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Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms

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Abstract

Free-sign pure discrete signomial (FPDS) terms are vital to and are frequently observed in many nonlinear programming problems, such as geometric programming, generalized geometric programming, and mixed-integer non-linear programming problems. In this study, all variables in the FPDS term are discrete variables. Any improvement to techniques for linearizing FPDS term contributes significantly to the solving of nonlinear programming problems; therefore, relative techniques have continually been developed. This study develops an improved exact method to linearize a FPDS term into a set of linear programs with minimal logarithmic numbers of zero-one variables and constraints. This method is tighter than current methods. Various numerical experiments demonstrate that the proposed method is significantly more efficient than current methods, especially when the problem scale is large.

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References

  1. Adams, W.P., Henry, S.M.: Base-2 expansions for linearizing products of functions of discrete variables. Oper. Res. 60(6), 701–713 (2012). doi:10.1287/opre.1120.1106

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergamini, M.L., Scenna, N.J., Aguirre, P.A.: Global optimal structures of heat exchanger networks by piecewise relaxation. Ind. Eng. Chem. Res. 46(6), 1752–1763 (2007). doi:10.1021/ie061288p

    Article  Google Scholar 

  3. Boyd, S.P., Kim, S.J., Patil, D.D., Horowitz, M.A.: Digital circuit optimization via geometric programming. Oper. Res. 53(6), 899–932 (2005). doi:10.1287/opre.1050.0254

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheng, H., Fang, S.C., Lavery, J.E.: A geometric programming framework for univariate cubic L1 smoothing splines. Ann. Oper. Res. 133(1–4), 229–248 (2005). doi:10.1007/s10479-004-5035-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Chiang, M.: Geometric Programming for Communication Systems. Now Publishers Inc., Boston (2005)

    MATH  Google Scholar 

  6. Creese, R.C.: Geometric Programming for Design and Cost Optimization (with Illustrative Case Study Problems and Solutions), 2nd edn. Morgan & Claypool Publishers, California (2010)

    Google Scholar 

  7. Dam, S., Mandal, P.: Iterative performance model upgradation in geometric programming based analog circuit sizing for improved design accuracy. VLSI Design, International Conference on 2012, pp. 376–381 (2012). doi:10.1109/VLSID.2012.100

  8. DasGupta, S., Mandal, P.: An improvised MOS transistor model suitable for geometric program based analog circuit sizing in sub-micron technology. VLSI Design, International Conference on 2010, pp. 294–299 (2010). doi:10.1109/VLSI.Design.2010.31

  9. Floudas, C.A.: Deterministic Global Optimization: Theory, Methods and Application. Kluwer, Boston (2000)

    Book  Google Scholar 

  10. Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009). doi:10.1007/s10898-008-9332-8

    Article  MathSciNet  MATH  Google Scholar 

  11. Floudas, C.A., Pardalos, P.M.: State of the Art in Global Optimization: Computational Methods and Applications. Kluwer, Boston (1996)

    Book  MATH  Google Scholar 

  12. Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer, Boston (1999)

    Book  MATH  Google Scholar 

  13. IBM/ILOG.: CPLEX 12.0 reference manual. http://www.ilog.com/products/cplex/ (2009)

  14. Kallrath, J.: Exact computation of global minima of a nonconvex portfolio optimization problem. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 237–254. Kluwer, Boston (2003)

    Google Scholar 

  15. Li, H.L., Lu, H.C.: Global optimization for generalized geometric programs with mixed free-sign variables. Oper. Res. 57(3), 701–713 (2009). doi:10.1287/opre.1080.0586

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, H.L., Lu, H.C., Huang, C.H., Hu, N.Z.: A superior representation method for piece-wise linear functions. INFORMS J. Comput. 21(2), 314–321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, H.L., Huang, Y.H., Fang, S.C.: A logarithmic method for reducing zero-one variables and inequality constraints in solving task assignment problems. INFORMS J. Comput. 25(4), 643–653 (2012). doi:10.1287/ijoc.1120.0527

    Article  Google Scholar 

  18. Li, W.: Energy efficient clustering algorithm in wireless sensor networks based on geometric programming. Second International Symposium on Electronic Commerce and Security, ISECS 2009, pp. 525–527 (2009). doi:10.1109/ISECS.2009.65

  19. Lin, M.H., Tsai, J.F.: Finding multiple optimal solutions of signomial discrete programming problems with free variables. Optim. Eng. 12(3), 425–443 (2011). doi:10.1007/s11081-011-9137-3

    Article  MathSciNet  MATH  Google Scholar 

  20. LINGO, Release 12. Lindo System Inc: Chicago (2010)

  21. Lu, H.C.: An efficient convexification method for solving generalized geometric problems. J. Ind. Manag. Optim. 8(2), 429–455 (2012). doi:10.3934/jimo.2012.8.429

    Article  MathSciNet  Google Scholar 

  22. Lu, H.C.: A logarithmic method for eliminating zero-one variables and constraints for the product of free-sign discrete functions. Discrete Optim. 10(1), 11–24 (2013). doi:10.1016/j.disopt.2012.10.001

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, H.C., Li, H.L., Gounaris, C.E., Floudas, C.A.: Convex relaxation for solving posynomial programs. J. Glob. Optim. 46(1), 147–154 (2010). doi:10.1007/s10898-009-9414-2

    Article  MathSciNet  MATH  Google Scholar 

  24. Lundell, A.: Transformation techniques for signomial functions in global optimization, Ph.D. Dissertation, Åbo Akademi University (2009)

  25. Lundell, A., Westerlund, J., Westerlund, T.: Some transformation techniques with applications in global optimization. J. Glob. Optim. 43(2), 391–405 (2009). doi:10.1007/0-387-30528-9_3

    Article  MathSciNet  MATH  Google Scholar 

  26. Lundell, A., Westerlund, T.: Global optimization of mixed-integer signomial problems in Mixed Integer Nonlinear Programming. IMA Vol. Math. Appl. 154, 349–369 (2012)

    Article  MATH  Google Scholar 

  27. Maranas, C.D., Floudas, C.A.: Global optimization in generalized geometric programming. Comput. Chem. Eng. 21(4), 351–370 (1997). doi:10.1016/S0098-1354(96)00282-7

    Article  Google Scholar 

  28. Mine, H., Ohno, K.: Decomposition of mathematical programming problems by dynamic programming and its applications to block diagonal geometric programming. J. Math. Anal. Appl. 32(2), 370–385 (1970). doi:10.1016/0022-247X(70)90303-3

    Article  MathSciNet  MATH  Google Scholar 

  29. Misener, R., Floudas, C.A.: A framework for globally optimizing mixed-integer signomial programs. J. Optim. Theory Appl. 161(3), 905–932 (2014). doi:10.1007/s10957-013-0396-3

    Article  MathSciNet  MATH  Google Scholar 

  30. Pörn, R., Harjunkosi, I., Westerlund, T.: Convexification of different classes of non-convex MINLP problems. Comput. Chem. Eng. 23(3), 439–448 (1999). doi:10.1016/S0098-1354(98)00305-6

    Article  Google Scholar 

  31. Pörn, R., Björk, K.M., Westerlund, T.: Global solution of optimization problems with signomial parts. Discrete Optim. 5(1), 108–120 (2008). doi:10.1016/j.disopt.2007.11.005

    Article  MathSciNet  MATH  Google Scholar 

  32. Sandgren, E.: Nonlinear integer and discrete programming in mechanical design optimization. J. Mech. Des. 112(2), 223–229 (1990). doi:10.1115/1.2912596

    Article  Google Scholar 

  33. Tsai, J.F., Li, H.L., Hu, N.Z.: Global optimization for signomial discrete programming problems in engineering design. Eng. Optim. 34(6), 613–622 (2002). doi:10.1080/03052150215719

    Article  Google Scholar 

  34. Tasi, J.F., Lin, M.H.: An optimization approach for solving signomial discrete programming problems with free variables. Comput. Chem. Eng. 30(8), 1256–1263 (2006). doi:10.1016/j.compchemeng.2006.02.013

    Article  Google Scholar 

  35. Tsai, J.F., Lin, M.H.: Global optimization of signomial mixed-integer nonlinear programming problems with free variables. J. Glob. Optim. 42(1), 39–49 (2008). doi:10.1007/s10898-007-9211-8

    Article  MathSciNet  MATH  Google Scholar 

  36. Tsai, J.F., Lin, M.H.: An improved framework for solving NLIPs with signomial terms in the objective or constraints to global optimality. Comput. Chem. Eng. 53(11), 44–54 (2013). doi:10.1016/j.compchemeng.2013.01.015

    Article  Google Scholar 

  37. Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of zero-one variables and constraints. Math. Program. 128(1–2), 49–72 (2011). doi:10.1007/s10107-009-0295-4

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This research is supported by the National Science Council of Taiwan under grants NSC 102-2410-H-030-063-MY3. The authors wish to thank the area editor, the associate editor, and the anonymous referees for providing insightful comments and suggestions, which have helped me improve the quality of the paper.

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  1. Department of Information Management, College of Management, Fu Jen Catholic University, No. 510, Jhongjheng Rd., Sinjhuang City, 242, Taipei County, Taiwan

    Hao-Chun Lu

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Lu, HC. Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms. J Glob Optim 68, 95–123 (2017). https://doi.org/10.1007/s10898-016-0451-3

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