Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=500 on pages 649-651 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05.
Also, it also tries to solve the corresponding integer problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 500
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 300
129 FOR KKQQ=1 TO 500
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 5
181 J=1+FIX(RND*500)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 500
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 500
359 PSUM=PSUM+(1/500)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 500
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=1 TO 500
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1799 PSUM=0
1857 FOR J44=1 TO 500
1859 PSUM=PSUM+(1/500)*( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1898 PRINT A(1),A(2),A(3),A(4),A(5)
1899 PRINT A(496),A(497),A(498),A(499),A(500),M,JJJJ,PPOBA2
1998 PRINT AA(1),AA(2),AA(3),AA(4),AA(5)
2000 PRINT AA(496),AA(497),AA(498),AA(499),AA(500),PSUM,JJJJ
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
-2.894538 -2.916732 -2.897156 -2.90236 -2.884825
-2.870931 -2.934536 -2.915289 -2.895821 -2.923359
78.32247 -32000 78.32247
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.99951 -32000
-2.881405 -2.883218 -2.903935 -2.9088 -2.920653
-2.89391 -2.918805 -2.890355 -2.92452 -2.913075
78.3227 -32000 78.3227
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.99951 -31999
-2.883065 -2.927211 -2.925028 -2.928296 -2.915845
-2.908983 -2.889557 -2.880188 -2.898715 -2.909385
78.26743 -31998 78.26743
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.93951 -31998
-2.939881 -2.884955 -2.892202 -2.880561 -2.9227
-2.902572 -2.919708 -2.88201 -2.891476 -2.911448
78.20945 -31997 78.20945
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.87951 -31997
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was 22 minutes.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26, 645-663.
« March 2012 | Main | May 2012 »
April 2012
04/18/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Test Function with 500 Continuous Variables, Second Edition
Posted at 10:17 AM | Permalink | Comments (0)
Testing the Domino Method of General Integer Nonlinear Programming with a One-Hundred-Variable Test Problem from the Literature
Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=100 on pages 649-651 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.01-(FIX(RND*2))*.01.
Also, it also tries to solve the corresponding integer problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 100
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*100)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 100
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 100
359 PSUM=PSUM+(1/100)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 100
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=1 TO 100
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1799 PSUM=0
1857 FOR J44=1 TO 100
1859 PSUM=PSUM+(1/100)*( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1871 LPRINT A(1),A(2),A(3),A(4),A(5)
1872 LPRINT A(6),A(7),A(8),A(9),A(10)
1873 LPRINT A(11),A(12),A(13),A(14),A(15)
1874 LPRINT A(16),A(17),A(18),A(19),A(20)
1875 LPRINT A(21),A(22),A(23),A(24),A(25)
1876 LPRINT A(26),A(27),A(28),A(29),A(30)
1877 LPRINT A(31),A(32),A(33),A(34),A(35)
1878 LPRINT A(36),A(37),A(38),A(39),A(40)
1879 LPRINT A(41),A(42),A(43),A(44),A(45)
1880 LPRINT A(46),A(47),A(48),A(49),A(50)
1881 LPRINT A(51),A(52),A(53),A(54),A(55)
1882 LPRINT A(56),A(57),A(58),A(59),A(60)
1883 LPRINT A(61),A(62),A(63),A(64),A(65)
1884 LPRINT A(66),A(67),A(68),A(69),A(70)
1885 LPRINT A(71),A(72),A(73),A(74),A(75)
1886 LPRINT A(76),A(77),A(78),A(79),A(80)
1887 LPRINT A(81),A(82),A(83),A(84),A(85)
1888 LPRINT A(86),A(87),A(88),A(89),A(90)
1889 LPRINT A(91),A(92),A(93),A(94),A(95)
1899 LPRINT A(96),A(97),A(98),A(99),A(100),M,JJJJ,PPOBA2
1981 LPRINT AA(1),AA(2),AA(3),AA(4),AA(5)
1982 LPRINT AA(6),AA(7),AA(8),AA(9),AA(10)
1983 LPRINT AA(11),AA(12),AA(13),AA(14),AA(15)
1984 LPRINT AA(16),AA(17),AA(18),AA(19),AA(20)
1985 LPRINT AA(21),AA(22),AA(23),AA(24),AA(25)
1986 LPRINT AA(26),AA(27),AA(28),AA(29),AA(30)
1987 LPRINT AA(31),AA(32),AA(33),AA(34),AA(35)
1988 LPRINT AA(36),AA(37),AA(38),AA(39),AA(40)
1989 LPRINT AA(41),AA(42),AA(43),AA(44),AA(45)
1990 LPRINT AA(46),AA(47),AA(48),AA(49),AA(50)
1991 LPRINT AA(51),AA(52),AA(53),AA(54),AA(55)
1992 LPRINT AA(56),AA(57),AA(58),AA(59),AA(60)
1993 LPRINT AA(61),AA(62),AA(63),AA(64),AA(65)
1994 LPRINT AA(66),AA(67),AA(68),AA(69),AA(70)
1995 LPRINT AA(71),AA(72),AA(73),AA(74),AA(75)
1996 LPRINT AA(76),AA(77),AA(78),AA(79),AA(80)
1997 LPRINT AA(81),AA(82),AA(83),AA(84),AA(85)
1998 LPRINT AA(86),AA(87),AA(88),AA(89),AA(90)
1999 LPRINT AA(91),AA(92),AA(93),AA(94),AA(95)
2000 LPRINT AA(96),AA(97),AA(98),AA(99),AA(100),PSUM,JJJJ
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31999 is partly shown below. What follows is a hand copy of a part of the output on paper; immediately below there is no rounding by hand.
-2.944537 -2.916732 -2.947156 -2.752361 -2.934828
-2.93068 -2.90087 -2.854549 -2.888273 -2.640314
-2.912057 -2.950721 -2.860551 -2.891105 -2.906807
-2.875113 -2.911443 -2.916216 -2.948119 -2.951232
-2.932936 -2.95214 -2.888903 -2.932449 -2.927424
-2.904305 -2.875746 -2.892661 -2.935703 -2.862717
-2.900761 -2.866887 -2.915224 -2.86757 -2.905979
-2.941909 -2.868139 -2.924882 -2.935846 -2.896885
-2.919468 -2.872118 -2.940473 -2.893989 -2.945194
-2.893814 -2.928761 -2.919025 -2.850415 -2.887194
-2.871518 -2.872053 -2.897437 -2.877811 -2.854598
-2.908012 -2.883936 -2.885157 -2.982018 -2.927364
-2.871056 -2.8383 -2.929085 -2.916683 -2.861544
-2.941498 -2.945516 -2.939318 -2.927868 -2.889835
-2.908567 -2.87599 -2.890361 -2.93483 -2.926166
-2.915433 -2.945317 -2.900087 -2.990017 -2.895014
-2.943143 -2.946891 -2.878754 -2.891358 -2.91647
-2.889857 -2.911033 -2.860968 -2.858273 -2.862544
-2.948691 -2.93086 -2.95001 -2.933074 -2.949839
-2.93918 -2.939747 -2.669775 -2.872014 -2.925835
78.24671 -32000 78.24671
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.99994 -32000
Also, at JJJJ=-31999, the objective function values are 78.10007 and -77.7995 for the non-integer solution and the integer solution, respectively.
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was thirty seconds (with a fast printer).
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26, 645-663.
Posted at 06:54 AM | Permalink | Comments (0)
04/17/2012
Testing the Domino Method of General Integer Nonlinear Programming with a Fifty-Variable Test Problem from the Literature
Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=50 on pages 649-651 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1.
Also, it also tries to solve the corresponding integer problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 50
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 50
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*50)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 50
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 50
359 PSUM=PSUM+(1/50)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 50
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=1 TO 50
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1799 PSUM=0
1857 FOR J44=1 TO 50
1859 PSUM=PSUM+(1/50)*( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1871 PRINT A(1),A(2),A(3),A(4),A(5)
1872 PRINT A(6),A(7),A(8),A(9),A(10)
1873 PRINT A(11),A(12),A(13),A(14),A(15)
1874 PRINT A(16),A(17),A(18),A(19),A(20)
1875 PRINT A(21),A(22),A(23),A(24),A(25)
1876 PRINT A(26),A(27),A(28),A(29),A(30)
1877 PRINT A(31),A(32),A(33),A(34),A(35)
1878 PRINT A(36),A(37),A(38),A(39),A(40)
1879 PRINT A(41),A(42),A(43),A(44),A(45)
1880 PRINT A(46),A(47),A(48),A(49),A(50),M,JJJJ,PPOBA2
1881 PRINT AA(1),AA(2),AA(3),AA(4),AA(5)
1882 PRINT AA(6),AA(7),AA(8),AA(9),AA(10)
1883 PRINT AA(11),AA(12),AA(13),AA(14),AA(15)
1884 PRINT AA(16),AA(17),AA(18),AA(19),AA(20)
1885 PRINT AA(21),AA(22),AA(23),AA(24),AA(25)
1886 PRINT AA(26),AA(27),AA(28),AA(29),AA(30)
1887 PRINT AA(31),AA(32),AA(33),AA(34),AA(35)
1888 PRINT AA(36),AA(37),AA(38),AA(39),AA(40)
1889 PRINT AA(41),AA(42),AA(43),AA(44),AA(45)
1890 PRINT AA(46),AA(47),AA(48),AA(49),AA(50),PSUM,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31999 is partly shown below. What follows is a hand copy of a part of the output on the computer monitor screen; immediately below there is no rounding by hand.
-2.944537 -2.916732 -2.947156 -2.952361 -2.934828
-2.93068 -2.90087 -2.854549 -2.888273 -2.840313
-2.912057 -2.950721 -2.860551 -2.891105 -2.906807
-2.875113 -2.911443 -2.916216 -2.948119 -2.951232
-2.932936 -2.95214 -2.888903 -2.932449 -2.927424
-2.904305 -2.875746 -2.892661 -2.935703 -2.862717
-2.900761 -2.866887 -2.915224 -2.86757 -2.905979
-2.941909 -2.868139 -2.924882 -2.935846 -2.896885
-2.919468 -2.872118 -2.940473 -2.893989 -2.945194
-2.893814 -2.928761 -2.919025 -2.850415 -2.887194
78.29903 -32000 78.29903
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-78 -32000
Also, at JJJJ=-31999, the objective function values are 78.28525 and -78 for the non-integer solution and the integer solution, respectively.
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was eight seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 07:00 PM | Permalink | Comments (0)
Testing the Domino Method of General Integer Nonlinear Programming with a Thirty-Variable Test Problem from the Literature
Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=30 on pages 649-651 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1.
Also, it also tries to solve the corresponding integer problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 30
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 30
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*30)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 30
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 30
359 PSUM=PSUM+(1/30)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 30
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=1 TO 30
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1799 PSUM=0
1857 FOR J44=1 TO 30
1859 PSUM=PSUM+(1/30)*( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1871 PRINT A(1),A(2),A(3),A(4),A(5)
1872 PRINT A(6),A(7),A(8),A(9),A(10)
1873 PRINT A(11),A(12),A(13),A(14),A(15)
1874 PRINT A(16),A(17),A(18),A(19),A(20)
1875 PRINT A(21),A(22),A(23),A(24),A(25)
1876 PRINT A(26),A(27),A(28),A(29),A(30),M,JJJJ,PPOBA2
1881 PRINT AA(1),AA(2),AA(3),AA(4),AA(5)
1882 PRINT AA(6),AA(7),AA(8),AA(9),AA(10)
1883 PRINT AA(11),AA(12),AA(13),AA(14),AA(15)
1884 PRINT AA(16),AA(17),AA(18),AA(19),AA(20)
1885 PRINT AA(21),AA(22),AA(23),AA(24),AA(25)
1886 PRINT AA(26),AA(27),AA(28),AA(29),AA(30),PSUM,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31999 is shown below. What follows is a hand copy of the output on the computer monitor screen; immediately below there is no rounding by hand.
-2.944537 -2.916732 -2.947156 -2.952361 -2.934828
-2.93068 -2.90087 -2.854549 -2.888273 -2.940313
-2.912057 -2.950721 2.739449 -2.891105 -2.906807
-2.875113 -2.911443 -2.916216 -2.948119 -2.951232
-2.932946 -2.95214 -2.88903 -2.932449 -2.927424
-2.904305 -2.875746 -2.892661 -2.935703 -2.862717
77.35726 -32000 77.35726
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-76.99998 -32000
-2.921306 -2.915736 -2.874718 -2.858452 -2.862712
-2.871065 -2.920109 -2.938157 -2.925991 -2.92405
-2.88269 -2.93581 -2.92683 -2.890129 -2.928642
-2.915385 -2.885802 -2.870098 -2.868082 -2.909031
-2.918983 -2.878227 -2.870698 -2.870172 -2.878579
-2.820383 -2.93138 -2.876653 -2.943559 -2.922173
78.30014 -31999 78.30014
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-77.99998 -31999
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was three seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 11:27 AM | Permalink | Comments (0)
Testing the Domino Method of General Integer Nonlinear Programming with a Test Problem from the Literature
Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=10 on pages 649-651 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1.
Also, it also tries to solve the corresponding integer problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 10
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*10)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.1-(FIX(RND*2))*.1
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 10
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 10
359 PSUM=PSUM+(1/10)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 10
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=1 TO 10
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1799 PSUM=0
1857 FOR J44=1 TO 10
1859 PSUM=PSUM+(1/10)*( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1881 PRINT A(1),A(2),A(3),A(4),A(5)
1882 PRINT A(6),A(7),A(8),A(9),A(10),M,JJJJ,PPOBA2
1887 PRINT AA(1),AA(2),AA(3),AA(4),AA(5)
1888 PRINT AA(6),AA(7),AA(8),AA(9),AA(10),PSUM,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31999 is shown below. What follows is a hand copy of the output on the computer monitor screen; immediately below there is no rounding by hand.
-2.944537 -2.916732 -2.947156 -2.952361 -2.934828
-2.93068 -2.90087 -2.854549 -2.888273 -2.940313
78.29106 -32000 78.29106
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-78 -32000
-2.869665 -2.910866 -2.943695 -2.863411 -2.887168
-2.854703 -2.873176 -2.942019 -2.90022 -2.918919
78.29885 -31999 78.29885
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-78 -31999
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was two seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 10:28 AM | Permalink | Comments (0)
04/16/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Test Function with 500 Continuous Variables
Jsun Yui Wong
The computer program below tries to solve the function optimization problem with m=500 on page 650 of Zhang and Xu [19]. Line 181 through line 1111 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(519),H(519),L(519),U(519),X(511),D(511),P(511),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 500
74 A(J44)=-RND*5
77 NEXT J44
126 IMAR=10+FIX(RND*10000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 500
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*499))
181 J=1+FIX(RND*500)
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 500
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 500
359 PSUM=PSUM+(1/500)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 500
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1888 PRINT M,JJJJ,A(1),A(499),A(500)
1889 GOTO 1999
1901 LPRINT A(1),A(2),A(3),A(4),A(5)
1925 LPRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is shown below. Because of line 1888, printing is very limited. What follows is a hand copy of the output on the computer monitor screen; immediately below there is no rounding by hand.
77.27566 -32000 -2.894538 -2.89582 -2.873359
77.48701 -31999 -2.860751 -2.890623 -2.605364
77.33158 -31998 -2.937055 -2.90249 -2.895322
77.15294 -31997 -2.83425 -2.82314 -2.863891
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was one hour.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 10:39 AM | Permalink | Comments (0)
04/15/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Test Function of Fifty Continuous Variables and Fifty General Integer Variables
Jsun Yui Wong
The computer program below tries to solve the function optimization problem on page 50 of Chen and Chen [2]. Line 181 through line 1111 of the following computer program partly describe the problem. Line 184 is noteworthy.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(119),N(119),A(119),H(119),L(119),U(119),X(111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 100
74 A(J44)=-FIX(RND*5)
77 NEXT J44
126 IMAR=10+FIX(RND*10000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*99))
181 J=1+FIX(RND*100)
182 IF J<50.5 THEN 183 ELSE 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.01-(FIX(RND*2))*.01
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 100
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 100
359 PSUM=PSUM+(1/100)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 100
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1901 LPRINT A(1),A(2),A(3),A(4),A(5)
1902 LPRINT A(6),A(7),A(8),A(9),A(10)
1903 LPRINT A(11),A(12),A(13),A(14),A(15)
1904 LPRINT A(16),A(17),A(18),A(19),A(20)
1905 LPRINT A(21),A(22),A(23),A(24),A(25)
1906 LPRINT A(26),A(27),A(28),A(29),A(30)
1907 LPRINT A(31),A(32),A(33),A(34),A(35)
1908 LPRINT A(36),A(37),A(38),A(39),A(40)
1909 LPRINT A(41),A(42),A(43),A(44),A(45)
1910 LPRINT A(46),A(47),A(48),A(49),A(50)
1911 LPRINT A(51),A(52),A(53),A(54),A(55)
1912 LPRINT A(56),A(57),A(58),A(59),A(60)
1913 LPRINT A(61),A(62),A(63),A(64),A(65)
1914 LPRINT A(66),A(67),A(68),A(69),A(70)
1915 LPRINT A(71),A(72),A(73),A(74),A(75)
1916 LPRINT A(76),A(77),A(78),A(79),A(80)
1917 LPRINT A(81),A(82),A(83),A(84),A(85)
1918 LPRINT A(86),A(87),A(88),A(89),A(90)
1919 LPRINT A(91),A(92),A(93),A(94),A(95)
1920 LPRINT A(96),A(97),A(98),A(99),A(100)
1925 LPRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is partly shown below. What follows is a hand copy of the output on paper; immediately below there is no rounding by hand.
-2.9 -2.899998 -2.9 -2.899998 -2.900001
-2.899999 -2.900001 -2.899999 -2.91 -2.899998
-2.899998 -2.900001 -2.899998 -2.9 -2.899998
-2.899998 -2.9 -2.899998 -2.9 -2.900001
-2.899998 -2.9 -2.899998 -2.899998 -2.9
-2.899999 -2.899998 -2.9 -2.899998 -2.9
-2.9 -2.899999 -2.900001 -2.900001 -2.9
-2.9 -2.899998 -2.9 -2.9 -2.899998
-2.9 -2.899998 -2.899999 -2.899998 -2.899999
-2.900001 -2.900001 -2.899999 -2.900001 -2.9
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 3
-3 -3 -3 -3 -3
-3 -3 3 -3 -3
-3 -3 3 -3 -3
-3 -3 -3 -3 -3
77.26589 -32000 77.26589
Also, at JJJJ=-31999, JJJJ=-31998, and JJJJ=-31997, the objective function values are 75.7659, 75.55515, and 75.46589, respectively.
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was two hours and ten minutes.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 09:28 PM | Permalink | Comments (0)
04/12/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature
Jsun Yui Wong
The two computer programs below try to solve the nonlinefar fractional programming problem on page 408 of Tsai [16]. Line 181 through line 1111 of the following computer program partly describe the problem.
The first computer program below tries to identify an active constraint among the three given constraints. See Oberlin and Wright [12]. The second computer program tests this newly-identified active constraint candidate.
The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [2] and after the domino method of solving nonlinear systems of equations [19]. Line 240 through line 411 are illustrative. Domino one, domino two, and domino three are X(6), X(7), and X(8) of line 240, line 259, and line 261, respectively. These three extra variables are slack variables and are important.
0 DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
64 FOR J44=1 TO 5
65 A(J44)=.1+RND*9.899999
66 NEXT J44
126 IMAR=10+FIX(RND*32000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*4))
181 J=1+FIX(RND*5)
183 R=(1-RND*2)*A(J)
187 X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
231 FOR J44=1 TO 5
233 IF X(J44)<.1 THEN 1670
234 IF X(J44)>10 THEN 1670
235 NEXT J44
240 X(6)= 8 -X(1)-X(3) -.5*X(5)
259 X(7)= 10+2*X(1)-X(3)+X(4)
261 X(8)=2- 8/( X(1)*( X(2)+3*X(4) )^2 ) -1/X(5)^3
331 FOR J44=6 TO 8
332 IF X(J44) <0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
333 NEXT J44
403 POBA2= -(2+X(5) )/( X(1)*X(2)* (2*X(3)+X(4) ) ) +X(5)^.5 *X(3)^1.5 -2*X(2) -X(4)
411 POBA= POBA2 -999999999#*(PS(4)+PS(5)+PS(6)+PS(7)+PS(8) )
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4)
1901 PRINT A(5),A(6),A(7),A(8)
1904 PRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31999 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
.3667392144088764 .8490795798993684 5.817254363987371
.8236178701070403
3.632012843207505 2.775557561562891D-17 5.739841934937422
1.281093287008872D-15
22.76577371855824 -32000 22.76577371855824
.4056319360335867 .8095094637108889 5.73949126310706
.7820763827354627
3.709753601718179 2.636502127728591D-13 5.853848991695576
8.716127819496933D-11
22.66465394729126 -31999 22.66465394729126
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was 70 seconds.
The value of the domino variable X(6)/A(6) at JJJJ=32000 of 2.775557561562891D-17 suggests that the corresponding constraint may be active. Using line 240 below, the following computer program tests the newly-identified active constraint candidate.
0 DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
64 FOR J44=1 TO 5
65 A(J44)=.1+RND*9.899999
66 NEXT J44
126 IMAR=10+FIX(RND*32000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*4))
181 J=1+FIX(RND*5)
183 R=(1-RND*2)*A(J)
187 X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
231 FOR J44=1 TO 5
233 IF X(J44)<.1 THEN 1670
234 IF X(J44)>10 THEN 1670
235 NEXT J44
240 X(1)= 8 -X(3) -.5*X(5)
241 IF X(1)<.1 THEN 1670
242 IF X(1)>10 THEN 1670
259 X(7)= 10+2*X(1)-X(3)+X(4)
261 X(8)=2- 8/( X(1)*( X(2)+3*X(4) )^2 ) -1/X(5)^3
331 FOR J44=6 TO 8
332 IF X(J44) <0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
333 NEXT J44
403 POBA2= -(2+X(5) )/( X(1)*X(2)* (2*X(3)+X(4) ) ) +X(5)^.5 *X(3)^1.5 -2*X(2) -X(4)
411 POBA= POBA2 -999999999#*(PS(4)+PS(5)+PS(6)+PS(7)+PS(8) )
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 8
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M<0 THEN 1999
1900 PRINT A(1),A(2),A(3),A(4)
1901 PRINT A(5),A(6),A(7),A(8)
1904 PRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31999 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
.2967260368006816 .9197208007985542 5.948885624822289
.9244289306182739
3.508776676754059 0 5.568995379397348
3.690624195140657D-15
22.84071669494471 -32000 22.84071669494471
.295554001172341 .9234059681098373 5.938441088002545
.9254973773086692
3.532009821650228 0 5.578164291650806
4.395789288125229D-15
22.84130658903192 -31999 22.84130658903192
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31999 was 80 seconds.
References
[1] M. Avriel, J. D. Barrette, Optimal Design of Pitched Laminated Wood Beams. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 291-203.
[2] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Jing-Fan Fu, Robert G. Fenton, William L. Cleghorn (1991): A Mixed Integer-Discrete-Continuous Programming Method and its Applicaion to Engineering Design Optimization, Engineering Optimization, 17:4. 263-280.
[6] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Chih-Tan Chou (1993): A Global Approach for Nonlinear Mixed Discrete Programming in Design Optimization, Engineering Optimization, 22:2, 109-122.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Tapabrata Ray, Pankaj Saini (2001), Engineering Design Optimization Using a Swarm with an Intelligent Information Sharing among Individuals, Engineering Optimization, 33:6, 735-748.
[14] M. J. Rijckaert, X. M. Martens, Comparison of Generalized Geometric Programming Algorithms. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 205-242.
[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[16] Jung-Fa Tsai (2005): Global Optimization of Nonlinear Fractional Programming Problems in Engineering Design, Engineering Optimization, 37:4, 399-409.
[17] Jung-Fa Tsai (2010): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 42:9, 833-843.
[18] S. Wang, K. L. Teo, H. W. J. Lee( 1998): A New Approach to Nonlineaar Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[19] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
Posted at 11:15 AM | Permalink | Comments (0)
04/11/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem, Part Two
Jsun Yui Wong
.3880204451466907 5.295802113661627D-02 9.664888997389421
1.059021533716693 1.040834085586084D-17 13.53198241450875
0
-1.269400652463785D-02 -31997 -1.269400652463785D-02
The zero of the candidate solution above from part one suggests that the constraint corresponding to line 264 of part one may be tight. Hence the following change has been made: While line 264 of part one is 264 X(7)=-1+( X(1)^3*X(3) ) / (71785!*X(2)^4 ), line 238 of the following computer program is 238 X(3)= (71785!*X(2)^4 ) /X(1)^3.
0 DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
65 A(1)=.25+RND*1.05
66 A(2)=.05+RND*1.95
67 A(3)=2+RND*13
126 IMAR=10+FIX(RND*32000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*2))
181 J=1+FIX(RND*3)
183 R=(1-RND*2)*A(J)
187 X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
223 IF X(1)<.25 THEN 1670
224 IF X(1)>1.3 THEN 1670
225 IF X(2)<.05 THEN 1670
226 IF X(2)>2 THEN 1670
229 IF X(3)<2 THEN 1670
230 IF X(3)>15 THEN 1670
237 X(4)=1.5-X(1)-X(2)
238 X(3)= (71785!*X(2)^4 ) /X(1)^3
259 X(5)=1- (4*X(1)^2-X(1)*X(2) ) /( 12566*( X(1)*X(2)^3 -X(2)^4 ) ) -1/(5108*X(2)^2 )
261 X(6)=-1+(140.45 *X(2) )/ ( X(2)* X(3) )
264 REM X(7)=-1+( X(1)^3*X(3) ) / (71785!*X(2)^4 )
331 FOR J44=4 TO 7
332 IF X(J44) <0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
333 NEXT J44
403 POBA2= -(X(3) +2 )* X(1)*X(2)^2
411 POBA= POBA2 -999999999#*(PS(4)+PS(5)+PS(6)+PS(7) )
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.01306 THEN 1999
1900 PRINT A(1),A(2),A(3)
1901 PRINT A(4),A(5),A(6),A(7)
1904 PRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31661 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
.3501404240422991 5.141416580111879D-02 11.68525299255173
1.098445410156582 8.673617379884036D-18 11.01942286040072
0
-1.266661794046643D-02 -31784 -1.266661794046643D-02
.3835300477759826 5.277981050611188D-02 9.874311593366765
1.063690141717906 5.204170427930421D-18 13.2237760698773
0
-1.268654864042479D-02 -31694 -1.268654864042479D-02
.3576182588974256 5.172646550625888D-02 11.23636573996081
1.090655275596316 2.081668171172169D-17 11.49959285756856
0
-1.266525828988787D-02 -31661 -1.266525828988787D-02
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31661 was seven minutes.
References
[1] M. Avriel, J. D. Barrette, Optimal Design of Pitched Laminated Wood Beams. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 291-203.
[2] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Jing-Fan Fu, Robert G. Fenton, William L. Cleghorn (1991): A Mixed Integer-Discrete-Continuous Programming Method and its Applicaion to Engineering Design Optimization, Engineering Optimization, 17:4. 263-280.
[6] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Chih-Tan Chou (1993): A Global Approach for Nonlinear Mixed Discrete Programming in Design Optimization, Engineering Optimization, 22:2, 109-122.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Tapabrata Ray, Pankaj Saini (2001), Engineering Design Optimization Using a Swarm with an Intelligent Information Sharing among Individuals, Engineering Optimization, 33:6, 735-748.
[13] M. J. Rijckaert, X. M. Martens, Comparison of Generalized Geometric Programming Algorithms. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 205-242.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Jung-Fa Tsai (2005): Global Optimization of Nonlinear Fractional Programming Problems in Engineering Design, Engineering Optimization, 37:4, 399-409.
[16] Jung-Fa Tsai (2010): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 42:9, 833-843.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlineaar Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
Posted at 11:58 AM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem
Jsun Yui Wong
The following computer program tries to solve the spring design problem considered by Ray and Saini [12, pp. 745-746] and Tsai [15, pp. 405-407], among others. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program is modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [2] and after the domino method of solving nonlinear systems of equations [16]. Line 240 through line 411 are illustrative. Domino one through domino four are X(4), X(5), X(6), and X(7) of line 240 through line 264, respectively. These four extra variables are slack variables and are important.
0 DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(111),PS(33)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
65 A(1)=.25+RND*1.05
66 A(2)=.05+RND*1.95
67 A(3)=2+RND*13
126 IMAR=10+FIX(RND*32000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 3
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO (1+FIX(RND*2))
181 J=1+FIX(RND*3)
183 R=(1-RND*2)*A(J)
187 X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
223 IF X(1)<.25 THEN 1670
224 IF X(1)>1.3 THEN 1670
225 IF X(2)<.05 THEN 1670
226 IF X(2)>2 THEN 1670
229 IF X(3)<2 THEN 1670
230 IF X(3)>15 THEN 1670
240 X(4)=1.5-X(1)-X(2)
259 X(5)=1- (4*X(1)^2-X(1)*X(2) ) /( 12566*( X(1)*X(2)^3 -X(2)^4 ) ) -1/(5108*X(2)^2 )
261 X(6)=-1+(140.45 *X(2) )/ ( X(2)* X(3) )
264 X(7)=-1+( X(1)^3*X(3) ) / (71785!*X(2)^4 )
331 FOR J44=4 TO 7
332 IF X(J44) <0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
333 NEXT J44
403 POBA2= -(X(3) +2 )* X(1)*X(2)^2
411 POBA= POBA2 -999999999#*(PS(4)+PS(5)+PS(6)+PS(7) )
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.01306 THEN 1999
1900 PRINT A(1),A(2),A(3)
1901 PRINT A(4),A(5),A(6),A(7)
1904 PRINT M,JJJJ,PPOBA2
1999 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31991 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
.3880204451466907 5.295802113661627D-02 9.664888997389421
1.059021533716693 1.040834085586084D-17 13.53198241450875
0
-1.269400652463785D-02 -31997 -1.269400652463785D-02
.4720346644787649 5.608676742895001D-02 6.753885080783585
.9718785680922851 9.367506770274758D-17 19.79543777667404
0
-1.299856930149464D-02 -31991 -1.299856930149464D-02
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31991 was four minutes.
References
[1] M. Avriel, J. D. Barrette, Optimal Design of Pitched Laminated Wood Beams. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 291-203.
[2] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] Jing-Fan Fu, Robert G. Fenton, William L. Cleghorn (1991): A Mixed Integer-Discrete-Continuous Programming Method and its Applicaion to Engineering Design Optimization, Engineering Optimization, 17:4. 263-280.
[6] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Chih-Tan Chou (1993): A Global Approach for Nonlinear Mixed Discrete Programming in Design Optimization, Engineering Optimization, 22:2, 109-122.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[12] Tapabrata Ray, Pankaj Saini (2001), Engineering Design Optimization Using a Swarm with an Intelligent Information Sharing among Individuals, Engineering Optimization, 33:6, 735-748.
[13] M. J. Rijckaert, X. M. Martens, Comparison of Generalized Geometric Programming Algorithms. Journal of Optimization Theory and Applications, Vol. 26, Number 2 (October 1978), pp. 205-242.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Jung-Fa Tsai (2005): Global Optimization of Nonlinear Fractional Programming Problems in Engineering Design, Engineering Optimization, 37:4, 399-409.
[16] Jung-Fa Tsai (2010): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 42:9, 833-843.
[17] S. Wang, K. L. Teo, H. W. J. Lee( 1998): A New Approach to Nonlineaar Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
Posted at 10:21 AM | Permalink | Comments (0)
