forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
test_opt.html
1553 lines (1510 loc) · 45.9 KB
/
test_opt.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<html>
<head>
<title>
TEST_OPT - Scalar Function Optimization Test Problems
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_OPT <br> Optimization of a Scalar Function <br> Test Problems
</h1>
<hr>
<p>
<b>TEST_OPT</b>
is a FORTRAN90 library which
defines test problems for the scalar function optimization problem.
</p>
<p>
The scalar function optimization problem is to find a value
for the N-dimensional vector <b>X</b> which minimizes the value of
the given scalar function <b>F(X)</b>. The function <b>F(X)</b> is
not usually defined as the sum of squares of other functions.
The minimum function value is not guaranteed to be zero.
</p>
<p>
Any system of M nonlinear functions in N unknowns can be turned into
a scalar optimization problem. One way to do this is to define the functional
<b>F(X)</b> to be the sum of the squares of the original nonlinear functions.
The minimizer of <b>F</b> will then minimize the sum of the squares of the
residuals. Since this process involves squaring, it can be less accurate
than dealing directly with the original nonlinear functions: that is to say,
the derived optimization problem may be more convenient to solve, but might
provide less accurate results than applying a nonlinear solver to the original
system.
</p>
<p>
If a function <b>F(X)</b> is differentiable, then at an optimum, the
gradient vector must vanish. Thus, it is also possible to start with an
optimization problem involving <b>F(X)</b> and turn it into a problem in
which we seek a zero of the nonlinear functions represented by the gradient
of <b>F</b>. Of course, the gradient must be zero at a mininum, but
the converse does not hold; thus unless we know more about <b>F</b>, it is not
safe to try to replace the optimization problem by a nonlinear function
solution.
</p>
<p>
For each test problem, routines are provided to evaluate the function,
gradient vector, and hessian matrix. Routines are also provided to
indicate the number of variables, the problem title, a suitable starting
point, and a minimizing solution, if known.
</p>
<p>
The functions defined include:
<ol>
<li>
The Fletcher-Powell helical valley function,<br>
N = 3.
</li>
<li>
The Biggs EXP6 function,<br>
N = 6.
</li>
<li>
The Gaussian function,<br>
N = 3.
</li>
<li>
The Powell badly scaled function,<br>
N = 2.
</li>
<li>
The Box 3-dimensional function,<br>
N = 3.
</li>
<li>
The variably dimensioned function,<br>
1 <= N.
</li>
<li>
The Watson function,<br>
2 <= N.
</li>
<li>
The penalty function #1,<br>
1 <= N.
</li>
<li>
The penalty function #2,<br>
1 <= N.
</li>
<li>
The Brown badly scaled function,<br>
N = 2.
</li>
<li>
The Brown and Dennis function,<br>
N = 4.
</li>
<li>
The Gulf R&D function,<br>
N = 3.
</li>
<li>
The trigonometric function,<br>
1 <= N.
</li>
<li>
The extended Rosenbrock parabolic valley function,<br>
1 <= N.
</li>
<li>
The extended Powell singular quartic function,<br>
4 <= N.
</li>
<li>
The Beale function,<br>
N = 2.
</li>
<li>
The Wood function,<br>
N = 4.
</li>
<li>
The Chebyquad function,<br>
1 <= N.
</li>
<li>
Leon's cubic valley function,<br>
N = 2.
</li>
<li>
Gregory and Karney's Tridiagonal Matrix Function,<br>
1 <= N.
</li>
<li>
The Hilbert function,<br>
1 <= N.
</li>
<li>
The De Jong Function F1,<br>
N = 3.
</li>
<li>
The De Jong Function F2,<br>
N = 2.
</li>
<li>
The De Jong Function F3 (discontinuous),<br>
N = 5.
</li>
<li>
The De Jong Function F4 (Gaussian noise),<br>
N = 30.
</li>
<li>
The De Jong Function F5,<br>
N = 2.
</li>
<li>
The Schaffer Function F6,<br>
N = 2.
</li>
<li>
The Schaffer Function F7,<br>
N = 2.
</li>
<li>
The Goldstein Price Polynomial,<br>
N = 2.
</li>
<li>
The Branin RCOS Function,<br>
N = 2.
</li>
<li>
The Shekel SQRN5 Function,<br>
N = 4.
</li>
<li>
The Shekel SQRN7 Function,<br>
N = 4.
</li>
<li>
The Shekel SQRN10 Function,<br>
N = 4.
</li>
<li>
The Six-Hump Camel-Back Polynomial,<br>
N = 2.
</li>
<li>
The Shubert Function,<br>
N = 2.
</li>
<li>
The Stuckman Function,<br>
N = 2.
</li>
<li>
The Easom Function,<br>
N = 2.
</li>
<li>
The Bohachevsky Function #1,<br>
N = 2.
</li>
<li>
The Bohachevsky Function #2,<br>
N = 2.
</li>
<li>
The Bohachevsky Function #3,<br>
N = 2.
</li>
<li>
The Colville Polynomial,<br>
N = 4.
</li>
<li>
The Powell 3D function,<br>
N = 3.
</li>
<li>
The Himmelblau function,<br>
N = 2.
</li>
</ol>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TEST_OPT</b> is available in
<a href = "../../f_src/test_opt/test_opt.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_opt/test_opt.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/asa047/asa047.html">
ASA047</a>,
a FORTRAN90 library which
minimizes a scalar function of several variables using the Nelder-Mead algorithm.
</p>
<p>
<a href = "../../f_src/brent/brent.html">
BRENT</a>,
a FORTRAN90 library which
contains Richard Brent's routines for finding the zero, local minimizer,
or global minimizer of a scalar function of a scalar argument, without
the use of derivative information.
</p>
<p>
<a href = "../../f_src/compass_search/compass_search.html">
COMPASS_SEARCH</a>,
a FORTRAN90 library which
seeks the minimizer of a scalar function of several variables
using compass search, a direct search algorithm that does not use derivatives.
</p>
<p>
<a href = "../../f_src/dqed/dqed.html">
DQED</a>,
a FORTRAN90 library which
solves constrained least squares problems.
</p>
<p>
<a href = "../../m_src/entrust/entrust.html">
ENTRUST</a>,
a MATLAB program which
minimizes a scalar function of several variables using trust region methods,
by Jeff Borggaard and Gene Cliff.
</p>
<p>
<a href = "../../f_src/minpack/minpack.html">
MINPACK</a>,
a FORTRAN90 library which
carries out the least squares minimization of the residual
of a set of linear or nonlinear equations.
</p>
<p>
<a href = "../../m_src/nelder_mead/nelder_mead.html">
NELDER_MEAD</a>,
a MATLAB program which
minimizes a scalar function of several variables using the Nelder-Mead algorithm.
</p>
<p>
<a href = "../../f_src/nl2sol/nl2sol.html">
NL2SOL</a>,
a FORTRAN90 library which
implements an adaptive nonlinear least-squares algorithm.
</p>
<p>
<a href = "../../f_src/praxis/praxis.html">
PRAXIS</a>,
a FORTRAN90 library which
minimizes a scalar
function of several variables.
</p>
<p>
<a href = "../../f_src/test_nls/test_nls.html">
TEST_NLS</a>,
a FORTRAN90 library which
defines a number of problems for nonlinear least squares solvers.
</p>
<p>
<a href = "../../f_src/test_nonlin/test_nonlin.html">
TEST_NONLIN</a>,
a FORTRAN90 library which
defines a number of problems for nonlinear equation solvers.
</p>
<p>
<a href = "../../f_src/test_opt_con/test_opt_con.html">
TEST_OPT_CON</a>,
a FORTRAN90 library which
defines test problems for the minimization of a scalar function
of several variables, with the search constrained to lie within a specified hyper-rectangle.
</p>
<p>
<a href = "../../f_src/test_optimization/test_optimization.html">
TEST_OPTIMIZATION</a>,
a FORTRAN90 library which
defines test problems for the minimization of a scalar function
of several variables, as described by Molga and Smutnicki.
</p>
<p>
<a href = "../../f_src/toms611/toms611.html">
TOMS611</a>,
a FORTRAN90 library which
minimizes a scalar functional of multiple variables.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Evelyn Beale,<br>
On an Iterative Method for Finding a Local Minimum of a Function
of More than One Variable,<br>
Technical Report 25, <br>
Statistical Techniques Research Group,<br>
Princeton University, 1958.
</li>
<li>
F K Branin,
A widely convergent method for finding multiple solutions of
simultaneous nonlinear equations,<br>
IBM Journal of Research and Development,<br>
pages 504-522, September 1972.
</li>
<li>
Richard Brent,<br>
Algorithms for Minimization without Derivatives,<br>
Dover, 2002,<br>
ISBN: 0-486-41998-3,<br>
LC: QA402.5.B74.
</li>
<li>
John Dennis, David Gay, Phuong Vu,<br>
A new nonlinear equations test problem,<br>
Technical Report 83-16,<br>
Mathematical Sciences Department,<br>
Rice University, 1983.
</li>
<li>
John Dennis, Robert Schnabel,<br>
Numerical Methods for Unconstrained Optimization
and Nonlinear Equations,<br>
SIAM, 1996,<br>
ISBN13: 978-0-898713-64-0,<br>
LC: QA402.5.D44.
</li>
<li>
Noel deVilliers, David Glasser,<br>
A continuation method for nonlinear regression,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 18, 1981, pages 1139-1154.
</li>
<li>
Eric Easom,<br>
A survey of global optimization techniques,<br>
Master of Engineering thesis,<br>
University of Louisville, Louisville, Kentucky, 1990.
</li>
<li>
Chris Fraley,<br>
Solution of nonlinear least-squares problems,<br>
Technical Report STAN-CS-1165,<br>
Computer Science Department,<br>
Stanford University, 1987.
</li>
<li>
Chris Fraley,<br>
Software performance on nonlinear least-squares problems,<br>
Technical Report SOL 88-17,<br>
Systems Optimization Laboratory,<br>
Department of Operations Research,<br>
Stanford University, 1988.
</li>
<li>
Allen Goldstein, J Price,<br>
On descent from local minima,<br>
Mathematics of Computation, <br>
Volume 25, Number 115, 1971.
</li>
<li>
David Himmelblau,<br>
Applied Nonlinear Programming,<br>
McGraw Hill, 1972,<br>
ISBN13: 978-0070289215,<br>
LC: T57.8.H55.
</li>
<li>
A Leon,<br>
A Comparison of Eight Known Optimizing Procedures,<br>
in Recent Advances in Optimization Techniques,<br>
edited by Abraham Lavi, Thomas Vogl,<br>
Wiley, 1966.
</li>
<li>
JJ McKeown,<br>
Specialized versus general-purpose algorithms for functions
that are sums of squared terms,<br>
Mathematical Programming,<br>
Volume 9, 1975, pages 57-68.
</li>
<li>
JJ McKeown,<br>
On algorithms for sums of squares problems,<br>
in Towards Global Optimization,<br>
edited by L Dixon, Gabor Szego,<br>
North-Holland, 1975, pages 229-257.
</li>
<li>
Zbigniew Michalewicz,<br>
Genetic Algorithms + Data Structures = Evolution Programs,<br>
Third Edition,<br>
Springer, 1996,<br>
ISBN: 3-540-60676-9,<br>
LC: QA76.618.M53.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 17-41.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Algorithm 566:
FORTRAN Subroutines for Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 136-140.
</li>
<li>
Michael Powell,<br>
An Efficient Method for Finding the Minimum of a Function of
Several Variables Without Calculating Derivatives,<br>
Computer Journal,<br>
Volume 7, Number 2, 1964, pages 155-162.
</li>
<li>
Douglas Salane,<br>
A continuation approach for solving large residual nonlinear
least squares problems,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 8, 1987, pages 655-671.
</li>
<li>
Bruno Shubert,<br>
A sequential method seeking the global maximum of a function,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 9, pages 379-388, 1972.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_opt.f90">test_opt.f90</a>, the source code.
</li>
<li>
<a href = "test_opt.sh">test_opt.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_opt_prb.f90">test_opt_prb.f90</a>, a sample calling
routine.
</li>
<li>
<a href = "test_opt_prb.sh">test_opt_prb.sh</a>, commands to
compile, link and run the calling routine.
</li>
<li>
<a href = "test_opt_prb_output.txt">test_opt_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>P00_F</b> evaluates the objective function for any problem.
</li>
<li>
<b>P00_G</b> evaluates the gradient for any problem.
</li>
<li>
<b>P00_GDIF</b> approximates the gradient via finite differences.
</li>
<li>
<b>P00_H</b> evaluates the Hessian for any problem.
</li>
<li>
<b>P00_HDIF</b> approximates the Hessian via finite differences.
</li>
<li>
<b>P00_PROBLEM_NUM</b> returns the number of problems available.
</li>
<li>
<b>P00_N</b> returns the number of variables for any problem.
</li>
<li>
<b>P00_SOL</b> returns the solution for any problem.
</li>
<li>
<b>P00_START</b> returns a starting point for optimization for any problem.
</li>
<li>
<b>P00_TITLE</b> returns a title for any problem.
</li>
<li>
<b>P01_F</b> evaluates the objective function for problem 1.
</li>
<li>
<b>P01_G</b> evaluates the gradient for problem 1.
</li>
<li>
<b>P01_H</b> evaluates the Hessian for problem 1.
</li>
<li>
<b>P01_N</b> returns the number of variables for problem 1.
</li>
<li>
<b>P01_SOL</b> returns the solution for problem 1.
</li>
<li>
<b>P01_START</b> returns a starting point for optimization for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns a title for problem 1.
</li>
<li>
<b>P02_F</b> evaluates the objective function for problem 2.
</li>
<li>
<b>P02_G</b> evaluates the gradient for problem 2.
</li>
<li>
<b>P02_H</b> evaluates the Hessian for problem 2.
</li>
<li>
<b>P02_N</b> returns the number of variables for problem 2.
</li>
<li>
<b>P02_SOL</b> returns the solution for problem 2.
</li>
<li>
<b>P02_START</b> returns a starting point for optimization for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns a title for problem 2.
</li>
<li>
<b>P03_F</b> evaluates the objective function for problem 3.
</li>
<li>
<b>P03_G</b> evaluates the gradient for problem 3.
</li>
<li>
<b>P03_YVEC</b> is an auxilliary routine for problem 3.
</li>
<li>
<b>P03_H</b> evaluates the Hessian for problem 3.
</li>
<li>
<b>P03_N</b> returns the number of variables for problem 3.
</li>
<li>
<b>P03_SOL</b> returns the solution for problem 3.
</li>
<li>
<b>P03_START</b> returns a starting point for optimization for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns a title for problem 3.
</li>
<li>
<b>P04_F</b> evaluates the objective function for problem 4.
</li>
<li>
<b>P04_G</b> evaluates the gradient for problem 4.
</li>
<li>
<b>P04_H</b> evaluates the Hessian for problem 4.
</li>
<li>
<b>P04_N</b> returns the number of variables for problem 4.
</li>
<li>
<b>P04_SOL</b> returns the solution for problem 4.
</li>
<li>
<b>P04_START</b> returns a starting point for optimization for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns a title for problem 4.
</li>
<li>
<b>P05_F</b> evaluates the objective function for problem 5.
</li>
<li>
<b>P05_G</b> evaluates the gradient for problem 5.
</li>
<li>
<b>P05_H</b> evaluates the Hessian for problem 5.
</li>
<li>
<b>P05_N</b> returns the number of variables for problem 5.
</li>
<li>
<b>P05_SOL</b> returns the solution for problem 5.
</li>
<li>
<b>P05_START</b> returns a starting point for optimization for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns a title for problem 5.
</li>
<li>
<b>P06_F</b> evaluates the objective function for problem 6.
</li>
<li>
<b>P06_G</b> evaluates the gradient for problem 6.
</li>
<li>
<b>P06_H</b> evaluates the Hessian for problem 6.
</li>
<li>
<b>P06_N</b> returns the number of variables for problem 6.
</li>
<li>
<b>P06_SOL</b> returns the solution for problem 6.
</li>
<li>
<b>P06_START</b> returns a starting point for optimization for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns a title for problem 6.
</li>
<li>
<b>P07_F</b> evaluates the objective function for problem 7.
</li>
<li>
<b>P07_G</b> evaluates the gradient for problem 7.
</li>
<li>
<b>P07_H</b> evaluates the Hessian for problem 7.
</li>
<li>
<b>P07_N</b> returns the number of variables for problem 7.
</li>
<li>
<b>P07_SOL</b> returns the solution for problem 7.
</li>
<li>
<b>P07_START</b> returns a starting point for optimization for problem 7.
</li>
<li>
<b>P07_TITLE</b> returns a title for problem 7.
</li>
<li>
<b>P08_F</b> evaluates the objective function for problem 8.
</li>
<li>
<b>P08_G</b> evaluates the gradient for problem 8.
</li>
<li>
<b>P08_H</b> evaluates the Hessian for problem 8.
</li>
<li>
<b>P08_N</b> returns the number of variables for problem 8.
</li>
<li>
<b>P08_SOL</b> returns the solution for problem 8.
</li>
<li>
<b>P08_START</b> returns a starting point for optimization for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns a title for problem 8.
</li>
<li>
<b>P09_F</b> evaluates the objective function for problem 9.
</li>
<li>
<b>P09_G</b> evaluates the gradient for problem 9.
</li>
<li>
<b>P09_H</b> evaluates the Hessian for problem 9.
</li>
<li>
<b>P09_N</b> returns the number of variables for problem 9.
</li>
<li>
<b>P09_SOL</b> returns the solution for problem 9.
</li>
<li>
<b>P09_START</b> returns a starting point for optimization for problem 9.
</li>
<li>
<b>P09_TITLE</b> returns a title for problem 9.
</li>
<li>
<b>P10_F</b> evaluates the objective function for problem 10.
</li>
<li>
<b>P10_G</b> evaluates the gradient for problem 10.
</li>
<li>
<b>P10_H</b> evaluates the Hessian for problem 10.
</li>
<li>
<b>P10_N</b> returns the number of variables for problem 10.
</li>
<li>
<b>P10_SOL</b> returns the solution for problem 10.
</li>
<li>
<b>P10_START</b> returns a starting point for optimization for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns a title for problem 10.
</li>
<li>
<b>P11_F</b> evaluates the objective function for problem 11.
</li>
<li>
<b>P11_G</b> evaluates the gradient for problem 11.
</li>
<li>
<b>P11_H</b> evaluates the Hessian for problem 11.
</li>
<li>
<b>P11_N</b> returns the number of variables for problem 11.
</li>
<li>
<b>P11_SOL</b> returns the solution for problem 11.
</li>
<li>
<b>P11_START</b> returns a starting point for optimization for problem 11.
</li>
<li>
<b>P11_TITLE</b> returns a title for problem 11.
</li>
<li>
<b>P12_F</b> evaluates the objective function for problem 12.
</li>
<li>
<b>P12_G</b> evaluates the gradient for problem 12.
</li>
<li>
<b>P12_H</b> evaluates the Hessian for problem 12.
</li>
<li>
<b>P12_N</b> returns the number of variables for problem 12.
</li>
<li>
<b>P12_SOL</b> returns the solution for problem 12.
</li>
<li>
<b>P12_START</b> returns a starting point for optimization for problem 12.
</li>
<li>
<b>P12_TITLE</b> returns a title for problem 12.
</li>
<li>
<b>P13_F</b> evaluates the objective function for problem 13.
</li>
<li>
<b>P13_G</b> evaluates the gradient for problem 13.
</li>
<li>
<b>P13_H</b> evaluates the Hessian for problem 13.
</li>
<li>
<b>P13_N</b> returns the number of variables for problem 13.
</li>
<li>
<b>P13_SOL</b> returns the solution for problem 13.
</li>
<li>
<b>P13_START</b> returns a starting point for optimization for problem 13.
</li>
<li>
<b>P13_TITLE</b> returns a title for problem 13.
</li>
<li>
<b>P14_F</b> evaluates the objective function for problem 14.
</li>
<li>
<b>P14_G</b> evaluates the gradient for problem 14.
</li>
<li>
<b>P14_H</b> evaluates the Hessian for problem 14.
</li>
<li>
<b>P14_N</b> returns the number of variables for problem 14.
</li>
<li>
<b>P14_SOL</b> returns the solution for problem 14.
</li>
<li>
<b>P14_START</b> returns a starting point for optimization for problem 14.
</li>
<li>
<b>P14_TITLE</b> returns a title for problem 14.
</li>
<li>
<b>P15_F</b> evaluates the objective function for problem 15.
</li>
<li>
<b>P15_G</b> evaluates the gradient for problem 15.
</li>
<li>
<b>P15_H</b> evaluates the Hessian for problem 15.
</li>
<li>
<b>P15_N</b> returns the number of variables for problem 15.
</li>
<li>
<b>P15_SOL</b> returns the solution for problem 15.
</li>
<li>
<b>P15_START</b> returns a starting point for optimization for problem 15.
</li>
<li>
<b>P15_TITLE</b> returns a title for problem 15.
</li>
<li>
<b>P16_F</b> evaluates the objective function for problem 16.
</li>
<li>
<b>P16_G</b> evaluates the gradient for problem 16.
</li>
<li>
<b>P16_H</b> evaluates the Hessian for problem 16.
</li>
<li>
<b>P16_N</b> returns the number of variables for problem 16.
</li>
<li>
<b>P16_SOL</b> returns the solution for problem 16.
</li>
<li>
<b>P16_START</b> returns a starting point for optimization for problem 16.
</li>
<li>
<b>P16_TITLE</b> returns a title for problem 16.
</li>
<li>
<b>P17_F</b> evaluates the objective function for problem 17.
</li>
<li>
<b>P17_G</b> evaluates the gradient for problem 17.
</li>
<li>
<b>P17_H</b> evaluates the Hessian for problem 17.
</li>
<li>
<b>P17_N</b> returns the number of variables for problem 17.
</li>
<li>
<b>P17_SOL</b> returns the solution for problem 17.
</li>
<li>
<b>P17_START</b> returns a starting point for optimization for problem 17.
</li>
<li>
<b>P17_TITLE</b> returns a title for problem 17.
</li>
<li>
<b>P18_F</b> evaluates the objective function for problem 18.
</li>
<li>
<b>P18_FVEC</b> is an auxilliary routine for problem 18.
</li>
<li>
<b>P18_G</b> evaluates the gradient for problem 18.
</li>
<li>
<b>P18_H</b> evaluates the Hessian for problem 18.
</li>
<li>
<b>P18_N</b> returns the number of variables for problem 18.
</li>
<li>
<b>P18_SOL</b> returns the solution for problem 18.
</li>
<li>
<b>P18_START</b> returns a starting point for optimization for problem 18.
</li>
<li>
<b>P18_TITLE</b> returns a title for problem 18.
</li>
<li>
<b>P19_F</b> evaluates the objective function for problem 19.
</li>
<li>
<b>P19_G</b> evaluates the gradient for problem 19.
</li>
<li>
<b>P19_H</b> evaluates the Hessian for problem 19.
</li>
<li>
<b>P19_N</b> returns the number of variables for problem 19.
</li>
<li>
<b>P19_SOL</b> returns the solution for problem 19.
</li>
<li>
<b>P19_START</b> returns a starting point for optimization for problem 19.
</li>
<li>
<b>P19_TITLE</b> returns a title for problem 19.
</li>
<li>
<b>P20_F</b> evaluates the objective function for problem 20.
</li>
<li>
<b>P20_G</b> evaluates the gradient for problem 20.
</li>