1 /* Copyright (c) 2007-2008 Massachusetts Institute of Technology
3 * Permission is hereby granted, free of charge, to any person obtaining
4 * a copy of this software and associated documentation files (the
5 * "Software"), to deal in the Software without restriction, including
6 * without limitation the rights to use, copy, modify, merge, publish,
7 * distribute, sublicense, and/or sell copies of the Software, and to
8 * permit persons to whom the Software is furnished to do so, subject to
9 * the following conditions:
11 * The above copyright notice and this permission notice shall be
12 * included in all copies or substantial portions of the Software.
14 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
15 * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
16 * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
17 * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
18 * LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
19 * OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
20 * WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
27 #include "nlopt-util.h"
32 #define MIN(a,b) ((a) < (b) ? (a) : (b))
33 #define MAX(a,b) ((a) > (b) ? (a) : (b))
35 /***************************************************************************/
36 /* basic data structure:
38 * a hyper-rectangle is stored as an array of length L = 2n+3, where [1]
39 * is the value (f) of the function at the center, [0] is the "size"
40 * measure (d) of the rectangle, [3..n+2] are the coordinates of the
41 * center (c), [n+3..2n+2] are the widths of the sides (w), and [2]
42 * is an "age" measure for tie-breaking purposes.
44 * we store the hyper-rectangles in a red-black tree, sorted by (d,f)
45 * in lexographic order, to allow us to perform quick convex-hull
46 * calculations (in the future, we might make this data structure
47 * more sophisticated based on the dynamic convex-hull literature).
49 * n > 0 always, of course.
52 /* parameters of the search algorithm and various information that
53 needs to be passed around */
55 int n; /* dimension */
56 int L; /* size of each rectangle (2n+3) */
57 double magic_eps; /* Jones' epsilon parameter (1e-4 is recommended) */
58 int which_diam; /* which measure of hyper-rectangle diam to use:
59 0 = Jones, 1 = Gablonsky */
60 int which_div; /* which way to divide rects:
61 0: orig. Jones (divide all longest sides)
62 1: Gablonsky (cubes divide all, rects longest)
63 2: Jones Encyc. Opt.: pick random longest side */
64 int which_opt; /* which rects are considered "potentially optimal"
65 0: Jones (all pts on cvx hull, even equal pts)
66 1: Gablonsky DIRECT-L (pick one pt, if equal pts)
67 2: ~ 1, but pick points randomly if equal pts
68 ... 2 seems to suck compared to just picking oldest pt */
70 const double *lb, *ub;
71 nlopt_stopping *stop; /* stopping criteria */
72 nlopt_func f; void *f_data;
73 double *work; /* workspace, of length >= 2*n */
74 int *iwork; /* workspace, length >= n */
75 double minf, *xmin; /* minimum so far */
77 /* red-black tree of hyperrects, sorted by (d,f,age) in
78 lexographical order */
80 int age; /* age for next new rect */
81 double **hull; /* array to store convex hull */
82 int hull_len; /* allocated length of hull array */
85 /***************************************************************************/
87 /* Evaluate the "diameter" (d) of a rectangle of widths w[n]
89 We round the result to single precision, which should be plenty for
90 the use we put the diameter to (rect sorting), to allow our
91 performance hack in convex_hull to work (in the Jones and Gablonsky
92 DIRECT algorithms, all of the rects fall into a few diameter
93 values, and we don't want rounding error to spoil this) */
94 static double rect_diameter(int n, const double *w, const params *p)
97 if (p->which_diam == 0) { /* Jones measure */
99 for (i = 0; i < n; ++i)
101 /* distance from center to a vertex */
102 return ((float) (sqrt(sum) * 0.5));
104 else { /* Gablonsky measure */
106 for (i = 0; i < n; ++i)
109 /* half-width of longest side */
110 return ((float) (maxw * 0.5));
114 #define ALLOC_RECT(rect, L) if (!(rect = (double*) malloc(sizeof(double)*(L)))) return NLOPT_OUT_OF_MEMORY
116 static int sort_fv_compare(void *fv_, const void *a_, const void *b_)
118 const double *fv = (const double *) fv_;
119 int a = *((const int *) a_), b = *((const int *) b_);
120 double fa = MIN(fv[2*a], fv[2*a+1]);
121 double fb = MIN(fv[2*b], fv[2*b+1]);
129 static void sort_fv(int n, double *fv, int *isort)
132 for (i = 0; i < n; ++i) isort[i] = i;
133 nlopt_qsort_r(isort, (unsigned) n, sizeof(int), fv, sort_fv_compare);
136 static double function_eval(const double *x, params *p) {
137 double f = p->f(p->n, x, NULL, p->f_data);
140 memcpy(p->xmin, x, sizeof(double) * p->n);
145 #define FUNCTION_EVAL(fv,x,p,freeonerr) fv = function_eval(x, p); if (p->minf < p->stop->minf_max) { free(freeonerr); return NLOPT_MINF_MAX_REACHED; } else if (nlopt_stop_evals((p)->stop)) { free(freeonerr); return NLOPT_MAXEVAL_REACHED; } else if (nlopt_stop_time((p)->stop)) { free(freeonerr); return NLOPT_MAXTIME_REACHED; }
147 #define THIRD (0.3333333333333333333333)
149 #define EQUAL_SIDE_TOL 5e-2 /* tolerance to equate side sizes */
151 /* divide rectangle idiv in the list p->rects */
152 static nlopt_result divide_rect(double *rdiv, params *p)
155 const const int n = p->n;
157 double *c = rdiv + 3; /* center of rect to divide */
158 double *w = c + n; /* widths of rect to divide */
160 int imax = 0, nlongest = 0;
163 for (i = 1; i < n; ++i)
166 for (i = 0; i < n; ++i)
167 if (wmax - w[i] <= wmax * EQUAL_SIDE_TOL)
169 if (p->which_div == 1 || (p->which_div == 0 && nlongest == n)) {
170 /* trisect all longest sides, in increasing order of the average
171 function value along that direction */
172 double *fv = p->work;
173 int *isort = p->iwork;
174 for (i = 0; i < n; ++i) {
175 if (wmax - w[i] <= wmax * EQUAL_SIDE_TOL) {
177 c[i] = csave - w[i] * THIRD;
178 FUNCTION_EVAL(fv[2*i], c, p, 0);
179 c[i] = csave + w[i] * THIRD;
180 FUNCTION_EVAL(fv[2*i+1], c, p, 0);
184 fv[2*i] = fv[2*i+1] = HUGE_VAL;
187 sort_fv(n, fv, isort);
188 if (!(node = rb_tree_find(&p->rtree, rdiv)))
189 return NLOPT_FAILURE;
190 for (i = 0; i < nlongest; ++i) {
192 w[isort[i]] *= THIRD;
193 rdiv[0] = rect_diameter(n, w, p);
195 node = rb_tree_resort(&p->rtree, node);
196 for (k = 0; k <= 1; ++k) {
199 memcpy(rnew, rdiv, sizeof(double) * L);
200 rnew[3 + isort[i]] += w[isort[i]] * (2*k-1);
201 rnew[1] = fv[2*isort[i]+k];
203 if (!rb_tree_insert(&p->rtree, rnew)) {
205 return NLOPT_OUT_OF_MEMORY;
212 if (nlongest > 1 && p->which_div == 2) {
213 /* randomly choose longest side */
214 i = nlopt_iurand(nlongest);
215 for (k = 0; k < n; ++k)
216 if (wmax - w[k] <= wmax * EQUAL_SIDE_TOL) {
217 if (!i) { i = k; break; }
222 i = imax; /* trisect longest side */
223 if (!(node = rb_tree_find(&p->rtree, rdiv)))
224 return NLOPT_FAILURE;
226 rdiv[0] = rect_diameter(n, w, p);
228 node = rb_tree_resort(&p->rtree, node);
229 for (k = 0; k <= 1; ++k) {
232 memcpy(rnew, rdiv, sizeof(double) * L);
233 rnew[3 + i] += w[i] * (2*k-1);
234 FUNCTION_EVAL(rnew[1], rnew + 3, p, rnew);
236 if (!rb_tree_insert(&p->rtree, rnew)) {
238 return NLOPT_OUT_OF_MEMORY;
242 return NLOPT_SUCCESS;
245 /***************************************************************************/
246 /* Convex hull algorithm, used later to find the potentially optimal
247 points. What we really have in DIRECT is a "dynamic convex hull"
248 problem, since we are dynamically adding/removing points and
249 updating the hull, but I haven't implemented any of the fancy
250 algorithms for this problem yet. */
252 /* Find the lower convex hull of a set of points (x,y) stored in a rb-tree
253 of pointers to {x,y} arrays sorted in lexographic order by (x,y).
255 Unlike standard convex hulls, we allow redundant points on the hull,
256 and even allow duplicate points if allow_dups is nonzero.
258 The return value is the number of points in the hull, with pointers
259 stored in hull[i] (should be an array of length >= t->N).
261 static int convex_hull(rb_tree *t, double **hull, int allow_dups)
265 double xmin, xmax, yminmin, ymaxmin;
268 /* Monotone chain algorithm [Andrew, 1979]. */
272 nmax = rb_tree_max(t);
279 do { /* include any duplicate points at (xmin,yminmin) */
280 hull[nhull++] = n->k;
282 } while (n && n->k[0] == xmin && n->k[1] == yminmin);
284 hull[nhull++] = n->k;
286 if (xmin == xmax) return nhull;
288 /* set nmax = min mode with x == xmax */
290 while (nmax->k[0] == xmax)
291 nmax = rb_tree_pred(nmax); /* non-NULL since xmin != xmax */
292 nmax = rb_tree_succ(nmax);
294 /* performance hack (see also below) */
297 kshift[0] = xmax * (1 - 1e-13);
298 kshift[1] = -HUGE_VAL;
299 nmax = rb_tree_find_gt(t, kshift); /* non-NULL since xmin != xmax */
303 ymaxmin = nmax->k[1];
304 minslope = (ymaxmin - yminmin) / (xmax - xmin);
306 /* set n = first node with x != xmin */
308 while (n->k[0] == xmin)
309 n = rb_tree_succ(n); /* non-NULL since xmin != xmax */
311 /* performance hack (see also below) */
314 kshift[0] = xmin * (1 + 1e-13);
315 kshift[1] = -HUGE_VAL;
316 n = rb_tree_find_gt(t, kshift); /* non-NULL since xmin != xmax */
320 for (; n != nmax; n = rb_tree_succ(n)) {
322 if (k[1] > yminmin + (k[0] - xmin) * minslope)
325 /* performance hack: most of the points in DIRECT lie along
326 vertical lines at a few x values, and we can exploit this */
327 if (nhull && k[0] == hull[nhull - 1][0]) { /* x == previous x */
328 if (k[1] > hull[nhull - 1][1]) {
330 /* because of the round to float in rect_diameter, above,
331 it shouldn't be possible for two diameters (x values)
332 to have a fractional difference < 1e-13. Note
333 that k[0] > 0 always in DIRECT */
334 kshift[0] = k[0] * (1 + 1e-13);
335 kshift[1] = -HUGE_VAL;
336 n = rb_tree_pred(rb_tree_find_gt(t, kshift));
339 else { /* equal y values, add to hull */
346 /* remove points until we are making a "left turn" to k */
348 double *t1 = hull[nhull - 1], *t2;
350 /* because we allow equal points in our hull, we have
351 to modify the standard convex-hull algorithm slightly:
352 we need to look backwards in the hull list until we
353 find a point t2 != t1 */
357 } while (it2 >= 0 && t2[0] == t1[0] && t2[1] == t1[1]);
360 /* cross product (t1-t2) x (k-t2) > 0 for a left turn: */
361 if ((t1[0]-t2[0]) * (k[1]-t2[1])
362 - (t1[1]-t2[1]) * (k[0]-t2[0]) >= 0)
370 do { /* include any duplicate points at (xmax,ymaxmin) */
371 hull[nhull++] = nmax->k;
372 nmax = rb_tree_succ(nmax);
373 } while (nmax && nmax->k[0] == xmax && nmax->k[1] == ymaxmin);
375 hull[nhull++] = nmax->k;
380 /***************************************************************************/
382 static int small(double *w, params *p)
385 for (i = 0; i < p->n; ++i)
386 if (w[i] > p->stop->xtol_abs[i] &&
387 w[i] > (p->ub[i] - p->lb[i]) * p->stop->xtol_rel)
392 static nlopt_result divide_good_rects(params *p)
396 int nhull, i, xtol_reached = 1, divided_some = 0;
397 double magic_eps = p->magic_eps;
399 if (p->hull_len < p->rtree.N) {
400 p->hull_len += p->rtree.N;
401 p->hull = (double **) realloc(p->hull, sizeof(double*)*p->hull_len);
402 if (!p->hull) return NLOPT_OUT_OF_MEMORY;
404 nhull = convex_hull(&p->rtree, hull = p->hull, p->which_opt != 1);
406 for (i = 0; i < nhull; ++i) {
407 double K1 = -HUGE_VAL, K2 = -HUGE_VAL, K;
410 /* find unequal points before (im) and after (ip) to get slope */
411 for (im = i-1; im >= 0 && hull[im][0] == hull[i][0]; --im);
412 for (ip = i+1; ip < nhull && hull[ip][0] == hull[i][0]; ++ip);
415 K1 = (hull[i][1] - hull[im][1]) / (hull[i][0] - hull[im][0]);
417 K1 = (hull[i][1] - hull[ip][1]) / (hull[i][0] - hull[ip][0]);
419 if (hull[i][1] - K * hull[i][0]
420 <= p->minf - magic_eps * fabs(p->minf) || ip == nhull) {
421 /* "potentially optimal" rectangle, so subdivide */
422 nlopt_result ret = divide_rect(hull[i], p);
424 if (ret != NLOPT_SUCCESS) return ret;
425 xtol_reached = xtol_reached && small(hull[i] + 3+n, p);
428 /* for the DIRECT-L variant, we only divide one rectangle out
429 of all points with equal diameter and function values
430 ... note that for p->which_opt == 1, i == ip-1 should be a no-op
431 anyway, since we set allow_dups=0 in convex_hull above */
432 if (p->which_opt == 1)
433 i = ip - 1; /* skip to next unequal point for next iteration */
434 else if (p->which_opt == 2) /* like DIRECT-L but randomized */
435 i += nlopt_iurand(ip - i); /* possibly do another equal pt */
438 if (magic_eps != 0) {
440 goto divisions; /* try again */
442 else { /* WTF? divide largest rectangle with smallest f */
443 /* (note that this code actually gets called from time
444 to time, and the heuristic here seems to work well,
445 but I don't recall this situation being discussed in
447 rb_node *max = rb_tree_max(&p->rtree);
449 double wmax = max->k[0];
450 do { /* note: this loop is O(N) worst-case time */
452 pred = rb_tree_pred(max);
453 } while (pred && pred->k[0] == wmax);
454 return divide_rect(max->k, p);
457 return xtol_reached ? NLOPT_XTOL_REACHED : NLOPT_SUCCESS;
460 /***************************************************************************/
462 /* lexographic sort order (d,f,age) of hyper-rects, for red-black tree */
463 int cdirect_hyperrect_compare(double *a, double *b)
465 if (a[0] < b[0]) return -1;
466 if (a[0] > b[0]) return +1;
467 if (a[1] < b[1]) return -1;
468 if (a[1] > b[1]) return +1;
469 if (a[2] < b[2]) return -1;
470 if (a[2] > b[2]) return +1;
471 return (int) (a - b); /* tie-breaker, shouldn't be needed */
474 /***************************************************************************/
476 nlopt_result cdirect_unscaled(int n, nlopt_func f, void *f_data,
477 const double *lb, const double *ub,
480 nlopt_stopping *stop,
481 double magic_eps, int which_alg)
486 nlopt_result ret = NLOPT_OUT_OF_MEMORY;
488 p.magic_eps = magic_eps;
489 p.which_diam = which_alg % 3;
490 p.which_div = (which_alg / 3) % 3;
491 p.which_opt = (which_alg / (3*3)) % 3;
492 p.lb = lb; p.ub = ub;
505 rb_tree_init(&p.rtree, cdirect_hyperrect_compare);
507 p.work = (double *) malloc(sizeof(double) * (2*n));
508 if (!p.work) goto done;
509 p.iwork = (int *) malloc(sizeof(int) * n);
510 if (!p.iwork) goto done;
511 p.hull_len = 128; /* start with a reasonable number */
512 p.hull = (double **) malloc(sizeof(double *) * p.hull_len);
513 if (!p.hull) goto done;
515 if (!(rnew = (double *) malloc(sizeof(double) * p.L))) goto done;
516 for (i = 0; i < n; ++i) {
517 rnew[3+i] = 0.5 * (lb[i] + ub[i]);
518 rnew[3+n+i] = ub[i] - lb[i];
520 rnew[0] = rect_diameter(n, rnew+3+n, &p);
521 rnew[1] = function_eval(rnew+3, &p);
523 if (!rb_tree_insert(&p.rtree, rnew)) {
528 ret = divide_rect(rnew, &p);
529 if (ret != NLOPT_SUCCESS) goto done;
532 double minf0 = p.minf;
533 ret = divide_good_rects(&p);
534 if (ret != NLOPT_SUCCESS) goto done;
535 if (p.minf < minf0 && nlopt_stop_f(p.stop, p.minf, minf0)) {
536 ret = NLOPT_FTOL_REACHED;
542 rb_tree_destroy_with_keys(&p.rtree);
551 /* in the conventional DIRECT-type algorithm, we first rescale our
552 coordinates to a unit hypercube ... we do this simply by
553 wrapping cdirect() around cdirect_unscaled(). */
555 double cdirect_uf(int n, const double *xu, double *grad, void *d_)
557 cdirect_uf_data *d = (cdirect_uf_data *) d_;
560 for (i = 0; i < n; ++i)
561 d->x[i] = d->lb[i] + xu[i] * (d->ub[i] - d->lb[i]);
562 f = d->f(n, d->x, grad, d->f_data);
564 for (i = 0; i < n; ++i)
565 grad[i] *= d->ub[i] - d->lb[i];
569 nlopt_result cdirect(int n, nlopt_func f, void *f_data,
570 const double *lb, const double *ub,
573 nlopt_stopping *stop,
574 double magic_eps, int which_alg)
578 const double *xtol_abs_save;
581 d.f = f; d.f_data = f_data; d.lb = lb; d.ub = ub;
582 d.x = (double *) malloc(sizeof(double) * n*4);
583 if (!d.x) return NLOPT_OUT_OF_MEMORY;
585 for (i = 0; i < n; ++i) {
586 x[i] = (x[i] - lb[i]) / (ub[i] - lb[i]);
589 d.x[3*n+i] = stop->xtol_abs[i] / (ub[i] - lb[i]);
591 xtol_abs_save = stop->xtol_abs;
592 stop->xtol_abs = d.x + 3*n;
593 ret = cdirect_unscaled(n, cdirect_uf, &d, d.x+n, d.x+2*n, x, minf, stop,
594 magic_eps, which_alg);
595 stop->xtol_abs = xtol_abs_save;
596 for (i = 0; i < n; ++i)
597 x[i] = lb[i]+ x[i] * (ub[i] - lb[i]);