GSL 求多元函式極值：simplex 演算法
阿新 • • 發佈：2022-05-17
GSL 手冊上有關於 gsl_multimin_fminimizer_nmsimplex2 的說明。它只需要被優化函式 \(f(\vec{x})\)，而不需要偏導數。它自己有個 Nelder-Mead 演算法（我不懂這個，還沒有研究過，只是試用過），會自己去取點判斷下降方向。
手冊上有測試程式碼：
#include<iostream> using namespace std; #include<gsl/gsl_multimin.h> #include<gsl/gsl_sf_bessel.h> int count_myf = 0; /* Paraboloid centered on (p[0],p[1]), with scale factors (p[2],p[3]) and minimum p[4] */ double my_f (const gsl_vector *v, void *params) { count_myf ++; double x, y; double *p = (double *)params; x = gsl_vector_get(v, 0); y = gsl_vector_get(v, 1); return p[2] * (x - p[0]) * (x - p[0]) + p[3] * (y - p[1]) * (y - p[1]) + p[4]; } int main(){ // define the multi-var function to be optimized gsl_multimin_function minex_func; /* Paraboloid center at (1,2), scale factors (10, 20), minimum value 30 */ double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 }; minex_func.n = 2; /* number of function components */ minex_func.f = my_f; minex_func.params = par; // set initial position x gsl_vector *x; x = gsl_vector_alloc(2); gsl_vector_set( x, 0, 5.0 ); gsl_vector_set( x, 1, 7.0 ); // set initial step size to 1 gsl_vector *ss = gsl_vector_alloc(2); gsl_vector_set_all( ss, 1.0 ); // set minimizer const gsl_multimin_fminimizer_type *T = gsl_multimin_fminimizer_nmsimplex2; gsl_multimin_fminimizer *s = gsl_multimin_fminimizer_alloc( T, 2 ); gsl_multimin_fminimizer_set( s, & minex_func, x, ss ); int iter = 0, status; double size; do{ iter++; status = gsl_multimin_fminimizer_iterate( s ); if( status ) break; size = gsl_multimin_fminimizer_size (s); status = gsl_multimin_test_size( size, 1E-8); if( status == GSL_SUCCESS ) printf( " Converged to minimum at :\n" ); printf( "%5d %.10f %.10f f() = %7.3f size = %.10f\n", iter, gsl_vector_get( s->x, 0 ), gsl_vector_get( s->x, 1 ), s->fval, size ); } while ( status == GSL_CONTINUE && iter < 100 ); cout<<" count_myf = "<<count_myf<<endl; gsl_vector_free(x); gsl_vector_free(ss); gsl_multimin_fminimizer_free(s); return status; }
編譯+執行：
g++ test_simplex.cpp -lgsl -lgslcblas
./a.out
得到結果
1 6.5000000000 5.0000000000 f() = 512.500 size = 1.1303883305 2 5.2500000000 4.0000000000 f() = 290.625 size = 1.4092945438 3 5.2500000000 4.0000000000 f() = 290.625 size = 1.4092945438 4 5.5000000000 1.0000000000 f() = 252.500 size = 1.4092945438 5 2.6250000000 3.5000000000 f() = 101.406 size = 1.8474832731 6 2.6250000000 3.5000000000 f() = 101.406 size = 1.8474832731 7 -0.0000000000 3.0000000000 f() = 60.000 size = 1.8474832731 8 2.0937500000 1.8750000000 f() = 42.275 size = 1.3213162892 9 0.2578125000 1.9062500000 f() = 35.684 size = 1.0689461829 10 0.5878906250 2.4453125000 f() = 35.664 size = 0.8409024133 11 1.2583007812 2.0253906250 f() = 30.680 size = 0.4761530328 12 1.2583007812 2.0253906250 f() = 30.680 size = 0.3672920466 13 1.0926208496 1.8494873047 f() = 30.539 size = 0.2995597113 14 0.8829574585 2.0041198730 f() = 30.137 size = 0.1724325444 15 0.8829574585 2.0041198730 f() = 30.137 size = 0.1261625606 16 0.9581913948 2.0604190826 f() = 30.090 size = 0.1062198609 17 1.0218096972 2.0041832924 f() = 30.005 size = 0.0626448959 18 1.0218096972 2.0041832924 f() = 30.005 size = 0.0433856521 19 1.0218096972 2.0041832924 f() = 30.005 size = 0.0433856521 20 1.0218096972 2.0041832924 f() = 30.005 size = 0.0274263092 21 1.0218096972 2.0041832924 f() = 30.005 size = 0.0218800866 22 0.9920430849 1.9969168063 f() = 30.001 size = 0.0156702845 23 0.9920430849 1.9969168063 f() = 30.001 size = 0.0133695954 24 0.9920430849 1.9969168063 f() = 30.001 size = 0.0079586570 25 0.9987625058 2.0047084907 f() = 30.000 size = 0.0079586570 26 1.0025252407 1.9999882754 f() = 30.000 size = 0.0053914447 27 1.0025252407 1.9999882754 f() = 30.000 size = 0.0034382764 28 1.0025252407 1.9999882754 f() = 30.000 size = 0.0027835269 29 0.9985776580 2.0003782319 f() = 30.000 size = 0.0020123788 30 0.9985776580 2.0003782319 f() = 30.000 size = 0.0017260798 31 1.0008632701 2.0003940352 f() = 30.000 size = 0.0010139823 32 0.9996159870 1.9995509089 f() = 30.000 size = 0.0010139823 33 0.9994086433 2.0001753520 f() = 30.000 size = 0.0007350933 34 1.0001877926 2.0001285828 f() = 30.000 size = 0.0004349777 35 1.0001877926 2.0001285828 f() = 30.000 size = 0.0003513674 36 1.0002168497 1.9998973397 f() = 30.000 size = 0.0002633416 37 0.9999547118 1.9999321997 f() = 30.000 size = 0.0001553283 38 0.9999547118 1.9999321997 f() = 30.000 size = 0.0001215448 39 0.9999601990 2.0000167371 f() = 30.000 size = 0.0000940103 40 0.9999601990 2.0000167371 f() = 30.000 size = 0.0000557365 41 0.9999601990 2.0000167371 f() = 30.000 size = 0.0000420072 42 0.9999914230 1.9999886035 f() = 30.000 size = 0.0000378037 43 1.0000114660 2.0000003713 f() = 30.000 size = 0.0000240432 44 1.0000114660 2.0000003713 f() = 30.000 size = 0.0000145614 45 0.9999911331 2.0000000498 f() = 30.000 size = 0.0000109784 46 0.9999963613 1.9999944070 f() = 30.000 size = 0.0000090450 47 1.0000026066 1.9999987999 f() = 30.000 size = 0.0000052764 48 1.0000026066 1.9999987999 f() = 30.000 size = 0.0000037733 49 1.0000026066 1.9999987999 f() = 30.000 size = 0.0000037733 50 0.9999986944 1.9999995432 f() = 30.000 size = 0.0000023683 51 0.9999986944 1.9999995432 f() = 30.000 size = 0.0000018371 52 1.0000012525 1.9999995221 f() = 30.000 size = 0.0000013433 53 1.0000005378 2.0000002391 f() = 30.000 size = 0.0000011223 54 0.9999997948 1.9999997119 f() = 30.000 size = 0.0000006582 55 0.9999997948 1.9999997119 f() = 30.000 size = 0.0000004499 56 1.0000001234 2.0000000980 f() = 30.000 size = 0.0000002732 57 1.0000001234 2.0000000980 f() = 30.000 size = 0.0000002028 58 1.0000001234 2.0000000980 f() = 30.000 size = 0.0000001210 59 0.9999998954 2.0000000247 f() = 30.000 size = 0.0000000827 60 0.9999999427 1.9999999776 f() = 30.000 size = 0.0000001100 61 0.9999999427 1.9999999776 f() = 30.000 size = 0.0000000599 62 0.9999999427 1.9999999776 f() = 30.000 size = 0.0000000599 63 1.0000000134 2.0000000198 f() = 30.000 size = 0.0000000543 64 1.0000000233 2.0000000006 f() = 30.000 size = 0.0000000398 65 0.9999999805 1.9999999939 f() = 30.000 size = 0.0000000213 66 0.9999999805 1.9999999939 f() = 30.000 size = 0.0000000187 67 1.0000000087 2.0000000009 f() = 30.000 size = 0.0000000143 Converged to minimum at : 68 0.9999999944 1.9999999993 f() = 30.000 size = 0.0000000077 count_myf = 134
可見，nmsimplex2 需要 68 次迭代，才真正收斂。
之前的帖子裡記錄了，共軛梯度法只需要 13 次迭代，就會收斂：https://www.cnblogs.com/luyi07/p/14560924.html
所以共軛梯度法收斂更快，雖然它需要偏導數。
但如果偏導數很昂貴，不妨拿 nmsimplex2 試一試。