Substance

A Computer Program about Markowitz Portfolio Selection, Second Edition

A Computer Program about Markowitz Portfolio Selection, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the small example about Markowitz portfolio selection [3] on pp. 888-890 of Li and Tsai [2].

Minimize    .0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3)

subject to    1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3) >= 115

X(1) + X(2) + X(3) = 100

where X(1), X(2), and X(3) are integers.

Below X(1) through X(3) are integer variables, and X(4) is a slack variable.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 10000, here line 128 is 128 FOR I = 1 TO 8000.

0 REM 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 3
65 A(J44) = RND * 10

66 NEXT J44
126 REM IMAR=10+FIX(RND*32000)
128 FOR I = 1 TO 8000

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
222 NEXT IPP

223 X(1) = 100 - X(2) - X(3)

224 FOR J33 = 1 TO 3

226 X(J33) = INT(X(J33))

227 NEXT J33

229 REM X(5) = 1
233 X(4) = -115 + 1.089 * X(1) + 1.214 * X(2) + 1.235 * X(3)

281 FOR J44 = 1 TO 3

283 IF X(J44) < 0 THEN 1670

284 IF X(J44) > 100 THEN 1670

285 NEXT J44

331 FOR J44 = 4 TO 4

332 IF X(J44) < 0 THEN PS(J44) = ABS(X(J44)) ELSE PS(J44) = 0
333 NEXT J44
403 POBA2 = -(.0108 * X(1) ^ 2 + .0584 * X(2) ^ 2 + .0942 * X(3) ^ 2 + .0248 * X(1) * X(2) + .0262 * X(1) * X(3) + .1108 * X(2) * X(3))

411 POBA = POBA2 - 1000000 * (PS(4))

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 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -224.647 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1904 PRINT M, JJJJ, PPOBA2
1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [5]. The complete output through JJJJ=-16563 is shown below:

53       35       12       .027
-224.6452       -29251       -224.6452

53       36       11       .006
-223.8916       -19156       -223.8916

53       35       12       .027
-224.6452       -18228       -224.6452

53       36       11       .006
-223.8916       -16563       -223.8916

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [5], the wall-clock time for obtaining the output through JJJJ=-16563 was 3 minutes and 20 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.

[2] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp.882-891.

[3] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[4] 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.

[5] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[6] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/

Using the general-purpose nonlinear programming solver used in this blog many times for over a decade instead of a dynamic programming approach used in Ecker and Kupferschmid [29], second edition

Using the general-purpose nonlinear programming solver used in this blog many times for over a decade instead of a dynamic programming approach used in Ecker and Kupferschmid [29], second edition   

Jsun Yui Wong

A loading problem  is presented in Ecker and Kupferschmid [29; please see p. 353].  Their problem is briefly summarized as follows:

           weight         value

           In tons         in thousands of $

k            w(k)            v(k)

1               2                 4

2               8               10

3               3                 6

4               4                 8

And the ship has a total weight capacity of ten tons.

Maximize    v(1)x(1)+…+v(4)x(4)

subject  to

w(1)x(1)+…+w(4)x(4)<=10

x(k)>=0,    x(k) integer k = 1,…, 4,

where x(k) = number of crates of product k selected.

“Thus, the branch-and-bound method of Chapter 8 could be used to solve this problem.  An alternate approach is to use dynamic programming,” Ecker and Kupferschmid [29, p. 353].

Because there is a chance that the long constraint above is binding, it is worthy to explore line 244, which is 244 X(1) = (-(8 * X(2) + 3 * X(3) + 4 * X(4)) + 10) / 2.

Instead of the dynamic programming approach used in Ecker and Kupferchmid [29, pp. 353-356], the following computer program, which is modeled after the general-purpose nonlinear programming solver used in this blog many times for over a decade, attempts to solve this loading problem in Ecker and Kupferschmid [29, p. 353].

 0 REM  DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ

    86 M = -3E+50

    118 FOR J44 = 1 TO 4

        119 A(J44) = 0 + (RND * 10)

    120 NEXT J44

    128 FOR I = 1 TO 3000

        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        151 FOR IPP = 1 TO FIX(1 + RND * 5)

            153 J = 1 + FIX(RND * 4)

            154 IF J < 5 THEN GOTO 156 ELSE GOTO 156

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 GOTO 169

            163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)

            164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

        169 NEXT IPP

        172 X(1) = INT(X(1))

        174 X(2) = INT(X(2))

        176 X(3) = INT(X(3))

        178 X(4) = INT(X(4))

        239 REM    IF 2 * X(1) + 8 * X(2) + 3 * X(3) + 4 * X(4) > 10 THEN 1670

        244 X(1) = (-(8 * X(2) + 3 * X(3) + 4 * X(4)) + 10) / 2

        249 X(1) = INT(X(1))

        326 FOR J44 = 1 TO 4

            328 IF X(J44) < 0 THEN 1670

        330 NEXT J44

        471 P = 4 * X(1) + 10 * X(2) + 6 * X(3) + 8 * X(4)

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

        1557 GOTO 128

    1670 NEXT I

    1889 IF M < 18 THEN 1999

    1933 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

 

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ=-31972  is shown below:

5      0      0      0      20

-31998

2      0      2      0      20

-31994

2      0      2      0      20

-31991

1      0      0      2      20

-31987

3      0      0      1      20

-31984

3      0      0      1      20

-31982

2      0      2      0      20

-31979

5      0      0      0      20

-31978

0      0      2      1      20

-31972

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  Five distinct solutions are shown above, same as the five solutions in Ecker and Kupferschmid [29, p. 356].  By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31972 was 2 seconds, counting from "Starting program...".   One can compare the computational procedure above with that in Ecker and Kupferschmid [29, pp. 353-356]. 

The computational results presented above were obtained from the following computer system: 

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz   2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

 

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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[90] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020).  Global optimization for mixed-discrete structural design.  Symmetry 2020, 12, 1529

One can get a Google view of this article.  www.mdpi.com/journal/symmetry.

[91] Jung-Fa Tsai, Ming-Hua Lin, Lei-Yi Peng (2020).  Finding all global optima of engineering design problems with discrete signomial terms.  Engineering Optimization, 2020, vol. 52, no. 1, 165-184.

[92] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[93] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990) 34:325-334.

[94] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

[95] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

[96] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[97] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

[98] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[99] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems ZOR – Methods and Models of Operations Research (1990): 34:325-334.

[100] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (101). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

[101] Eric W. Weisstein, "Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[102] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[103] Wayne L. Winston, (1987), Operations Research:  Applications and Algorithms, PWS Publishers, Boston, Massachusettes 02116.

[104] Helen Wu, (2015), Geometric Programming

https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan.  Signomial programs with equality constraints: numerical solution and applications.  Ph. D. thesis. University of British Columbia, 1976.

One can easily view this dissertation on Google.

Using the general-purpose nonlinear programming solver used in this blog many times for over a decade instead of a dynamic programming approach used in Ecker and Kupferschmid [29]

Using the general-purpose nonlinear programming solver used in this blog many times for over a decade instead of a dynamic programming approach used in Ecker and Kupferschmid [29]   

Jsun Yui Wong

A loading problem  is presented in Ecker and Kupferschmid [29; please see p. 353].  Their problem is briefly summarized as follows:

           weight         value

           in tons         in thousands of $

k            w(k)            v(k)

1               2                 4

2               8               10

3               3                 6

4               4                 8

And the ship has a total weight capacity of ten tons.

Maximize    v(1)x(1)+…+v(4)x(4)

subject  to

w(1)x(1)+,,,+w(4)x(4)<=10

x(k)>=0,    x(k) integer k = 1,…, 4,

where x(k) = number of crates of product k selected.

“Thus, the branch-and-bound method of Chapter 8 could be used to solve this problem, an alternate approach is to use dynamic programming,” Ecker and Kupferschmid [29, p. 353].

Instead of the dynamic programming approach used in Ecker and Kupfchmid [29, pp.353-356], the following computer program, modeled after the general-purpose nonlinear programming solver used in this blog many times for over a decade, attempts to solve this loading problem from Ecker and Kupferschmid [29, p. 353].    

0 REM  DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ

    86 M = -3E+50

    118 FOR J44 = 1 TO 4

        119 A(J44) = 0 + (RND * 10)

    120 NEXT J44

    128 FOR I = 1 TO 3000

        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        151 FOR IPP = 1 TO FIX(1 + RND * 5)

            153 J = 1 + FIX(RND * 4)

            154 IF J < 5 THEN GOTO 156 ELSE GOTO 156

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 GOTO 169

            163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)

            164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

        169 NEXT IPP

        172 X(1) = INT(X(1))

        174 X(2) = INT(X(2))

        176 X(3) = INT(X(3))

        178 X(4) = INT(X(4))

        239 IF 2 * X(1) + 8 * X(2) + 3 * X(3) + 4 * X(4) > 10 THEN 1670

        326 FOR J44 = 1 TO 4

            328 IF X(J44) < 0 THEN 1670

        330 NEXT J44

        471 P = 4 * X(1) + 10 * X(2) + 6 * X(3) + 8 * X(4)

        472

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

        1557 GOTO 128

    1670 NEXT I

    1889 IF M < 18 THEN 1999

    1933 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its complete output of one run through JJJJ= -30831 is shown below:

5      0      0      0      20

-31998

0      0      3      0      18

-31950

3      0      0      1      20

-31948

0      0      3      0      18

-31943

0      0      2      1      20

-31926

.

.

.

2      0      2      0      20

-31816

.

.

.

1      0      0      2      20

-30831

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -30831 was  10 seconds, counting from "Starting program...".   One can compare the computational procedure above with that in Ecker and Kupferschmid [29, pp. 353-356]. 

The computational results presented above were obtained from the following computer system: 

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz   2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming

[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.

[106] James Yan.  Signomial programs with equality constraints: numerical solution and applications.  Ph. D. thesis. University of British Columbia, 1976.

One can easily view this dissertation on Google.

Solving a nonlinear programming problem from the literature: another test problem in Visweswaran and Floudas [101]

Solving a nonlinear programming problem from the literature: another test problem in Visweswaran and Floudas [101]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following test problem from Visweswaran and Floudas [101, pp. 1431-1432, Example 8].

Minimize

-12 * X(1) - 7 * X(2) + X(2) ^ 2

subject to

-2 * X(1) ^ 4 + 2 - X(2) =0
0 <= X(1) <= 2
0 <= X(2) <= 3.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
111 A(1) = 0 + RND * 2
112 A(2) = 0 + RND * 3

128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

141 B = 1 + FIX(RND * 2)

142 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

175 REM FOR J44 = 1 TO 6

177 REM X(J44) = INT(X(J44))
179 REM NEXT J44

255 X(2) = -2 * X(1) ^ 4 + 2

453 IF X(1) < 0 THEN 1670

454 IF X(1) > 2 THEN 1670
473 IF X(2) < 0 THEN 1670

484 IF X(2) > 3 THEN 1670

922 PD1 = 12 * X(1) + 7 * X(2) - X(2) ^ 2

999 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -999999 THEN 1999
1888 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31995 is shown below:

.7175208          1.469524          16.73889          -32000

.7176438 1.469524 16.73889 -31999

.7176516 1.469501 16.73889 -31998

.7175084 1.469524 16.73889 -31997

.7176572 1.469524 16.73889 -31996

.7177044 1.469524 16.73889 -31995

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 2 seconds, counting from "Starting program...". One can compare the computational results above with those in Visweswaran and Floudas [101, pp. 1431-1432, Example 8].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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One can easily view this dissertation on Google.

Solving a nonlinear programming problem in the literature: an illustration with a test problem from Visweswaran and Floudas [101]

Solving a nonlinear programming problem in the literature: an illustration with a test problem from Visweswaran and Floudas [101]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following test problem from Visweswaran and Floudas [101, p. 1431, Example 7].

Minimize
(1 / 6) * X(1) ^ 6 - (52 / 25) * X(1) ^ 5 + (39 / 80) * X(1) ^ 4 + (71 / 10) * X(1) ^ 3 - (79 / 20) * X(1) ^ 2 - X(1) + 1 / 10

subject to
-2 <= X(1) <= 11.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
99 FOR J44 = 1 TO 1
108 A(J44) = FIX(-2 + RND * 14)
114 NEXT J44
128 FOR i = 1 TO 10000
129 FOR KKQQ = 1 TO 1
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * .4)
141 B = 1 + FIX(RND * .4)

142 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

411 FOR J44 = 1 TO 1
433 IF X(J44) < -2 THEN 1670

434 IF X(J44) > 11 THEN 1670

436 NEXT J44

974 PD1 = -((1 / 6) * X(1) ^ 6 - (52 / 25) * X(1) ^ 5 + (39 / 80) * X(1) ^ 4 + (71 / 10) * X(1) ^ 3 - (79 / 20) * X(1) ^ 2 - X(1) + 1 / 10)

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 1
1455 A(KLX) = X(KLX)

1456 NEXT KLX
1670 NEXT i
1778 IF M < -999999999 THEN 1999
1888 PRINT A(1), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31990 is shown below:

9.999958          29763.23          -32000

0 -.1 -31999

9.99967 29763.23 -31998

9.999469 29763.23 -31997

9.999559 29763.23 -31996

.5000096 .5319271 -31995

9.999383 29763.23 -31994

9.999378 29763.23 -31993

.4999892 .5319271 -31992

9.999587 29763.23 -31991

10.00023 29763.23 -31990

 

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Visweswaran and Floudas [101, p. 1431, Example 7].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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A direct way to solve bilinear programming problems: an illustration from Visweswaran and Floudas [101]

A direct way to solve bilinear programming problems: an illustration from Visweswaran and Floudas [101]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following test problem (bilinear) from Visweswaran and Floudas [101, pp. 1420-1422, Example 1; H. Konno is an early bilinear programming researcher].

Minimize

-1* ( -X(1) + X(2) + X(3) + (X(1) - X(2)) * (X(3) - X(4)) )

subject to

X(1) + 4 * X(2) <= 8
4 * X(1) + X(2) <= 12
3 * X(1) + 4 * X(2) <= 12
2 * X(3) + X(4) <= 8
X(3) + 2 * X(4) <= 8
X(3) + 2 * X(4) <= 5
X(1), X(2), X(3), X(4) >=0.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
99 FOR J44 = 1 TO 4

108 A(J44) = 0 + RND * 10

114 NEXT J44

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 4)

141 B = 1 + FIX(RND * 4)

142 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

175 REM FOR J44 = 1 TO 4

177 REM X(J44) = INT(X(J44))
179 REM NEXT J44

255 IF X(1) + 4 * X(2) > 8 THEN 1670
259 IF 4 * X(1) + X(2) > 12 THEN 1670

265 IF 3 * X(1) + 4 * X(2) > 12 THEN 1670

269 IF 2 * X(3) + X(4) > 8 THEN 1670

275 IF X(3) + 2 * X(4) > 8 THEN 1670
279 IF X(3) + 2 * X(4) > 5 THEN 1670

451 FOR J44 = 1 TO 4

453 IF X(J44) < 0 THEN 1670

454 IF X(J44) > 10 THEN 1670

466 NEXT J44

922 PD1 = -X(1) + X(2) + X(3) + (X(1) - X(2)) * (X(3) - X(4))
999 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 4

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < 10 THEN 1999

1888 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output of one run through JJJJ= -31981 is shown below:

 

3          6.782655E-08          4          4.163211E-08
13          -31998

3 1.089493E-07 4 2.050244E-07
13 -31988

3 1.031992E-07 4 7.546745E-08
13 -31983

3 1.083157E-07 4 4.010428E-08
13 -31982

3 1.920373E-08 4 1.372317E-07
13 -31981

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31981 was 4 seconds, counting from "Starting program...". One can compare the computational results above with those in Visweswaran and Floudas [101, pp. 1420-1422].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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http://bazziahmad.com/

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A computer program to solve geometric programming problems: an illustration from Rijckaert and Martens [76, Problem 11]

A computer program to solve geometric programming problems: an illustration from Rijckaert and Martens [76, Problem 11]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following nonlinear programming problem from Rijckaert and Martens [76, p. 231, Problem 11]:

Minimize

• 
  - X(1) + .4 * X(1) ^ .67 * X(3) ^ -.67
• 
  subject to

.05882 * X(3) * X(4) + .1 * X(1) <= 1

4 * X(2) * X(4) ^ -1 + 2 * X(2) ^ -.71 * X(4) ^ -1 + .05582 * X(2) ^ -1.3 * X(3) <= 1.

 

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
99 FOR J44 = 1 TO 4

108 A(J44) = 0 + RND * 20

114 NEXT J44
128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 4)

141 B = 1 + FIX(RND * 4)

142 REM IF RND < .5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

229 X(1) = (1 - .05882 * X(3) * X(4)) / .1

236 IF 4 * X(2) * X(4) ^ -1 + 2 * X(2) ^ -.71 * X(4) ^ -1 + .05582 * X(2) ^ -1.3 * X(3) > 1 THEN 1670

411 FOR J44 = 1 TO 4
433 IF X(J44) < 0 THEN 1670

434 IF X(J44) > 20 THEN 1670

436 NEXT J44
972 PD1 = X(1) - .4 * X(1) ^ .67 * X(3) ^ -.67

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 4

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < 5.7222 THEN 1999
1888 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ

 

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31992 is shown below:

 

8.173226 .6131424 .553552 5.610498 5.744114
-31999

8.103439 .6128766 .5734213 5.622999 5.74464
-31994

8.112604        .6094353        .5708106        5.62142        5.74479
-31993

8.073374 .6098516 .5819437 5.628485 5.743585
-31992

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31992 was 2 seconds, counting from "Starting program...". One can compare the computational results above with those in R and Martens [76, p. 231, Problem 11].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71], Second and Easier Solution

A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71], Second and Easier Solution

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following classical bilinear problem in Porn et al. [71, p.111]:
Minimize
X(1) * X(2) - X(1) - X(2) )
subject to
-6 * X(1) + 8 * X(2) <= 3
3 * X(1) - X(2) <= 3
0 <= X(1), X(2) <= 1.5
X(1), X(2) epsilon R.
Unlike the earlier edition, the present edition does not need one to specify the active constraint; see line 233 and line 266, which are 233 IF -6 * X(1) + 8 * X(2) > 3 THEN 1670 and 236 IF 3 * X(1) - X(2) > 3 THEN 1670.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO -31990

87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 0 + RND * 1.5

111 A(2) = 0 + RND * 1.5

128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

141 B = 1 + FIX(RND * 2)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

226 REM X(2) = (3 * X(1) - 3)

233 IF -6 * X(1) + 8 * X(2) > 3 THEN 1670

236 IF 3 * X(1) - X(2) > 3 THEN 1670

433 IF X(1) < 0 THEN 1670

434 IF X(1) > 1.5 THEN 1670

437 IF X(2) < 0 THEN 1670

439 IF X(2) > 1.5 THEN 1670

972 PD1 = -X(1) * X(2) + X(1) + X(2)

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -999999999 THEN 1999
1888 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31990 is shown below:

1.166129       .4983885       1.083333       -32000
1.158768 .4763027 1.083146 -31999
1.176475 .5294256 1.083045 -31998
1.144613 .4338305 1.081874 -31997
1.146815 .4404464 1.082151 -31996
1.155732 .4671971 1.082975 -31995
1.18116 .5434815 1.082703 -31994
1.176988 .5309641 1.083014 -31993
1.164246 .4927368 1.083316 -31992
1.167029 .5010881 1.083333 -31991
1.167616 .5028473 1.083331 -31990

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Porn, Bjork, and Westerlund [71, p. 112].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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A computer program to solve nonconvex mixed-integer nonlinear programming problems: an illustration from Lundell, Skjal, and Westerlund [58]

A computer program to solve nonconvex mixed-integer nonlinear programming problems: an illustration from Lundell, Skjal, and Westerlund [58]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below attempts to solve the following mixed-integer nonlinear programming (MINLP) problem from Lundell, Skjal, and Westerlund [58, p. 136]:

Minimize

(2 * X(1) - 4) ^ 2 + (X(2) - 13 / 2) ^ 2

subject to

X(1) * COS(X(2)) ^ 2 + X(2) * SIN(X(1)) ^ 2 - 3 / X(2) + X(1) / 2 - 5 / 2 <= 0

2<= X(1) <= 4
2 <= X(2) <= 8
X(1) epsilon R
X(2) epsilon Z.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 2 + RND * 2

111 A(2) = 2 + FIX(RND * 7)

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

141 B = 1 + FIX(RND * 2)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

172 X(2) = INT(X(2))

246 IF X(1) * COS(X(2)) ^ 2 + X(2) * SIN(X(1)) ^ 2 - 3 / X(2) + X(1) / 2 - 5 / 2 > 0 THEN 1670

433 IF X(1) < 2 THEN 1670

434 IF X(1) > 4 THEN 1670

437 IF X(2) < 2 THEN 1670

439 IF X(2) > 8 THEN 1670

974 PD1 = -(2 * X(1) - 4) ^ 2 - (X(2) - 13 / 2) ^ 2

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -999999999 THEN 1999
1888 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31990 is shown below:

2.534119       5       -3.391133       -32000
2.534119 5 -3.391133 -31999
2.000422 2 -20.25 -31998
2.000283 2 -20.25 -31997
2.000102 2 -20.25 -31996
2.534119 5 -3.391133 -31995
2.000348 2 -20.25 -31994
2.534119 5 -3.391133 -31993
2.000318 2 -20.25 -31992
2.534119 5 -3.391133 -31991
2.534119 5 -3.391133 -31990

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Lundell, Skjal, and Westerlund [58, p. 137, Table 1].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[70] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[71] O. Perez, I. Amaya, R. Correa (2013), Numerical solution of certain exponential and nonlinear diophantine systems of equations by using a discrete particles swarm optimization algorithm. Applied Mathematics and Computation, Volume 225, 1 December, 2013, pp. 737-746.

[72] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation and Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.

[73] Rajgopal, Geometric Programming. https://sites.pitt.edu/~jrclass/notes6.pdf

[74] Singiresu S. Rao, Engineering Optimization, Fifth Edition, Wiley, New York, 2020.

[75] John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.

[76] M. J. Rijckaert, X. M. Martens, Comparison of generalized geometric programming algorithms, J. of Optimization, Theory and Applications, 26 (2) 205-242 (1978).

[77] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[78] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[79] Shafiullah, Irfan Ali, Abdul Bari (2015). Fuzzy geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp. 021-035 (2015).

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[81] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[82] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.

[83] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes’ Method of Multipliers to Resolve Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

[84] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. Journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Jung-Fa Tsai, Ming-Hua Lin (2006). An optimization approach for solving signomial discrete programming problems with free variables. Computers and Chemical Engineering 30 (2006) 1256-1263.

[88] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) 10-19.

[89] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming problems with free variables. Journal of Global Optimization (2008) 42:39-49.

[90] Jung-Fa Tsai (2009). Treatng free variables in generalized geometric programming problems. Computers and Chemical Enginering 33 (2009) 239-243.

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[103] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002.

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A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71]

A computer program to solve bilinear problems: an illustration using a classical bilinear problem in Porn, Bjork, and Westerlund [71]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following classical bilinear problem in Porn et al. [71, p. 111]:
Minimize
X(1) * X(2) - X(1) - X(2) )
subject to
-6 * X(1) + 8 * X(2) <= 3
3 * X(1) - X(2) <= 3
0 <= X(1), X(2) <= 1.5
X(1), X(2) epsilon R.

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 0 + RND * 1.5

111 A(2) = 0 + RND * 1.5

128 FOR i = 1 TO 10000

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 2)

141 B = 1 + FIX(RND * 2)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

172 REM
195 REM
197 REM
222 REM
226 X(2) = (3 * X(1) - 3)

233 IF -6 * X(1) + 8 * X(2) > 3 THEN 1670

433 IF X(1) < 0 THEN 1670

434 IF X(1) > 1.5 THEN 1670

437 IF X(2) < 0 THEN 1670

439 IF X(2) > 1.5 THEN 1670

972 PD1 = -X(1) * X(2) + X(1) + X(2)

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 2

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -999999999 THEN 1999
1888 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31993 is shown below:

1.166651       .4999523       1.083333       -32000

1.166767 .5003003 1.083333 -31998

1.166744 .5002327 1.083333 -31996

1.166796 .5003872 1.083333 -31995

1.166441 .499324 1.083333 -31994

1.166796 .5003886 1.083333 -31993

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31993 was 3 seconds, counting from "Starting program...". One can compare the computational results above with those in Porn, Bjork, and Westerlund [71, p. 112].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[92] Jung-Fa Tsai, Ming-Hua Lin, Duan-Yi Wen (16 September 2020). Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529
One can get a Google view of this article. www.mdpi.com/journal/symmetry.

[93] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

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[96] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

[97] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[98] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

[99] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

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[104] Wayne L. Winston, (2004), Operations Research--Applications and Algorithms, Fourth Edition, Brooks/Cole--Thomson Learning, Belmont, California 94002.

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One can easily view this dissertation on Google.

A computer program to solve mixed-integer nonlinear programming (MINLP) problems: an illustration using Example 3 in Lundell and Westerlund [58]

A computer program to solve mixed-integer nonlinear programming (MINLP) problems: an illustration using Example 3 in Lundell and Westerlund [58]

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve the following noncovex mixed-integer nonlinear programming (MINLP) example in Lundell and Westerlund [58, p. 129, Example 3]:

Minimize
(X(1) - 2) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 4.5) ^ 2

subject to
X(1) ^ 2 * X(2) - X(2) ^ -1 + X(2) ^ 2 * X(3) + 5 * SIN(X(2)) * COS(X(3)) <= 12
1 <= X(1), X(2) <= 3
1 <= X(3) <= 8
X(1), X(2) EPSILON R, X(3) EPSILON Z.

 

0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 32000
87 RANDOMIZE JJJJ
88 M = -3D+30
108 A(1) = 1 + RND * 2

111 A(2) = 1 + RND * 2

115 A(3) = 1 + FIX(RND * 8)

128 FOR i = 1 TO 50000

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 3)

141 B = 1 + FIX(RND * 3)

142 IF RND < 1.5 THEN 160 ELSE GOTO 167

160 r = (1 - RND * 2) * A(B)

162 X(B) = A(B) + (RND ^ (RND * 15)) * r
164 GOTO 168

167 REM IF RND < .5 THEN X(B) = A(B) - 1 ELSE X(B) = A(B) + 1

168 NEXT IPP

172 X(3) = INT(X(3))

195 IF ((X(2) ^ -1 - X(2) ^ 2 * X(3) - 5 * SIN(X(2)) * COS(X(3)) + 12) / X(2)) < 0 THEN 1670

197 X(1) = ((X(2) ^ -1 - X(2) ^ 2 * X(3) - 5 * SIN(X(2)) * COS(X(3)) + 12) / X(2)) ^ .5

433 IF X(1) < 1 THEN 1670

434 IF X(1) > 3 THEN 1670

437 IF X(2) < 1 THEN 1670

439 IF X(2) > 3 THEN 1670
444 IF X(3) < 1 THEN 1670

446 IF X(3) > 8 THEN 1670

972 PD1 = -(X(1) - 2) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 4.5) ^ 2

979 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 3

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT i

1778 IF M < -10 THEN 1999
1888 PRINT A(1), A(2), A(3), M, JJJJ
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [103]. Its complete output of one run through JJJJ= -31990 is shown below:

1.860231       1.606857       4       -.4240965       -31998
1.860309 1.60683 4 -.4240965 -31997
1.860341 1.606818 4 -.4240965 -31995
1.860216 1.606863 4 -.4240965 -31994
1.86032 1.606826 4 -.4240965 -31991
1.860316 1.606827 4 -.4240965 -31990

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31990 was 4 seconds, counting from "Starting program...". One can compare the computational results above with those in Lundell and Westerlund [58, p. 130, Example 3].

The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.

Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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One can easily view this dissertation on Google.