.6363418e-01 0.00e+00 3.14e-03 -11.0 1.02e-02 -0.5 1.00e+00 3.45e-01f 1
94 4.6314545e-01 0.00e+00 3.14e-03 -11.0 6.17e-02 -1.0 1.00e+00 4.60e-02f 1
95 4.6235630e-01 0.00e+00 3.12e-03 -11.0 1.13e-02 -0.6 1.00e+00 1.70e-01f 1
96 4.6143809e-01 0.00e+00 3.11e-03 -9.7 1.03e-01 -1.1 1.00e+00 7.93e-02f 1
97 4.6069624e-01 0.00e+00 3.10e-03 -10.6 1.38e-02 -0.6 1.00e+00 1.47e-01f 1
98 4.5824023e-01 0.00e+00 3.68e-03 -8.6 3.14e-01 -1.1 8.52e-01 1.81e-01f 1
99 4.5822224e-01 0.00e+00 3.68e-03 -9.9 2.27e-02 -0.7 1.00e+00 3.30e-03f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 4.5747364e-01 0.00e+00 3.71e-03 -11.0 7.25e-03 -0.3 1.00e+00 3.33e-01f 1
101 4.5651156e-01 0.00e+00 3.77e-03 -11.0 2.65e-02 -0.8 1.00e+00 1.62e-01f 1
102 4.5624508e-01 0.00e+00 3.79e-03 -11.0 8.36e-03 -0.3 1.00e+00 1.09e-01f 1
103 4.5460694e-01 0.00e+00 3.91e-03 -10.5 3.10e-02 -0.8 1.00e+00 2.54e-01f 1
104 4.5415701e-01 0.00e+00 3.92e-03 -11.0 9.64e-03 -0.4 1.00e+00 1.69e-01f 1
105 4.5379967e-01 0.00e+00 3.95e-03 -11.0 3.54e-02 -0.9 1.00e+00 5.13e-02f 1
106 4.5282822e-01 0.00e+00 3.99e-03 -11.0 1.10e-02 -0.4 1.00e+00 3.33e-01f 1
107 4.5236345e-01 0.00e+00 4.03e-03 -11.0 4.03e-02 -0.9 1.00e+00 6.16e-02f 1
108 4.5158337e-01 0.00e+00 4.06e-03 -11.0 1.24e-02 -0.5 1.00e+00 2.45e-01f 1
109 4.4983066e-01 0.00e+00 4.19e-03 -8.9 4.54e-02 -1.0 1.00e+00 2.15e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110 4.4976166e-01 0.00e+00 4.20e-03 -10.3 1.39e-02 -0.5 1.00e+00 2.01e-02f 1
111 4.4733643e-01 0.00e+00 4.34e-03 -8.3 4.72e-02 -1.0 9.30e-01 2.78e-01f 1
112 4.4674741e-01 0.00e+00 4.36e-03 -9.7 1.51e-02 -0.6 1.00e+00 1.60e-01f 1
113 4.4600496e-01 0.00e+00 4.39e-03 -8.6 4.25e-02 -1.1 1.00e+00 8.06e-02f 1
114 4.4596146e-01 0.00e+00 4.39e-03 -10.0 1.64e-02 -0.6 1.00e+00 1.09e-02f 1
115 4.4157951e-01 0.00e+00 4.49e-03 -8.4 4.50e-02 -1.1 1.00e+00 4.49e-01f 1
116 4.4137682e-01 0.00e+00 4.48e-03 -9.6 1.60e-02 -0.7 1.00e+00 4.90e-02f 1
117 4.3873388e-01 0.00e+00 4.47e-03 -7.5 3.70e-02 -1.2 9.19e-01 2.63e-01f 1
118 4.3815153e-01 0.00e+00 4.45e-03 -8.7 1.61e-02 -0.7 1.00e+00 1.33e-01f 1
119 4.3569743e-01 0.00e+00 4.41e-03 -7.5 3.98e-02 -1.2 1.00e+00 2.34e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120 4.3444019e-01 0.00e+00 4.36e-03 -8.6 1.67e-02 -0.8 1.00e+00 2.71e-01f 1
121 4.3145745e-01 0.00e+00 4.27e-03 -7.3 4.24e-02 -1.3 1.00e+00 2.74e-01f 1
122 4.2975412e-01 0.00e+00 4.19e-03 -8.0 1.81e-02 -0.8 1.00e+00 3.55e-01f 1
123 4.2433966e-01 0.00e+00 3.98e-03 -6.9 4.56e-02 -1.3 1.00e+00 4.92e-01f 1
124 4.2414838e-01 0.00e+00 3.96e-03 -7.5 1.92e-02 -0.9 1.00e+00 4.01e-02f 1
125 4.2194620e-01 0.00e+00 3.87e-03 -6.9 4.80e-02 -1.4 1.00e+00 1.96e-01f 1
126 4.1778190e-01 0.00e+00 3.63e-03 -7.7 2.08e-02 -0.9 1.00e+00 8.34e-01f 1
127 4.1737428e-01 0.00e+00 2.94e-03 -7.1 4.72e-02 -1.4 1.00e+00 3.66e-02f 1
128 4.1315312e-01 0.00e+00 2.54e-03 -7.8 2.22e-02 -1.0 1.00e+00 8.46e-01f 1
129 4.1173197e-01 0.00e+00 2.51e-03 -7.1 6.31e-02 -1.5 1.00e+00 1.27e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130 4.0915489e-01 0.00e+00 2.46e-03 -7.7 2.32e-02 -1.0 1.00e+00 5.13e-01f 1
131 4.0515277e-01 0.00e+00 2.38e-03 -6.9 9.97e-02 -1.5 1.00e+00 3.50e-01f 1
132 4.0343660e-01 0.00e+00 2.34e-03 -7.3 2.38e-02 -1.1 1.00e+00 3.43e-01f 1
133 3.9516682e-01 0.00e+00 2.17e-03 -6.8 1.68e-01 -1.6 1.00e+00 7.24e-01f 1
134 3.9098427e-01 0.00e+00 2.06e-03 -7.2 2.34e-02 -1.1 1.00e+00 9.07e-01f 1
135 3.9058959e-01 0.00e+00 2.05e-03 -7.0 5.50e-02 -1.6 1.00e+00 3.97e-02f 1
136 3.8628795e-01 0.00e+00 1.93e-03 -7.7 2.39e-02 -1.2 1.00e+00 9.42e-01f 1
137 3.8288703e-01 0.00e+00 1.84e-03 -7.2 6.76e-02 -1.7 1.00e+00 3.49e-01f 1
138 3.8113960e-01 0.00e+00 1.79e-03 -7.8 2.37e-02 -1.2 1.00e+00 4.02e-01f 1
139 3.7895865e-01 0.00e+00 1.74e-03 -7.3 8.98e-02 -1.7 1.00e+00 2.23e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140 3.7630588e-01 0.00e+00 1.65e-03 -7.9 2.56e-02 -1.3 1.00e+00 6.04e-01f 1
141 3.7040736e-01 0.00e+00 2.44e-03 -7.4 4.40e-01 -1.8 1.00e+00 5.52e-01f 1
142 3.6744301e-01 0.00e+00 2.20e-03 -8.1 5.47e-02 -1.3 1.00e+00 6.87e-01f 1
143 3.6668338e-01 0.00e+00 2.11e-03 -9.7 1.27e-02 -0.9 1.00e+00 4.33e-01f 1
144 3.6563026e-01 0.00e+00 2.04e-03 -10.0 1.45e-01 -1.4 1.00e+00 2.33e-01f 1
145 3.6473497e-01 0.00e+00 1.94e-03 -11.0 1.73e-02 -1.0 1.00e+00 4.85e-01f 1
146 3.6423323e-01 0.00e+00 2.14e-03 -11.0 2.77e-01 -1.5 1.00e+00 1.00e-01f 1
147 3.6397151e-01 0.00e+00 2.15e-03 -10.1 2.02e-02 -1.0 1.00e+00 1.32e-01f 1
148 3.6196968e-01 0.00e+00 2.50e-03 -8.5 1.56e-01 -1.5 1.00e+00 3.92e-01f 1
149 3.6176499e-01 0.00e+00 2.49e-03 -9.6 1.65e-02 -1.1 1.00e+00 1.00e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150 3.6069639e-01 0.00e+00 2.45e-03 -8.4 3.49e-02 -1.6 1.00e+00 2.18e-01f 1
151 3.6041434e-01 0.00e+00 2.43e-03 -9.4 1.63e-02 -1.1 1.00e+00 1.30e-01f 1
152 3.5921303e-01 0.00e+00 2.37e-03 -8.5 3.78e-02 -1.6 1.00e+00 2.34e-01f 1
153 3.5869485e-01 0.00e+00 2.33e-03 -9.4 1.64e-02 -1.2 1.00e+00 2.27e-01f 1
154 3.5699552e-01 0.00e+00 2.25e-03 -7.9 4.06e-02 -1.7 1.00e+00 3.20e-01f 1
155 3.5674149e-01 0.00e+00 2.23e-03 -8.7 1.77e-02 -1.2 1.00e+00 1.07e-01f 1
156 3.5471902e-01 0.00e+00 2.11e-03 -7.5 4.33e-02 -1.7 1.00e+00 3.71e-01f 1
157 3.5335745e-01 0.00e+00 1.99e-03 -8.3 1.91e-02 -1.3 1.00e+00 5.64e-01f 1
158 3.5245420e-01 0.00e+00 1.94e-03 -7.5 4.53e-02 -1.8 1.00e+00 1.67e-01f 1
159 3.5021073e-01 0.00e+00 1.73e-03 -8.3 2.05e-02 -1.3 1.00e+00 9.16e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160 3.4610138e-01 0.00e+00 1.04e-03 -7.3 4.72e-02 -1.8 1.00e+00 7.94e-01f 1
161 3.4469412e-01 0.00e+00 9.99e-04 -8.0 2.07e-02 -1.4 1.00e+00 6.40e-01f 1
162 3.4399767e-01 0.00e+00 1.02e-03 -7.7 4.72e-02 -1.9 1.00e+00 1.42e-01f 1
163 3.4260897e-01 0.00e+00 9.60e-04 -8.1 2.22e-02 -1.4 1.00e+00 6.21e-01f 1
164 3.4129508e-01 0.00e+00 9.37e-04 -7.4 4.92e-02 -1.9 1.00e+00 2.68e-01f 1
165 3.4016827e-01 0.00e+00 9.15e-04 -8.0 2.34e-02 -1.5 1.00e+00 5.07e-01f 1
166 3.3825082e-01 0.00e+00 8.81e-04 -7.2 5.10e-02 -2.0 1.00e+00 3.98e-01f 1
167 3.3610567e-01 0.00e+00 8.39e-04 -7.7 2.44e-02 -1.5 1.00e+00 1.00e+00f 1
168 3.3460677e-01 0.00e+00 8.12e-04 -7.1 5.92e-02 -2.0 1.00e+00 3.31e-01f 1
169 3.3266040e-01 0.00e+00 7.74e-04 -7.7 2.51e-02 -1.6 1.00e+00 9.50e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
170 3.3037125e-01 0.00e+00 7.34e-04 -7.1 8.30e-02 -2.1 1.00e+00 5.31e-01f 1
171 3.2873890e-01 0.00e+00 7.02e-04 -7.7 2.53e-02 -1.6 1.00e+00 8.56e-01f 1
172 3.2622239e-01 0.00e+00 6.57e-04 -7.2 5.31e-02 -2.1 1.00e+00 6.32e-01f 1
173 3.2470308e-01 0.00e+00 6.28e-04 -7.8 2.53e-02 -1.7 1.00e+00 8.69e-01f 1
174 3.2372187e-01 0.00e+00 6.27e-04 -7.5 5.26e-02 -2.2 1.00e+00 2.63e-01f 1
175 3.2283458e-01 0.00e+00 5.92e-04 -8.2 2.60e-02 -1.7 1.00e+00 5.20e-01f 1
176 3.2183252e-01 0.00e+00 6.17e-04 -7.5 5.47e-02 -2.2 1.00e+00 2.71e-01f 1
177 3.2046827e-01 0.00e+00 5.47e-04 -8.1 2.70e-02 -1.8 1.00e+00 8.08e-01f 1
178 3.1975003e-01 0.00e+00 6.03e-04 -7.1 5.85e-02 -2.3 1.00e+00 2.04e-01f 1
179 3.1900035e-01 0.00e+00 5.19e-04 -7.6 2.77e-02 -1.8 1.00e+00 4.53e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
180 3.1615561e-01 0.00e+00 4.90e-04 -6.8 6.32e-02 -2.3 1.00e+00 8.53e-01f 1
181 3.1524761e-01 0.00e+00 4.49e-04 -7.3 2.68e-02 -1.9 1.00e+00 5.85e-01f 1
182 3.1326122e-01 0.00e+00 4.87e-04 -7.0 6.24e-02 -2.4 9.98e-01 6.26e-01f 1
183 3.1296679e-01 0.00e+00 4.09e-04 -7.7 2.63e-02 -1.9 1.00e+00 2.01e-01f 1
184 3.1055716e-01 0.00e+00 4.34e-04 -6.8 6.57e-02 -2.4 1.00e+00 8.22e-01f 1
185 3.1020674e-01 0.00e+00 3.60e-04 -7.5 2.70e-02 -2.0 1.00e+00 2.54e-01f 1
186 3.0929569e-01 0.00e+00 4.68e-04 -7.0 6.86e-02 -2.5 1.00e+00 3.18e-01f 1
187 3.0819030e-01 0.00e+00 3.23e-04 -7.8 2.94e-02 -2.0 1.00e+00 8.02e-01f 1
188 3.0763044e-01 0.00e+00 4.47e-04 -7.2 8.21e-02 -2.5 1.00e+00 2.02e-01f 1
189 3.0725472e-01 0.00e+00 3.07e-04 -8.0 3.11e-02 -2.1 1.00e+00 2.85e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
190 3.0578606e-01 0.00e+00 4.05e-04 -7.5 2.33e-01 -2.6 1.00e+00 5.11e-01f 1
191 3.0495222e-01 0.00e+00 3.43e-04 -8.1 2.91e-02 -2.2 1.00e+00 6.59e-01f 1
192 3.0413633e-01 0.00e+00 1.17e-03 -7.6 4.93e-01 -2.6 1.00e+00 2.84e-01f 1
193 3.0331078e-01 0.00e+00 9.60e-04 -7.7 3.14e-02 -2.2 1.00e+00 6.81e-01f 1
194 3.0239991e-01 0.00e+00 8.65e-04 -6.7 7.67e-02 -2.7 9.99e-01 4.00e-01f 1
195 3.0137954e-01 0.00e+00 6.77e-04 -7.5 3.36e-02 -2.3 1.00e+00 8.52e-01f 1
196 3.0056119e-01 0.00e+00 6.24e-04 -7.5 8.05e-02 -2.7 1.00e+00 3.28e-01f 1
197 3.0014178e-01 0.00e+00 5.65e-04 -8.3 3.58e-02 -2.3 1.00e+00 3.62e-01f 1
198 2.9923633e-01 0.00e+00 5.17e-04 -8.2 8.53e-02 -2.8 1.00e+00 3.59e-01f 1
199 2.9888005e-01 0.00e+00 4.75e-04 -8.9 3.83e-02 -2.4 1.00e+00 3.15e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
200 2.9848396e-01 0.00e+00 4.57e-04 -8.1 8.98e-02 -2.8 1.00e+00 1.58e-01f 1
201 2.9821853e-01 0.00e+00 4.30e-04 -8.8 4.15e-02 -2.4 1.00e+00 2.29e-01f 1
202 2.9738245e-01 0.00e+00 3.96e-04 -7.2 9.62e-02 -2.9 9.99e-01 3.39e-01f 1
203 2.9665614e-01 0.00e+00 3.36e-04 -7.8 4.41e-02 -2.5 1.00e+00 6.30e-01f 1
204 2.9578199e-01 0.00e+00 3.07e-04 -6.8 1.02e-01 -2.9 1.00e+00 3.87e-01f 1
205 2.9465429e-01 0.00e+00 2.39e-04 -7.4 4.54e-02 -2.5 1.00e+00 1.00e+00f 1
206 2.9301415e-01 0.00e+00 1.99e-04 -6.7 1.07e-01 -3.0 1.00e+00 8.01e-01f 1
207 2.9292261e-01 0.00e+00 1.94e-04 -7.3 4.47e-02 -2.6 1.00e+00 8.17e-02f 1
208 2.9132376e-01 0.00e+00 2.05e-04 -6.8 1.11e-01 -3.0 1.00e+00 7.08e-01f 1
209 2.9068030e-01 0.00e+00 1.46e-04 -7.4 4.72e-02 -2.6 1.00e+00 5.83e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
210 2.8989341e-01 0.00e+00 2.15e-04 -6.8 1.19e-01 -3.1 1.00e+00 3.71e-01f 1
211 2.8905846e-01 0.00e+00 1.25e-04 -7.4 5.00e-02 -2.7 1.00e+00 7.99e-01f 1
212 2.8817010e-01 0.00e+00 2.35e-04 -6.9 1.32e-01 -3.1 1.00e+00 4.36e-01f 1
213 2.8741077e-01 0.00e+00 1.25e-04 -7.5 5.21e-02 -2.7 1.00e+00 7.41e-01f 1
214 2.8705020e-01 0.00e+00 2.96e-04 -7.2 1.57e-01 -3.2 1.00e+00 1.65e-01f 1
215 2.8639044e-01 0.00e+00 1.29e-04 -8.0 5.54e-02 -2.8 1.00e+00 6.44e-01f 1
216 2.8567025e-01 0.00e+00 2.13e-04 -7.4 4.30e-01 -3.2 1.00e+00 3.16e-01f 1
217 2.8519078e-01 0.00e+00 1.22e-04 -8.1 6.05e-02 -2.8 1.00e+00 4.79e-01f 1
218 2.8508336e-01 0.00e+00 1.93e-04 -7.5 3.80e-01 -3.3 1.00e+00 4.58e-02f 1
219 2.8459190e-01 0.00e+00 1.16e-04 -8.3 6.70e-02 -2.9 1.00e+00 4.75e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
220 2.8382163e-01 0.00e+00 1.63e-04 -7.2 3.30e-01 -3.3 1.00e+00 3.29e-01f 1
221 2.8341130e-01 0.00e+00 1.28e-04 -7.8 7.27e-02 -2.9 1.00e+00 3.93e-01f 1
222 2.8284739e-01 0.00e+00 1.56e-04 -7.0 2.21e-01 -3.4 1.00e+00 2.45e-01f 1
223 2.8183563e-01 0.00e+00 1.19e-04 -7.7 7.96e-02 -3.0 1.00e+00 9.45e-01f 1
224 2.8117611e-01 0.00e+00 1.40e-04 -6.9 4.77e-01 -3.4 1.00e+00 2.96e-01f 1
225 2.8033847e-01 0.00e+00 1.04e-04 -7.6 8.59e-02 -3.0 1.00e+00 7.96e-01f 1
226 2.8005033e-01 0.00e+00 9.73e-05 -9.1 3.26e-02 -2.6 1.00e+00 6.19e-01f 1
Number of Iterations....: 226
(scaled) (unscaled)
Objective...............: 2.8005032523583379e-01 2.8005032523583379e-01
Dual infeasibility......: 9.7272674831558608e-05 9.7272674831558608e-05
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.2315149852880829e-07 3.2315149852880829e-07
Overall NLP error.......: 9.7272674831558608e-05 9.7272674831558608e-05
Number of objective function evaluations = 227
Number of objective gradient evaluations = 227
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 228
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 1
Number of Lagrangian Hessian evaluations = 226
Total CPU secs in IPOPT (w/o function evaluations) = 3403.326
Total CPU secs in NLP function evaluations = 214.977
EXIT: Optimal Solution Found.
....在这个分辨率下时间是可以的，但是在更高的分辨率下，线性求解变得非常大并且非常慢。所以我想我的问题是:我是否可以期望这样的事情(在更高分辨率下，具有约 500 000 个参数)需要几天时间才能解决，或者是否有任何设置可以调整以增加收敛/减少迭代次数需要。顺便说一句，最初的猜测实际上相当不错，实际上我正在寻找的只是局部最优。
最佳答案
看来您的方法非常彻底。使用 ~500k 变量和 100k 约束，你有很大的自由度。内点方法(其中 IPOPT 非常好)的主要替代方法是事件集方法。主动集方法在自由度较低的情况下往往效果更好，因此 IPOPT 是您最好的选择。
IPOPT 输出表明了一些事情:
Hessian 矩阵不是正定的(几乎每次迭代都需要正则化)
正则化后，问题在变量边界内足够凸(无回溯线搜索)
IPOPT 时间是 >> 函数调用时间(3403.326 : 214.977 非常大)。大部分时间都在进行矩阵分解。
几乎每次迭代都会因为变量边界而被截断。
据我了解，当您的问题变得太大时，它会超出硬件限制(CPU 缓存)并且线性代数时间会耗尽。这可能是你的主要问题。
出于这些原因，我建议尝试 BFGS方法(IPOPT option)。通过直接计算 Hessian 的近似逆矩阵，您可以避免困难的矩阵分解/求解。此外，BFGS 方法可以保证正定 Hessian 矩阵。通常，当 Hessian 不可用时使用 BFGS 近似，因为精确的 Hessian 应该提供更准确的步骤。但是通过正则化、昂贵的分解和截断的步骤，BFGS 可能几乎一样好。期待更多的迭代(226 是非常小的)，但每一次都应该快得多。
您可能还想放宽变量范围。内点方法旨在避免越界。有那么多界限，它可能会减慢进度。
关于mathematical-optimization - 一个非常大的非线性最小二乘优化的收敛，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/51274069/