5 #include "nlopt_config.h"
7 #define UNUSED(x) (void) x
9 static double sqr(double x)
14 int testfuncs_verbose = 0;
15 int testfuncs_counter = 0;
17 static double testfuncs_status(unsigned n, const double *x, double f)
20 if (testfuncs_verbose) {
22 printf("f_%d (%g", testfuncs_counter, x[0]);
23 for (i = 1; i < n; ++i)
25 printf(") = %g\n", f);
30 #define RETURN(f) return testfuncs_status(n, x, f);
32 #define PI2 6.283185307179586 /* 2*pi */
33 #define PI3 9.424777960769379 /* 3*pi */
34 #define PI4 12.5663706143592 /* 4*pi */
36 /****************************************************************************/
37 static double rosenbrock_f(unsigned n, const double *x, double *grad, void *data)
39 double a = x[1] - x[0] * x[0], b = 1 - x[0];
42 grad[0] = -400 * a * x[0] - 2 * b;
45 RETURN(100 * sqr(a) + sqr(b));
48 static const double rosenbrock_lb[2] = { -2, -2 };
49 static const double rosenbrock_ub[2] = { 2, 2 };
50 static const double rosenbrock_xmin[2] = { 1, 1 };
52 /****************************************************************************/
53 static double mccormic_f(unsigned n, const double *x, double *grad, void *data)
55 double a = x[0] + x[1], b = x[0] - x[1];
58 grad[0] = cos(a) + 2 * b - 1.5;
59 grad[1] = cos(a) - 2 * b + 2.5;
61 RETURN(sin(a) + sqr(b) - 1.5 * x[0] + 2.5 * x[1] + 1);
64 static const double mccormic_lb[2] = { -1.5, -3 };
65 static const double mccormic_ub[2] = { 4, 4 };
66 static const double mccormic_xmin[2] = { -0.547197553, -1.54719756 };
68 /****************************************************************************/
69 static double boxbetts_f(unsigned n, const double *x, double *grad, void *data)
75 grad[0] = grad[1] = grad[2] = 0;
76 for (i = 1; i <= 10; ++i) {
77 double e0 = exp(-0.1 * i * x[0]);
78 double e1 = exp(-0.1 * i * x[1]);
79 double e2 = exp(-0.1 * i) - exp(-1.0 * i);
80 double g = e0 - e1 - e2 * x[2];
83 grad[0] += (2 * g) * (-0.1 * i * e0);
84 grad[1] += (2 * g) * (0.1 * i * e1);
85 grad[2] += -(2 * g) * e2;
91 static const double boxbetts_lb[3] = { 0.9, 9, 0.9 };
92 static const double boxbetts_ub[3] = { 1.2, 11.2, 1.2 };
93 static const double boxbetts_xmin[3] = { 1, 10, 1 };
95 /****************************************************************************/
96 static double paviani_f(unsigned n, const double *x, double *grad, void *data)
99 double f = 0, prod = 1;
102 for (i = 0; i < 10; ++i)
104 for (i = 0; i < 10; ++i) {
105 double ln1 = log(x[i] - 2);
106 double ln2 = log(10 - x[i]);
107 f += sqr(ln1) + sqr(ln2);
109 grad[i] += 2 * ln1 / (x[i] - 2) - 2 * ln2 / (10 - x[i]);
112 f -= (prod = pow(prod, 0.2));
114 for (i = 0; i < 10; ++i)
115 grad[i] -= 0.2 * prod / x[i];
119 static const double paviani_lb[10] = { 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001 };
120 static const double paviani_ub[10] = { 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999, 9.999 };
121 static const double paviani_xmin[10] = { 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583, 9.35026583 };
123 /****************************************************************************/
124 static double grosenbrock_f(unsigned n, const double *x, double *grad, void *data)
131 for (i = 0; i < 29; ++i) {
132 double a = x[i + 1] - x[i] * x[i], b = 1 - x[i];
134 grad[i] += -400 * a * x[i] - 2 * b;
135 grad[i + 1] = 200 * a;
137 f += 100 * sqr(a) + sqr(b);
142 static const double grosenbrock_lb[30] = { -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30, -30 };
143 static const double grosenbrock_ub[30] = { 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30 };
144 static const double grosenbrock_xmin[30] = { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };
146 /****************************************************************************/
147 static double goldsteinprice_f(unsigned n, const double *x, double *grad, void *data)
149 double x0, x1, a1, a12, a2, b1, b12, b2;
154 a2 = 19 - 14 * x0 + 3 * x0 * x0 - 14 * x1 + 6 * x0 * x1 + 3 * x1 * x1;
155 b1 = 2 * x0 - 3 * x1;
157 b2 = 18 - 32 * x0 + 12 * x0 * x0 + 48 * x1 - 36 * x0 * x1 + 27 * x1 * x1;
160 grad[0] = (1 + a12 * a2) * (2 * b1 * 2 * b2 + b12 * (-32 + 24 * x0 - 36 * x1))
161 + (2 * a1 * a2 + a12 * (-14 + 6 * x0 + 6 * x1)) * (30 + b12 * b2);
162 grad[1] = (1 + a12 * a2) * (2 * b1 * (-3) * b2 + b12 * (48 - 36 * x0 + 54 * x1))
163 + (2 * a1 * a2 + a12 * (-14 + 6 * x0 + 6 * x1)) * (30 + b12 * b2);
165 RETURN((1 + a12 * a2) * (30 + b12 * b2));
168 static const double goldsteinprice_lb[2] = { -2, -2 };
169 static const double goldsteinprice_ub[2] = { 2, 2 };
170 static const double goldsteinprice_xmin[2] = { 0, -1 };
172 /****************************************************************************/
173 static double shekel_f(unsigned n, const double *x, double *grad, void *data)
175 static const double A[10][4] = { {4, 4, 4, 4},
186 static const double c[10] = { .1, .2, .2, .4, .4, .6, .3, .7, .5, .5 };
189 unsigned m = *((unsigned *) data);
191 for (i = 0; i < n; ++i)
193 for (i = 0; i < m; ++i) {
194 double fi = 1.0 / (c[i]
195 + sqr(x[0] - A[i][0])
196 + sqr(x[1] - A[i][1])
197 + sqr(x[2] - A[i][2])
198 + sqr(x[3] - A[i][3]));
201 grad[0] += (2 * fi * fi) * (x[0] - A[i][0]);
202 grad[1] += (2 * fi * fi) * (x[1] - A[i][1]);
203 grad[2] += (2 * fi * fi) * (x[2] - A[i][2]);
204 grad[3] += (2 * fi * fi) * (x[3] - A[i][3]);
210 static unsigned shekel_m[3] = { 5, 7, 10 };
211 static const double shekel_lb[4] = { 0, 0, 0, 0 };
212 static const double shekel_ub[4] = { 10, 10, 10, 10 };
213 static const double shekel0_xmin[4] = { 4.000037154, 4.000133276, 4.000037154, 4.000133276 };
214 static const double shekel1_xmin[4] = { 4.000572917, 4.000689366, 3.999489709, 3.999606158 };
215 static const double shekel2_xmin[4] = { 4.000746531, 4.000592935, 3.999663399, 3.999509801 };
217 /****************************************************************************/
218 static double levy_f(unsigned n, const double *x, double *grad, void *data)
221 double a = x[n - 1] - 1, b = 1 + sqr(sin(PI2 * x[n - 1]));
222 double f = sqr(sin(PI3 * x[0])) + a * b;
225 for (i = 0; i < n; ++i)
227 grad[0] = 2 * PI3 * sin(PI3 * x[0]) * cos(PI3 * x[0]);
228 grad[n - 1] += b + a * 2 * PI2 * sin(PI2 * x[n - 1]) * cos(PI2 * x[n - 1]);
230 for (i = 0; i < n - 1; ++i) {
232 b = 1 + sqr(sin(PI3 * x[i + 1]));
235 grad[i] += 2 * a * b;
236 grad[i + 1] += 2 * PI3 * sqr(a) * sin(PI3 * x[i + 1]) * cos(PI3 * x[i + 1]);
242 static const double levy_lb[7] = { -5, -5, -5, -5, -5, -5, -5 };
243 static const double levy_ub[7] = { 5, 5, 5, 5, 5, 5, 5 };
244 static const double levy_xmin[7] = { 1, 1, 1, 1, 1, 1, -4.75440246 };
245 static const double levy4_lb[4] = { -10, -10, -10, -10 };
246 static const double levy4_ub[4] = { 10, 10, 10, 10 };
247 static const double levy4_xmin[4] = { 1, 1, 1, -9.75235596 };
249 /****************************************************************************/
250 static double griewank_f(unsigned n, const double *x, double *grad, void *data)
255 for (i = 0; i < n; ++i) {
256 f += sqr(x[i]) * 0.00025;
257 p *= cos(x[i] / sqrt(i + 1.));
259 grad[i] = x[i] * 0.0005;
263 for (i = 0; i < n; ++i)
264 grad[i] += p * tan(x[i] / sqrt(i + 1.)) / sqrt(i + 1.);
268 static const double griewank_lb[10] = { -500, -500, -500, -500, -500, -500, -500, -500, -500, -500 };
269 static const double griewank_ub[10] = { 600, 600, 600, 600, 600, 600, 600, 600, 600, 600 };
270 static const double griewank_xmin[10] = { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 };
272 /****************************************************************************/
273 static double sixhumpcamel_f(unsigned n, const double *x, double *grad, void *data)
277 grad[0] = 8 * x[0] - 2.1 * 4 * pow(x[0], 3.) + 2 * pow(x[0], 5.) + x[1];
278 grad[1] = x[0] - 8 * x[1] + 16 * pow(x[1], 3.);
280 RETURN(4 * sqr(x[0]) - 2.1 * pow(x[0], 4.) + pow(x[0], 6.) / 3. + x[0] * x[1] - 4 * sqr(x[1]) + 4 * pow(x[1], 4.));
283 static const double sixhumpcamel_lb[2] = { -5, -5 };
284 static const double sixhumpcamel_ub[2] = { 5, 5 };
285 static const double sixhumpcamel_xmin[2] = { 0.08984201317, -0.7126564032 };
287 /****************************************************************************/
288 static double convexcosh_f(unsigned n, const double *x, double *grad, void *data)
293 for (i = 0; i < n; ++i)
294 f *= cosh((x[i] - i) * (i + 1));
296 for (i = 0; i < n; ++i)
297 grad[i] = f * tanh((x[i] - i) * (i + 1)) * (i + 1);
301 static const double convexcosh_lb[10] = { -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 };
302 static const double convexcosh_ub[10] = { 2, 3, 6, 7, 8, 10, 11, 13, 14, 16 };
303 static const double convexcosh_xmin[10] = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 };
305 /****************************************************************************/
306 static double branin_f(unsigned n, const double *x, double *grad, void *data)
308 double a = 1 - 2 * x[1] + 0.05 * sin(PI4 * x[1]) - x[0];
309 double b = x[1] - 0.5 * sin(PI2 * x[0]);
312 grad[0] = -2 * a - cos(PI2 * x[0]) * PI2 * b;
313 grad[1] = 2 * a * (0.05 * PI4 * cos(PI4 * x[1]) - 2) + 2 * b;
315 RETURN(sqr(a) + sqr(b));
318 static const double branin_lb[2] = { -10, -10 };
319 static const double branin_ub[2] = { 10, 10 };
320 static const double branin_xmin[2] = { 1, 0 };
322 /****************************************************************************/
323 static double shubert_f(unsigned n, const double *x, double *grad, void *data)
328 for (j = 1; j <= 5; ++j)
329 for (i = 0; i < n; ++i)
330 f -= j * sin((j + 1) * x[i] + j);
332 for (i = 0; i < n; ++i) {
334 for (j = 1; j <= 5; ++j)
335 grad[i] -= j * (j + 1) * cos((j + 1) * x[i] + j);
341 static const double shubert_lb[2] = { -10, -10 };
342 static const double shubert_ub[2] = { 10, 10 };
343 static const double shubert_xmin[2] = { -6.774576143, -6.774576143 };
345 /****************************************************************************/
346 static double hansen_f(unsigned n, const double *x, double *grad, void *data)
351 for (i = 1; i <= 5; ++i)
352 a += i * cos((i - 1) * x[0] + i);
353 for (i = 1; i <= 5; ++i)
354 b += i * cos((i + 1) * x[1] + i);
357 for (i = 1; i <= 5; ++i)
358 grad[0] -= i * (i - 1) * sin((i - 1) * x[0] + i);
361 for (i = 1; i <= 5; ++i)
362 grad[1] -= i * (i + 1) * sin((i + 1) * x[1] + i);
368 static const double hansen_lb[2] = { -10, -10 };
369 static const double hansen_ub[2] = { 10, 10 };
370 static const double hansen_xmin[2] = { -1.306707704, -1.425128429 };
372 /****************************************************************************/
373 static double osc1d_f(unsigned n, const double *x, double *grad, void *data)
375 double y = *x - 1.23456789;
378 grad[0] = y * 0.02 + sin(y - 2 * sin(3 * y)) * (1 - 6 * cos(3 * y));
379 RETURN(sqr(y * 0.1) - cos(y - 2 * sin(3 * y)));
382 static const double osc1d_lb[1] = { -5 };
383 static const double osc1d_ub[1] = { 5 };
384 static const double osc1d_xmin[1] = { 1.23456789 };
386 /****************************************************************************/
387 static double corner4d_f(unsigned n, const double *x, double *grad, void *data)
389 double u = x[0] + x[1] * x[2] * sin(2 * x[3]);
390 double v = x[0] + 2 * sin(u);
394 grad[0] = 2 * v * (1 + 2 * cos(u));
395 grad[1] = 2 * v * 2 * cos(u) * x[2] * sin(2 * x[3]) + 0.1;
396 grad[2] = 2 * v * 2 * cos(u) * x[1] * sin(2 * x[3]) + 0.1;
397 grad[3] = 2 * v * 2 * cos(u) * x[1] * x[2] * cos(2 * x[3]) * 2 + 0.1;
399 RETURN(1 + v * v + 0.1 * (x[1] + x[2] + x[3]));
402 static const double corner4d_lb[4] = { 0, 0, 0, 0 };
403 static const double corner4d_ub[4] = { 1, 1, 1, 1 };
404 static const double corner4d_xmin[4] = { 0, 0, 0, 0 };
406 /****************************************************************************/
407 static double side4d_f(unsigned n, const double *x, double *grad, void *data)
409 double x0, x1, x2, x3, d0, d1, d2, d3;
410 const double w0 = 0.1, w1 = 0.2, w2 = 0.3, w3 = 0.4;
413 x0 = +0.4977 * x[0] - 0.3153 * x[1] - 0.5066 * x[2] - 0.4391 * x[3];
414 x1 = -0.3153 * x[0] + 0.3248 * x[1] - 0.4382 * x[2] - 0.4096 * x[3];
415 x2 = -0.5066 * x[0] - 0.4382 * x[1] + 0.3807 * x[2] - 0.4543 * x[3];
416 x3 = -0.4391 * x[0] - 0.4096 * x[1] - 0.4543 * x[2] + 0.5667 * x[3];
418 d0 = -1. / (x0 * x0 + w0 * w0);
419 d1 = -1. / (x1 * x1 + w1 * w1);
420 d2 = -1. / (x2 * x2 + w2 * w2);
421 d3 = -1. / (x3 * x3 + w3 * w3);
424 grad[0] = 2 * (x0 * d0 * d0 * +0.4977 + x1 * d1 * d1 * -0.3153 + x2 * d2 * d2 * -0.5066 + x3 * d3 * d3 * -0.4391);
425 grad[1] = 2 * (x0 * d0 * d0 * -0.3153 + x1 * d1 * d1 * +0.3248 + x2 * d2 * d2 * -0.4382 + x3 * d3 * d3 * -0.4096);
426 grad[2] = 2 * (x0 * d0 * d0 * -0.5066 + x1 * d1 * d1 * -0.4382 + x2 * d2 * d2 * +0.3807 + x3 * d3 * d3 * -0.4543);
427 grad[3] = 2 * (x0 * d0 * d0 * -0.4391 + x1 * d1 * d1 * -0.4096 + x2 * d2 * d2 * -0.4543 + x3 * d3 * d3 * +0.5667);
429 RETURN(d0 + d1 + d2 + d3);
432 static const double side4d_lb[4] = { 0.1, -1, -1, -1 };
433 static const double side4d_ub[4] = { 1, 1, 1, 1 };
434 static const double side4d_xmin[4] = { 0.1, 0.102971169, 0.0760520641, -0.0497098571 };
436 /****************************************************************************/
437 /****************************************************************************/
439 const testfunc testfuncs[NTESTFUNCS] = {
440 {rosenbrock_f, NULL, 1, 2,
441 rosenbrock_lb, rosenbrock_ub, rosenbrock_xmin,
442 0.0, "Rosenbrock function"},
443 {mccormic_f, NULL, 1, 2,
444 mccormic_lb, mccormic_ub, mccormic_xmin,
445 -1.91322295, "McCormic function"},
446 {boxbetts_f, NULL, 1, 3,
447 boxbetts_lb, boxbetts_ub, boxbetts_xmin,
448 0.0, "Box and Betts exponential quadratic sum"},
449 {paviani_f, NULL, 1, 10,
450 paviani_lb, paviani_ub, paviani_xmin,
451 -45.7784697, "Paviani function"},
452 {grosenbrock_f, NULL, 1, 30,
453 grosenbrock_lb, grosenbrock_ub, grosenbrock_xmin,
454 0.0, "Generalized Rosenbrock function"},
455 {goldsteinprice_f, NULL, 1, 2,
456 goldsteinprice_lb, goldsteinprice_ub, goldsteinprice_xmin,
457 3.0, "Goldstein and Price function"},
458 {shekel_f, shekel_m + 0, 1, 4,
459 shekel_lb, shekel_ub, shekel0_xmin,
460 -10.15319968, "Shekel m=5 function"},
461 {shekel_f, shekel_m + 1, 1, 4,
462 shekel_lb, shekel_ub, shekel1_xmin,
463 -10.40294057, "Shekel m=7 function"},
464 {shekel_f, shekel_m + 2, 1, 4,
465 shekel_lb, shekel_ub, shekel2_xmin,
466 -10.53640982, "Shekel m=10 function"},
468 levy4_lb, levy4_ub, levy4_xmin,
469 -21.50235596, "Levy n=4 function"},
471 levy_lb, levy_ub, levy_xmin + 2,
472 -11.50440302, "Levy n=5 function"},
474 levy_lb, levy_ub, levy_xmin + 1,
475 -11.50440302, "Levy n=6 function"},
477 levy_lb, levy_ub, levy_xmin,
478 -11.50440302, "Levy n=7 function"},
479 {griewank_f, NULL, 1, 10,
480 griewank_lb, griewank_ub, griewank_xmin,
481 0.0, "Griewank function"},
482 {sixhumpcamel_f, NULL, 1, 2,
483 sixhumpcamel_lb, sixhumpcamel_ub, sixhumpcamel_xmin,
484 -1.031628453, "Six-hump camel back function"},
485 {convexcosh_f, NULL, 1, 10,
486 convexcosh_lb, convexcosh_ub, convexcosh_xmin,
487 1.0, "Convex product of cosh functions"},
488 {branin_f, NULL, 1, 2,
489 branin_lb, branin_ub, branin_xmin,
490 -.0, "Branin function"},
491 {shubert_f, NULL, 1, 2,
492 shubert_lb, shubert_ub, shubert_xmin,
493 -24.06249888, "Shubert function"},
494 {hansen_f, NULL, 1, 2,
495 hansen_lb, hansen_ub, hansen_xmin,
496 -176.5417931367, "Hansen function"},
497 {osc1d_f, NULL, 1, 1,
498 osc1d_lb, osc1d_ub, osc1d_xmin,
499 -1.0, "1d oscillating function with a single minimum"},
500 {corner4d_f, NULL, 1, 4,
501 corner4d_lb, corner4d_ub, corner4d_xmin,
502 1.0, "4d function with minimum at corner"},
503 {side4d_f, NULL, 1, 4,
504 side4d_lb, side4d_ub, side4d_xmin,
505 -141.285020472, "4d function with minimum at side"}