Searching for Multiple Solutions of a Signomial Integer/Discrete Programming Problem

Jsun Yui Wong

The computer program listed below aims to find multiple solutions of the following nonlinear programming problem from Lin and Tsai [17, pp. 438-439]:
   
Minimize     X(17) + X(18) + X(19) + X(20) + .1 * X(21) + .2 * X(22) + .3 * X(23) + .4 * X(24)

subject to

         X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)>=9,
         X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)>=7,
         X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)>=12,
         X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)>=11,


        -( -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13))<=0,
        -( -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14))<=0,
        -( -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15))<=0,
         -(-1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16))<=0,

         -(1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13))<=0,
         -(1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14))<=0,
         -(1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15))<=0,
         -(1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16))<=0,


        -( -X(21) + X(1) + X(5) + X(9) + X(13))<=0,
        -( -X(22) + X(2) + X(6) + X(10) + X(14))<=0,
        -( -X(23) + X(3) + X(7) + X(11) + X(15))<=0,
        -( -X(24) + X(4) + X(8) + X(12) + X(16))<=0,

        -( 5 * X(21) - X(1) - X(5) - X(9) - X(13))<=0,
        -( 5 * X(22) - X(2) - X(6) - X(10) - X(14))<=0,
        -( 5 * X(23) - X(3) - X(7) - X(11) - X(15))<=0,
        -( 5 * X(24) - X(4) - X(8) - X(12) - X(16))<=0,


        X(21) - X(17)<=0,
        X(22) - X(18)<=0,
        X(23) - X(19)<=0,
        X(24) - X(20)<=0,


        - 15 * X(21) + X(17)<=0,
        - 12 * X(22) + X(18)<=0,
        - 9 * X(23) + X(19)<=0,

        -6 * X(24) + X(20)<=0,

        X(17) + X(18) + X(19) + X(20)>=8,


         X(21) - X(22)>=0,
         X(22) - X(23)>=0,
         X(23) - X(24)>=0,


         X(17) - X(18)>=0
         X(18) - X(19)>=0,
         X(19) - X(20)>=0,


        0<=  X(i) <=5, i=1, 2, 3,..., 16,

        0<= X(17) <= 15,

         0<=X(18) <= 12,
     
         0<= X(19) <= 9,

        0<= X(20) <= 6,

       
        X(21) through X(24) are 0-1 variables.
       

        0<=  X(i) <=5, i=1,2,3,..., 16,

        0<= X(17) <= 15,

         0<=X(18) <= 12,
     
         0<= X(19) <= 9,

        0<= X(20) <= 6,
       
        X(21) through X(24) are 0-1 variables,
       
where X(1) through X(20) are integer variables and X(21) through X(24) are 0-1 variables.
     
X(25) through X(59) below are slack variables.

One notes line 111 and line 191, which are 111 IF RND < .333 THEN SOFMS = 8 ELSE IF RND < .5 THEN SOFMS = 9 ELSE SOFMS = 10
and 191 X(20) = SOFMS - X(17) - X(18) - X(19).



0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(99)


12 FOR JJJJ = -32000 TO 31997.00 STEP .01


    14 RANDOMIZE JJJJ
    16 M = -1D+37

    22 FOR J55 = 1 TO 16

        37 A(J55) = INT(RND * 5)


    38 NEXT J55
    71 A(17) = 2 + INT(RND * 3)
    72 A(18) = 2 + INT(RND * 3)

    73 A(19) = 2 + INT(RND * 3)
    74 A(20) = 2 + INT(RND * 3)


    91 IF RND < .5 THEN A(21) = 0 ELSE A(21) = 1

    97 IF RND < .5 THEN A(22) = 0 ELSE A(22) = 1
    98 IF RND < .5 THEN A(23) = 0 ELSE A(23) = 1
    99 IF RND < .5 THEN A(24) = 0 ELSE A(24) = 1
    111 IF RND < .333 THEN SOFMS = 8 ELSE IF RND < .5 THEN SOFMS = 9 ELSE SOFMS = 10



    128 FOR I = 1 TO 8000000



        129 FOR KKQQ = 1 TO 24

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 16))

            151 J = 1 + FIX(RND * 16)

            152 X(J) = INT(RND * 5)


        153 NEXT IPP
        155 IF RND < .25 THEN GOTO 160 ELSE IF RND < .333 THEN GOTO 164 ELSE IF RND < .5 THEN GOTO 168 ELSE GOTO 173


        160 X(17) = INT(RND * 15)
        163 IF RND < .5 THEN 164 ELSE GOTO 168

        164 X(18) = INT(RND * 12)
        167 IF RND < .5 THEN 168 ELSE GOTO 173

        168 X(19) = INT(RND * 9)
        169 IF RND < .5 THEN 173 ELSE GOTO 175

        173 X(20) = INT(RND * 6)

        175 FOR IPQ = 1 TO (1 + FIX(RND * 4))

            177 J = 21 + FIX(RND * 4)

            179 X(J) = INT(RND * 1)


        181 NEXT IPQ



        183 REM r = (1 - RND * 2) * A(J)
        187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r

        191 X(20) = SOFMS - X(17) - X(18) - X(19)

        192 IF X(17) > 0 THEN X(21) = 1
        193 IF X(18) > 0 THEN X(22) = 1
        194 IF X(19) > 0 THEN X(23) = 1

        195 IF X(20) > 0 THEN X(24) = 1

        201 IF X(1) < 0 THEN 1670
        202 IF X(1) > 5 THEN 1670
        203 IF X(2) < 0 THEN 1670
        204 IF X(2) > 5 THEN 1670

        205 IF X(3) < 0 THEN 1670
        206 IF X(3) > 5 THEN 1670


        207 IF X(4) < 0 THEN 1670
        208 IF X(4) > 5 THEN 1670
        209 IF X(5) < 0 THEN 1670
        210 IF X(5) > 5 THEN 1670

        211 IF X(6) < 0 THEN 1670
        212 IF X(6) > 5 THEN 1670

        213 IF X(7) < 0 THEN 1670
        214 IF X(7) > 5 THEN 1670
        215 IF X(8) < 0 THEN 1670
        216 IF X(8) > 5 THEN 1670

        217 IF X(9) < 0 THEN 1670
        218 IF X(9) > 5 THEN 1670

        219 IF X(10) < 0 THEN 1670
        220 IF X(10) > 5 THEN 1670
        221 IF X(11) < 0 THEN 1670
        222 IF X(11) > 5 THEN 1670

        223 IF X(12) < 0 THEN 1670
        224 IF X(12) > 5 THEN 1670


        225 IF X(13) < 0 THEN 1670
        226 IF X(13) > 5 THEN 1670

        227 IF X(14) < 0 THEN 1670
        228 IF X(14) > 5 THEN 1670
        229 IF X(15) < 0 THEN 1670
        230 IF X(15) > 5 THEN 1670

        231 IF X(16) < 0 THEN 1670
        232 IF X(16) > 5 THEN 1670

        233 IF X(17) < 0 THEN 1670
        234 IF X(17) > 15 THEN 1670
        235 IF X(18) < 0 THEN 1670
        236 IF X(18) > 12 THEN 1670

        237 IF X(19) < 0 THEN 1670
        238 IF X(19) > 9 THEN 1670


        239 IF X(20) < 0 THEN 1670
        240 IF X(20) > 6 THEN 1670
        241 IF X(21) < 0 THEN 1670
        242 IF X(21) > 1 THEN 1670

        243 IF X(22) < 0 THEN 1670
        244 IF X(22) > 1 THEN 1670



        245 IF X(23) < 0 THEN 1670
        246 IF X(23) > 1 THEN 1670



        247 IF X(24) < 0 THEN 1670
        248 IF X(24) > 1 THEN 1670



        301 X(25) = -9 + X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)
        302 X(26) = -7 + X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)
        303 X(27) = -12 + X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)
        304 X(28) = -11 + X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)




        305 X(29) = -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13)
        306 X(30) = -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14)
        307 X(31) = -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15)
        308 X(32) = -1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16)

        309 X(33) = 1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13)
        310 X(34) = 1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14)
        311 X(35) = 1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15)
        312 X(36) = 1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16)



        313 X(37) = -X(21) + X(1) + X(5) + X(9) + X(13)
        314 X(38) = -X(22) + X(2) + X(6) + X(10) + X(14)
        315 X(39) = -X(23) + X(3) + X(7) + X(11) + X(15)
        316 X(40) = -X(24) + X(4) + X(8) + X(12) + X(16)

        317 X(41) = 5 * X(21) - X(1) - X(5) - X(9) - X(13)
        318 X(42) = 5 * X(22) - X(2) - X(6) - X(10) - X(14)
        319 X(43) = 5 * X(23) - X(3) - X(7) - X(11) - X(15)
        320 X(44) = 5 * X(24) - X(4) - X(8) - X(12) - X(16)





        321 X(45) = -X(21) + X(17)
        322 X(46) = -X(22) + X(18)
        323 X(47) = -X(23) + X(19)
        324 X(48) = -X(24) + X(20)


        325 X(49) = 15 * X(21) - X(17)
        326 X(50) = 12 * X(22) - X(18)
        327 X(51) = 9 * X(23) - X(19)

        328 X(52) = 6 * X(24) - X(20)

        329 X(53) = -8 + X(17) + X(18) + X(19) + X(20)



        330 X(54) = X(21) - X(22)
        331 X(55) = X(22) - X(23)
        332 X(56) = X(23) - X(24)


        333 X(57) = X(17) - X(18)
        334 X(58) = X(18) - X(19)
        335 X(59) = X(19) - X(20)




        425 FOR J99 = 25 TO 59



            426 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

        427 NEXT J99

        431 SUMP = 0
        434 FOR J66 = 25 TO 59
            437 SUMP = SUMP + X(J66)
        440 NEXT J66



        459 POBA = -X(17) - X(18) - X(19) - X(20) - .1 * X(21) - .2 * X(22) - .3 * X(23) - .4 * X(24) + 1000000 * SUMP



        466 P = POBA


        1111 IF P <= M THEN 1670



        1452 M = P
        1454 FOR klx = 1 TO 59



            1459 A(klx) = X(klx)
        1460 NEXT klx
        1557 REM GOTO 128

    1670 NEXT I

    1889 IF M < -8.9 THEN 1999



    1900 PRINT A(1), A(2), A(3), A(4), A(5)
    1903 PRINT A(6), A(7), A(8), A(9), A(10)

    1950 PRINT A(11), A(12), A(13), A(14), A(15)
    1953 PRINT A(16), A(17), A(18), A(19), A(20)
    1955 PRINT A(21), A(22), A(23), A(24)


    1957 PRINT M, JJJJ

1999 NEXT JJJJ



This BASIC computer program was run with qb64v1000-win [34]. The complete output through JJJJ = -31998.09000000031 is shown below:   

1      1      2      0      0
2      1      0      3      0
0      0      1      2      2
0      4      3      1      0
1      1      1      0
-8.6      -31999.21000000013     

1      2      0      0      1
0      3      0      2      1
1      0      1      2      1
0      4      3      1      0
1      1      1      0
-8.6      -31998.72000000021

1      2      0      0      0
0      4      0      3      1
0      0      1      2      1
0      3      3      2      0
1      1      1      0
-8.6      -31998.52000000024

1      1      2      0      0
2      1      0      3      0
0      0      1      2      2
0      4      3      1      0
1      1      1      0
-8.6      -31998.45000000025

1      2      0      0      0
1      4      0      3      0
0      0      1      2      1
0      4      3      1      0
1      1      1      0
-8.6      -31998.09000000031

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 [34], the wall-clock time for obtaining the output through JJJJ= -31998.09000000031 was 2 hours and 40 minutes.  One can compare the computational results here with those in Table 4 of Lin and Tsai  [17, p. 440]. 

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.  http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3]  S. S. Chadha (2002).   Fractional programming with absolute-value functions.   European Journal of Operational Research 141 (2002) pp. 233-238.
[4]  Ching-Ter Chang (2002).   On the posynomial fractional programming problems.  European Journal of Operational Research 143 (2002) pp. 42-52. 
[5]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[12]  Ali Husseinzadeh Kashan  (2011).  An effective algorithm for constrained global optimization and application to mechanical engineering design:  League championship algorithm (LCA).  Computer-Aided Design 43 (2011) 1769-1792. 

[13]  Ali Husseinzadeh Kashan  (2015).  An effective algorithm for constrained optimization based on optics inspired optimization (OIO).  Computer-Aided Design 63 (2015) 52-71. 

[14]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[15] 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.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016).  An enhanced logarithmic method for signomial programming with discrete variables.  European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2011).  Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[18] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[19]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[20]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[21]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms.  Journal of Global Optimization (2017) 68, pp. 95-123.
[22] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[23] 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.
[24] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[25] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[26] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[27] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[28]  Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei.  A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem.  Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[29] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[30] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[31] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[32] 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) pp. 10-19.
[33] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[34] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[35] 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/
[36]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[37] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[38] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[39] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Solving a Speed Reducer Design Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following speed-reducer weight minimization problem in Li et al. [16, p. 932], Kashan [13, p. 68], and Kashan [12, p. 1790].

Minimize     

.7854 * X(1) * X(2) ^ 2 * (3.3333 * X(3) ^ 2 + 14.9334 * X(3) - 43.0934) - 1.508 * X(1) * (X(6) ^ 2 + X(7) ^ 2) + 7.4777 * (X(6) ^ 3 + X(7) ^ 3) + .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2)

subject to

         27 / (X(1) * X(2) ^ 2 * X(3))  <=1,

         397.5 / (X(1) * X(2) ^ 2 * X(3) ^ 2)  <=1,

         1.93 * X(4) ^ 3 / (X(2) * X(3) * X(6) ^ 4) <=1,

        1.93 * X(5) ^ 3 / (X(2) * X(3) * X(7) ^ 4) <=1,


         (1 / (110 * X(6) ^ 3)) * (((((745 * X(4)) / (X(2) * X(3))) ^ 2 + 16900000))) ^ .5  <=1,


         (1 / (85 * X(7) ^ 3)) * (((((745 * X(5)) / (X(2) * X(3))) ^ 2 + 157500000))) ^ .5  <=1,


        X(2) * X(3) / 40  <=1,

         5 * X(2) / X(1) <=1,

        X(1) / (12 * X(2)) <=1,

         (1.5 * X(6) + 1.9) / X(4) <=1,

         (1.1 * X(7) + 1.9) / X(5)  <=1,

       2.6 <=  X(1) <= 3.6,
         .7 <= X(2) <= .8 ,

         17 <= X(3) <= 28,
         7.3 <= X(4) <= 8.3,
         7.3 <= X(5) <= 8.3,

         2.9 <= X(6) <= 3.9,

        5<=  X(7) <= 5.5,

where X(3), the number of teeth, is an integer, and the other six variables are continuous.

X(8) through X(18) below are slack variables.

One notes line 193, which is 193 X(3) = INT(X(3)), where X(3) is the number of teeth, Kashan [13, p. 68].


0 DEFDBL A-Z

2 DEFINT 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 STEP .01

    14 RANDOMIZE JJJJ
    16 M = -1D+37

    87 A(1) = 2.6 + FIX(RND * 1000) * .001

    89 A(2) = .7 + FIX(RND * 1000) * .0001


    91 A(3) = 17 + FIX(RND * 1000) * .011

    94 A(4) = 7.3 + FIX(RND * 1000) * .001

    96 A(5) = 7.3 + FIX(RND * 1000) * .001



    98 A(6) = 2.9 + FIX(RND * 1000) * .001

    99 A(7) = 5 + FIX(RND * 1000) * .0005



    128 FOR I = 1 TO 100000


        129 FOR KKQQ = 1 TO 7

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 7))


            151 J = 1 + FIX(RND * 7)
            155 IF J = 1 GOTO 167 ELSE IF J = 2 THEN GOTO 169 ELSE IF J = 3 THEN GOTO 172 ELSE IF J = 4 THEN GOTO 174 ELSE IF J = 5 THEN GOTO 176 ELSE IF J = 6 THEN GOTO 178 ELSE IF J = 7 THEN GOTO 180

            167 X(1) = 2.6 + FIX(RND * 1000) * .001
            168 IF RND < .5 THEN 170

            169 X(2) = .7 + FIX(RND * 1000) * .0001

            170 IF RND < .5 THEN 173
            172 X(3) = 17 + FIX(RND * 1000) * .011
            173 IF RND < .5 THEN 175
            174 X(4) = 7.3 + FIX(RND * 1000) * .001
            175 IF RND < .5 THEN 177
            176 X(5) = 7.3 + FIX(RND * 1000) * .001

            177 IF RND < .5 THEN 179
            178 X(6) = 2.9 + FIX(RND * 1000) * .001
            179 IF RND < .5 THEN 181
            180 X(7) = 5 + FIX(RND * 1000) * .0005
            181 REM

        191 NEXT IPP
        193 X(3) = INT(X(3))

        201 IF X(1) < 2.6 THEN 1670
        203 IF X(1) > 3.6 THEN 1670
        211 IF X(2) < .7 THEN 1670
        213 IF X(2) > .8 THEN 1670


        231 IF X(3) < 17 THEN 1670
        233 IF X(3) > 28 THEN 1670
        235 IF X(4) < 7.3 THEN 1670
        237 IF X(4) > 8.3 THEN 1670
        239 IF X(5) < 7.3 THEN 1670
        241 IF X(5) > 8.3 THEN 1670

        247 IF X(6) < 2.9 THEN 1670
        249 IF X(6) > 3.9 THEN 1670

        251 IF X(7) < 5 THEN 1670
        253 IF X(7) > 5.5 THEN 1670



        305 X(8) = 1 - 27 / (X(1) * X(2) ^ 2 * X(3))
        306 X(9) = 1 - 397.5 / (X(1) * X(2) ^ 2 * X(3) ^ 2)


        307 X(10) = 1 - 1.93 * X(4) ^ 3 / (X(2) * X(3) * X(6) ^ 4)

        309 X(11) = 1 - 1.93 * X(5) ^ 3 / (X(2) * X(3) * X(7) ^ 4)


        320 X(12) = 1 - (1 / (110 * X(6) ^ 3)) * (((((745 * X(4)) / (X(2) * X(3))) ^ 2 + 16900000))) ^ .5



        322 X(13) = 1 - (1 / (85 * X(7) ^ 3)) * (((((745 * X(5)) / (X(2) * X(3))) ^ 2 + 157500000))) ^ .5



        323 X(14) = 1 - X(2) * X(3) / 40

        325 X(15) = 1 - 5 * X(2) / X(1)

        327 X(16) = 1 - X(1) / (12 * X(2))

        329 X(17) = 1 - (1.5 * X(6) + 1.9) / X(4)

        330 X(18) = 1 - (1.1 * X(7) + 1.9) / X(5)

        335 FOR J99 = 8 TO 18



            340 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

        341 NEXT J99



        359 POBA = -.7854 * X(1) * X(2) ^ 2 * (3.3333 * X(3) ^ 2 + 14.9334 * X(3) - 43.0934) + 1.508 * X(1) * (X(6) ^ 2 + X(7) ^ 2) - 7.4777 * (X(6) ^ 3 + X(7) ^ 3) - .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2) + 1000000 * (X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) + X(16) + X(17) + X(18))



        466 P = POBA

        1111 IF P <= M THEN 1670



        1452 M = P
        1454 FOR KLX = 1 TO 18



            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM GOTO 128

    1670 NEXT I

    1889 IF M < -2996 THEN 1999

    1900 PRINT A(1), A(2), A(3), A(4), A(5)
    1903 PRINT A(6), A(7), A(8), A(9), A(10)

    1950 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ =  -31999.74000000004 is shown below:   

3.5      .7      17      7.301       7.718
3.351     5.287    0    0     0
0      0    0    0    0
0    0     0     -2994.958416306808
-31999.99

3.5      .7      17      7.301       7.716
3.351     5.287    0    0     0
0      0    0    0    0
0    0     0     -2994.914508725582
-31999.98

3.5      .7      17      7.301       7.716
3.351     5.287    0    0     0
0      0    0    0    0
0    0     0     -2994.914508725582
-31999.78000000004

3.5      .7      17      7.301       7.718
3.352     5.287    0    0     0
0      0    0    0    0
0    0     0     -2995.213455221353
-31999.74000000004

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 [33], the wall-clock time for obtaining the output through
JJJJ=-31999.74000000004 was 15 seconds, not including the time for creating the .EXE file.   One can compare the computational results here with those in Table 6 of Li et al. [16, p. 933].     

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.  http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3]  S. S. Chadha (2002).   Fractional programming with absolute-value functions.   European Journal of Operational Research 141 (2002) pp. 233-238.
[4]  Ching-Ter Chang (2002).   On the posynomial fractional programming problems.  European Journal of Operational Research 143 (2002) pp. 42-52. 
[5]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[12]  Ali Husseinzadeh Kashan  (2011).  An effective algorithm for constrained global optimization and application to mechanical engineering design:  League championship algorithm (LCA).  Computer-Aided Design 43 (2011) 1769-1792. 

[13]  Ali Husseinzadeh Kashan  (2015).  An effective algorithm for constrained optimization based on optics inspired optimization (OIO).  Computer-Aided Design 63 (2015) 52-71. 

[14]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[15] 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.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016).  An enhanced logarithmic method for signomial programming with discrete variables.  European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[19]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[20]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms.   Journal of Global Optimization (2017) 68, pp. 95-123.
[21] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[22] 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.
[23] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[26] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[27]  Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei.  A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem.  Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[28] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[29] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[30] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[31] 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) pp. 10-19.
[32] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] 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/
[35]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[37] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[38] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Solving in Integers Another Fractional Nonlinear Integer Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following integer problem:

Maximize           n1x / d1x + n2x / d2x + n3x / d3x

where

         n1x = X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 16 * X(2) + X(3) ^ 2 - 16 * X(3) + X(4) ^ 2 - 16 * X(4) + 214,


         n2x = X(1) ^ 2 - 16 * X(1) + 2 * X(2) ^ 2 - 20 * X(2) + 3 * X(3) ^ 2 - 60 * X(3) + 4 * X(4) ^ 2 - 56 * X(4) + 586,


         n3x = X(1) ^ 2 - 20 * X(1) + X(2) ^ 2 - 20 * X(2) + X(3) ^ 2 - 20 * X(3) + X(4) ^ 2 - 20 * X(4) + 324,

         d1x = 2 * X(1) - X(2) - X(3) + X(4) + 2,

         d2x = -X(1) + X(2) + X(3) - X(4) + 10,

         d3x = X(1) ^ 2 - 4 * X(4),

subject to
     
        6<= X(1) <= 10,
       
        4<= X(2) <= 6,
     
        8<= X(3) <= 12,
     
        6<=X(4) <= 8,

        X(1) + X(2) + X(3) + X(4)<=26,

X(1) through X(4) are integer variables.

The integer problem above is based  on Example 4 in Shen, Duan, and Pei [23, p. 155].

X(5) below is a slack variable.


0 DEFDBL A-Z

2 DEFINT 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 STEP .01


    14 RANDOMIZE JJJJ
    16 M = -1D+37


    75 A(1) = 6 + RND * 4

    77 A(2) = 4 + RND * 2

    78 A(3) = 8 + RND * 4

    79 A(4) = 6 + RND * 2

    128 FOR I = 1 TO 2000




        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 3))



            181 J = 1 + FIX(RND * 4)

            183 r = (1 - RND * 2) * A(J)
            187 X(J) = A(J) + (RND ^ (RND * 10)) * r

        191 NEXT IPP
        193 REM GOTO 209
        196 FOR J99 = 1 TO 4

            199 X(J99) = INT(X(J99))


        204 NEXT J99


        209 IF X(1) < 6 THEN 1670

        212 IF X(1) > 10 THEN 1670

        214 IF X(2) < 4 THEN 1670

        216 IF X(2) > 6 THEN 1670
        218 IF X(3) < 8 THEN 1670

        222 IF X(3) > 12 THEN 1670

        229 IF X(4) < 6 THEN 1670
        23 IF X(4) > 8 THEN 1670


        306 X(5) = 26 - X(1) - X(2) - X(3) - X(4)



        327 XX(5) = X(5)



        330 IF X(5) < 0 THEN X(5) = X(5) ELSE X(5) = 0


        340 n1x = X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 16 * X(2) + X(3) ^ 2 - 16 * X(3) + X(4) ^ 2 - 16 * X(4) + 214


        342 n2x = X(1) ^ 2 - 16 * X(1) + 2 * X(2) ^ 2 - 20 * X(2) + 3 * X(3) ^ 2 - 60 * X(3) + 4 * X(4) ^ 2 - 56 * X(4) + 586


        346 n3x = X(1) ^ 2 - 20 * X(1) + X(2) ^ 2 - 20 * X(2) + X(3) ^ 2 - 20 * X(3) + X(4) ^ 2 - 20 * X(4) + 324

        349 d1x = 2 * X(1) - X(2) - X(3) + X(4) + 2

        352 d2x = -X(1) + X(2) + X(3) - X(4) + 10

        355 d3x = X(1) ^ 2 - 4 * X(4)



        357 POBA = n1x / d1x + n2x / d2x + n3x / d3x + 1000000 * (X(5))



        466 P = POBA

        1111 IF P <= M THEN 1670



        1452 M = P
        1454 FOR KLX = 1 TO 5



            1456 XXX(KLX) = XX(KLX)

            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM GOTO 128


    1670 NEXT I



    1889 IF M < -99999 THEN 1999

    1900 PRINT A(1), A(2), A(3), A(4), A(5), XXX(5)

    1902 PRINT M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [29]. The complete output through JJJJ =  -31999.9600000001  is  shown below:

6   4   8   6   0
2
-2.783333333333333   -32000       

6   4   8   6   0
2
-2.783333333333333   -31999.99   

6   4   8   6   0
2
-2.783333333333333   -31999.98   

6   4   8   6   0
2
-2.783333333333333   -31999.9700000001     

6   4   8   6   0
2
-2.783333333333333   -31999.9600000001     

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  The candidate solution above at JJJJ=-32000, for example, is optimal, Senn, Duan, and Pei [23, p. 156].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [29], the wall-clock time for obtaining the output through JJJJ=  -31999.9600000001  was 2 seconds, not including the time for creating the .EXE file.   
   
Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.  http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3]  S. S. Chadha (2002).   Fractional programming with absolute-value functions.   European Journal of Operational Research 141 (2002) pp. 233-238.
[4]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[5] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[6] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[7] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[10]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.
[11]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[12] 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.
[13] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[14]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[15]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[16]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[17] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[18] 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.
[19] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[20] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[21] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[22] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[23]  Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei.  A simplicial branch and duality bound algorithm for the sum of convex-convex ratios problem.  Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[24] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[25] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[26] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[27] 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) pp. 10-19.
[28] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[29] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[30] 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/
[31]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[32] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[33] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[35] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Erratum in: Solving in Integers a Fractional Nonlinear Integer Programming Problem

Jsun Yui Wong

The computer output of the preceding paper should read as follows:

1   1   5   1   4
0   0   0   0 
1.484375   8   0   8
18.83350950079081      -31999.96000000001

1   1   5   1   4
0   0   0   0 
1.484375   8   0   8
18.83350950079081      -31999.93000000001

1   1   5   1   4
0   0   0   0 
1.484375   8   0   8
18.83350950079081      -31999.83000000003

2   1   5   1   2
0   0   0   0 
1.625   10   0   57
12.65060107369884      -31999.82000000003

Solving in Integers a Fractional Nonlinear Integer Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following integer problem:

Minimize 

(2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)  - (((X(3) + 2) * X(1) * X(2) ^ 2) / (200 * (2 ^ .5) * X(1) + 100 * X(2)))

subject to

        8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / (X(5) ^ 3)<=2,

        - 2 * X(1) + X(3) - X(4)<=10,

         X(1) + X(3) + .5 * X(5)<=8,

         X(1) ^ 3 + X(2) ^ 3 + X(3) ^ 3 + X(4) ^ 3 + X(5) ^ 3<=200,

         .1 <= X(i) <= 10, i=1, 2, 3, 4, 5,

X(1) through X(5) are integer variables.

The integer problem above is based mainly on Example 3 in Tsai [24, p. 408].

X(6) through X(9) below are slack variables.


0 DEFDBL A-Z

2 DEFINT 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 STEP .01

    14 RANDOMIZE JJJJ
    16 M = -1D+37
    70 FOR J44 = 1 TO 5

        72 A(J44) = .1 + RND * 9.9



    73 NEXT J44



    128 FOR I = 1 TO 1000



        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

        191 NEXT IPP

        196 FOR J99 = 1 TO 5

            199 X(J99) = INT(X(J99))

            201 IF X(J99) < .1 THEN 1670
            203 IF X(J99) > 10 THEN 1670
        204 NEXT J99

        305 X(6) = 2 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / (X(5) ^ 3)



        306 X(7) = 10 + 2 * X(1) - X(3) + X(4)



        307 X(8) = 8 - X(1) - X(3) - .5 * X(5)

        311 X(9) = 200 - X(1) ^ 3 - X(2) ^ 3 - X(3) ^ 3 - X(4) ^ 3 - X(5) ^ 3

        325 FOR J99 = 6 TO 9

            327 XX(J99) = X(J99)



            330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

        331 NEXT J99



        357 POBA = -(2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4) + 1000000 * (X(6) + X(7) + X(8) + X(9)) + (((X(3) + 2) * X(1) * X(2) ^ 2) / (200 * (2 ^ .5) * X(1) + 100 * X(2)))



        466 P = POBA

        1111 IF P <= M THEN 1670



        1452 M = P
        1454 FOR KLX = 1 TO 9



            1456 XXX(KLX) = XX(KLX)

            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 GOTO 128

    1670 NEXT I



    1889 IF M < 10 THEN 1999

    1900 PRINT A(1), A(2), A(3), A(4), A(5)

    1902 PRINT A(6), A(7), A(8), X(9)

    1903 PRINT XXX(6), XXX(7), XXX(8), XXX(9)

    1912 PRINT M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [28]. The complete output through JJJJ =-31999.82000000003 is  shown below:

1   1   5   1   4
0   0   0   0   0
1.484375   8   0   8
18.83350950079081      -31999.96000000001

1   1   5   1   4
0   0   0   0   0
1.484375   8   0   8
18.83350950079081      -31999.93000000001

1   1   5   1   4
0   0   0   0   0
1.484375   8   0   8
18.83350950079081      -31999.83000000003

2   1   5   1   2
0   0   0   0   0
1.625   10   0   57
12.65060107369884      -31999.82000000003

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 [28], the wall-clock time for obtaining the output through JJJJ=-31999.82000000003 was 2 seconds, not including the time for creating the .EXE file.   
   
Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.  http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3]  S. S. Chadha (2002).   Fractional programming with absolute-value functions.   European Journal of Operational Research 141 (2002) pp. 233-238.
[4]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[5] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[6] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[7] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[10]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[11]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[12] 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.
[13] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[14]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[15]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[16]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms.   Journal of Global Optimization (2017) 68, pp. 95-123.
[17] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[18] 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.
[19] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[20] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[21] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[22] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[23] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[24] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[25] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[26] 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) pp. 10-19.
[27] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[28] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[29] 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/
[30]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[31] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[32] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[33] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Solving a Fractional Nonlinear Integer Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Example 4 in Chang [4, p. 383]:

Minimize 
   
 (    ABS(X(5) ^ 1.8) * X(1) * X(3) ^ .3 * X(4) ^ 1.5 + X(2) * X(3) ^ 1.8 * X(4) ^ 1.3 * X(5) + X(1) * X(2) * X(3) ^ 1.7    )  /  (    X(1) * X(3) * X(4) + X(2) * X(3) ^ 2.3 * X(4) ^ 1.4  )

subject to

         2 * X(1) + X(1) * X(3) ^ 1.6>=5,

         X(1) + X(2)>=1,

         X(2) + X(4)<=6,

         X(1) + X(2) + X(5)>=3,

        1 <=X(3)<= 7 ,

         1 <= X(4) <= 6,

         1 <= X(5) <= 5 ,

X(1) and X(2) are binary variables,

X(3) and X(4) are continuous variables,

X(5) is absolute continuous variable.


X(6) through X(9) below are slack variables.



0 DEFDBL A-Z

2 DEFINT K

3 DIM B(199), N(199), A(199), H(199), L(199), U(199), X(1111), D(111), P(111), PS(133)

12 FOR JJJJ = -32000 TO 32000 STEP .01

    14 RANDOMIZE JJJJ
    16 M = -1D+37
    71 IF RND < .5 THEN A(1) = 0 ELSE A(1) = 1

    73 IF RND < .5 THEN A(2) = 0 ELSE A(2) = 1

    84 A(3) = 1 + (RND * 7)
    85 A(4) = 1 + (RND * 6)
    86 A(5) = 1 + (RND * 5)

    128 FOR I = 1 TO 30000


        129 FOR KKQQ = 1 TO 5



            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        132 REM GOTO 162

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


            151 J = 3 + FIX(RND * 3)


            153 r = (1 - RND * 2) * A(J)
            157 X(J) = A(J) + (RND ^ (RND * 10)) * r


        161 NEXT IPP

        162 REM IF RND < .5 THEN 164
        163 REM X(1) = INT(X(1))
        164 REM IF RND < .5 THEN 197



        165 REM X(2) = INT(X(2))



        197 IF X(3) < 1 THEN 1670

        199 IF X(3) > 7 THEN 1670

        201 IF X(4) < 1 THEN 1670

        203 IF X(4) > 6 THEN 1670


        205 IF X(5) < 1 THEN 1670

        207 IF X(5) > 5 THEN 1670

        261 X(6) = -5 + 2 * X(1) + X(1) * X(3) ^ 1.6


        264 X(7) = -1 + X(1) + X(2)

        267 X(8) = 6 - X(2) - X(4)

        269 X(9) = -3 + X(1) + X(2) + X(5)



        281 FOR J99 = 6 TO 9

            283 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0


        285 NEXT J99

        311 ups = ABS(X(5) ^ 1.8) * X(1) * X(3) ^ .3 * X(4) ^ 1.5 + X(2) * X(3) ^ 1.8 * X(4) ^ 1.3 * X(5) + X(1) * X(2) * X(3) ^ 1.7

        322 dow = X(1) * X(3) * X(4) + X(2) * X(3) ^ 2.3 * X(4) ^ 1.4


        365 POBA = -ups / dow + 1000000 * (X(6) + X(7) + X(8) + X(9))


        466 P = POBA

        1111 IF P <= M THEN 1670


        1452 M = P
        1454 FOR KLX = 1 TO 9



            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM  GOTO 128

    1670 NEXT I


    1777 FOR J44 = 6 TO 9

        1778 IF A(J44) < 0 THEN 1999
    1779 NEXT J44

    1889 IF M < -.5 THEN 1999


    1908 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [28]. The complete output through JJJJ =  -31999.88000000002 is  shown below:

1      1      6.999999999990646      4.9999999999972
1.000000000004951      0      0      0
0      -.3632313212606416      -31999.96000000001

1      1      6.99999999999997       4.999999999991044
1.000000000003382      0      0      0
0      -.3632313212599075      -31999.90000000002

1      1      6.9999999999965         4.99999999999999
1.000000000001283      0      0      0
0      -.3632313212591524      -31999.88000000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  The computational results above can be compared with the results in Chang [4, p. 385].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [28], the wall-clock time for obtaining the output through JJJJ=   -31999.88000000002 was 3 seconds, not including the time for creating the .EXE file.   
   

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.  http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3]  S. S. Chadha (2002).   Fractional programming with absolute-value functions.   European Journal of Operational Research 141 (2002) pp. 233-238.
[4]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[5] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[6] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[7] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[10]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[11]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[12] 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.
[13] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[14]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[15]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[16]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[17] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[18] 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.
[19] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[20] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[21] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[22] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[23] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[24] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[25] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[26] 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) pp. 10-19.
[27] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[28] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[29] 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/
[30]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[31] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[32] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[33] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Solving a Nonlinear Programming Problem Involving Noncovex Posynomial and Signomial Terms

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear programming formulation from page 452 of Lu [14]: 

Minimize            .622 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3)

subject to

        -X(1)+ .0193 * X(3)<=0,

        - X(2) + .00954 * X(3)<=0,

        1296000 - 3.141592654 * X(3) ^ 2 * X(4) - (4 / 3) * 3.141592654 * X(3) ^ 3<=0,

        - 240+ X(2)<=0,

          .0625<=X(i) <= 6.1875, i=1, 2,

           10<=X(3) <= 200, i=3, 4.

X(1) through X(4) are continuous.

X(5) through X(8) below are slack variables.


0 DEFDBL A-Z

2 DEFINT K

3 DIM B(199), N(199), A(199), H(199), L(199), U(199), X(1111), D(111), P(111), PS(133)

12 FOR JJJJ = -32000 TO 32000 STEP .01

    14 RANDOMIZE JJJJ
    16 M = -1D+37

    22 FOR J55 = 1 TO 2


        67 A(J55) = .0625 + RND * 6.125


    71 NEXT J55
    82 FOR J55 = 3 TO 4


        86 A(J55) = 10 + RND * 190


    88 NEXT J55


    128 FOR I = 1 TO 60000


        129 FOR KKQQ = 1 TO 4


            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        132 REM GOTO 163

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


            151 J = 1 + FIX(RND * 4)

            153 r = (1 - RND * 2) * A(J)
            157 X(J) = A(J) + (RND ^ (RND * 10)) * r


        161 NEXT IPP
        163 REM

        164 REM


        197 IF X(1) < .0625 THEN 1670

        199 IF X(1) > 6.1875 THEN 1670

        201 IF X(2) < .0625 THEN 1670

        203 IF X(2) > 6.1875 THEN 1670

        205 IF X(3) < 10 THEN 1670

        207 IF X(3) > 200 THEN 1670

        209 IF X(4) < 10 THEN 1670

        211 IF X(4) > 200 THEN 1670


        261 X(5) = X(1) - .0193 * X(3)

        264 X(6) = X(2) - .00954 * X(3)

        267 X(7) = -1296000 + 3.141592654 * X(3) ^ 2 * X(4) + (4 / 3) * 3.141592654 * X(3) ^ 3

        269 X(8) = 240 - X(2)
        281 FOR J99 = 5 TO 8

            283 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0


        285 NEXT J99

        365 POBA = -.622 * X(1) * X(3) * X(4) - 1.7781 * X(2) * X(3) ^ 2 - 3.1661 * X(1) ^ 2 * X(4) - 19.84 * X(1) ^ 2 * X(3) + 1000000 * (X(8) + X(7) + X(6) + X(5))

        466 P = POBA

        1111 IF P <= M THEN 1670


        1452 M = P
        1454 FOR KLX = 1 TO 8



            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM GOTO 128

    1670 NEXT I


    1889 IF M < -5884.5 THEN 1999


    1900 PRINT A(1), A(2), A(3), A(4), A(5)

    1911 PRINT A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [27]. The complete output through JJJJ = -31951.60000000775 is  shown below:

.7791371869593762        .3851279152120704         40.36980243312682
199.3025835450308           0
0       0       0       -5884.48288302348
-31993.1700000011

.7785102423305716        .38481801615722         40.33731825546528
199.7537562076116          0
0       0       0       -5883.4077896634
-31951.60000000775

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  The computational results above can be compared with the results in Table 7 of Lu [14, p.  453].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [27], the wall-clock time for obtaining the output through
JJJJ=  -31951.60000000775 was 10 minutes, total.   
   
Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .
[3]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[4] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[5] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[6] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[9]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[10]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[11] 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.
[12] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[13]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[14]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[15]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[16] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[17] 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.
[18] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[19] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[20] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[21] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[22] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[23] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[24] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[25] 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) pp. 10-19.
[26] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[27] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[28] 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/
[29]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[30] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[31] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[32] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.

Solving a Nonlinear Programming Problem Involving 3 Noncovex Posynomial Terms in Constraints

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Lu [14, p. 449]: 

Minimize       - X(1) + 2 * X(2) - 3 * X(3) + X(4) + X(5) ^ 2 - 5 * X(6) ^ .6

subject to

         X(1) + X(3) + X(5) + X(7) - 2 * X(4)<=18,

         X(3) - X(4) + X(7) + 2 * X(6) - 3 * X(2)=12,

         .5 * X(1) * X(2) * X(4) - 3 * X(7)<=5,

         .8 * X(1) * X(3) * X(4) - 3 * X(6)<=4,

         1.3 * X(2) * X(3) * X(6) - 3 * X(2)<=5,

        1<= X(i) <= 10, i=1, 5, 6, 7,

        1<= X(i) <= 1000, i=2, 3, 4.

One notes that the following computer program's line 283, which is 283 IF X(J99) < -2.55D-06 THEN X(J99) = X(J99) ELSE X(J99) = 0, uses the feasibility tolerance of
2.55*10^(-06) from Lu [14, p. 450].   


0 DEFDBL A-Z

2 DEFINT K

3 DIM B(199), N(199), A(199), H(199), L(199), U(199), X(1111), D(111), P(111), PS(133)

12 FOR JJJJ = -32000 TO 32000 STEP .01


    14 RANDOMIZE JJJJ
    16 M = -1D+37

    22 FOR J55 = 1 TO 7


        67 A(J55) = 1 + RND * 3


    71 NEXT J55
 

    128 FOR I = 1 TO 200000


        129 FOR KKQQ = 1 TO 7


            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        133 FOR IPP = 1 TO (1 + FIX(RND * 7))


            151 J = 1 + FIX(RND * 7)

            153 r = (1 - RND * 2) * A(J)
            157 X(J) = A(J) + (RND ^ (RND * 10)) * r
         

        161 NEXT IPP
        163 REM

        164 X(3) = 12 + X(4) - X(7) - 2 * X(6) + 3 * X(2)
        197 IF X(1) < 1 THEN 1670

        199 IF X(1) > 10 THEN 1670

        201 IF X(2) < 1 THEN 1670

        203 IF X(2) > 1000 THEN 1670

        205 IF X(3) < 1 THEN 1670

        207 IF X(3) > 1000 THEN 1670

        209 IF X(4) < 1 THEN 1670

        211 IF X(4) > 1000 THEN 1670

        213 IF X(5) < 1 THEN 1670

        215 IF X(5) > 10 THEN 1670

        217 IF X(6) < 1 THEN 1670

        219 IF X(6) > 10 THEN 1670

        221 IF X(7) < 1 THEN 1670

        223 IF X(7) > 10 THEN 1670


        261 X(8) = 18 - X(1) - X(3) - X(5) - X(7) + 2 * X(4)

        264 X(9) = 5 - .5 * X(1) * X(2) * X(4) + 3 * X(7)

        267 X(10) = 4 - .8 * X(1) * X(3) * X(4) + 3 * X(6)

        269 X(11) = 5 - 1.3 * X(2) * X(3) * X(6) + 3 * X(2)
        281 FOR J99 = 8 TO 11

            283 IF X(J99) < -2.55D-06 THEN X(J99) = X(J99) ELSE X(J99) = 0


        285 NEXT J99


        365 POBA = X(1) - 2 * X(2) + 3 * X(3) - X(4) - X(5) ^ 2 + 5 * X(6) ^ .6 + 1000000 * (X(8) + X(9) + X(10) + X(11))

        466 P = POBA

        1111 IF P <= M THEN 1670


        1452 M = P
        1454 FOR KLX = 1 TO 11



            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM GOTO 128

    1670 NEXT I


    1889 IF M < 23.86 THEN 1999

    1900 PRINT A(1), A(2), A(3), A(4), A(5)

    1911 PRINT A(6), A(7), A(8), A(9), A(10), A(11), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with qb64v1000-win [27]. The complete output through JJJJ = -31995.52000000072 is  shown below:

9.999999994672203        1.00034612016621         1.000660271878407
1.001888834118436        1.000000008183936
6.148457689470717        2.705351543797224          0
0         0         0              23.86661799397463
-31999.64000000006

9.999999999978774        1.000048194661112         1.00135793334377
1.001998889919505        1.000000002143264
6.145317132756756        2.71015127504556          0
0          0          0            23.86463992982906
-31995.52000000072

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  The computational results above can be compared with the results in Table 5 of Lu [14, p.  450].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [27], the wall-clock time for obtaining the output through
JJJJ= -31995.52000000072 was 190 seconds, total.   
   
 
Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529. .
[3]  Ching-Ter Chang (2006).   Formulating the mixed integer fractional posynomial programming,  European Journal of Operational Research 173 (2006) pp. 370-386.       
[4] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[5] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[6] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[9]  Chrysanthos E. Gounaris, Christodoulos A. Floudas.  Tight convex underestimators for Csquare-continuous problems: II. multivariate functions.  Journal of Global Optimization (2008) 42, pp. 69-89.

[10]  Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008).  Convex underestimating for posynomial functions of postive variables.  Optimization Letters 2, 333-340 (2008).
[11] 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.
[12] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[13]  Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010).  Convex relaxation for solving posynomial problems.  Journal of Global Optimization (2010) 46, pp. 147-154.
[14]  Hao-Chun Lu (2012).  An efficient convexification method for solving generalized geometric problems.  Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[15]  Hao-Chun Lu (2017).  Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms.  Journal of Global Optimization (2017) 68, pp. 95-123.
[16] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[17] 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.
[18] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[19] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[20] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[21] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[22] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[23] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[24] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[25] 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) pp. 10-19.
[26] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[27] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[28] 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/
[29]  Helen Wu (2015).  Geometric Programming.  https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[30] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms.  https://arxiv.org/pdf/1403.7793.pdf.
[31] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[32] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization.  Computers and Structures 82 (2004) 241-256.