Abstract
With the increasing reliance on mathematical programming based approaches in various fields, signomial discrete programming (SDP) problems occur frequently in real applications. Since free variables are often introduced to model problems and alternative optima are practical for decision making among multiple strategies, this paper proposes a generalized method to find multiple optimal solutions of SDP problems with free variables. By means of variable substitution and convexification strategies, an SDP problem with free variables is first converted into another convex mixed-integer nonlinear programming problem solvable to obtain an exactly global optimum. Then a general cut is utilized to exclude the previous solution and an algorithm is developed to locate all alternative optimal solutions. Finally, several illustrative examples are presented to demonstrate the effectiveness of the proposed approach.
Similar content being viewed by others
References
Arora JS (1989) Introduction to optimum design. McGraw-Hill, New York
Athier G, Floquet P, Pibouleau L, Domenech S (1997) Synthesis of heat exchanger network by simulated annealing and NLP procedures. AIChE J 43:3007–3019
Balas E, Jeroslow R (1972) Canonical cuts on the unit hypercube. SIAM J Appl Math 23(1):61–69
Biegler LT, Grossmann IE (2004) Retrospective on optimization. Comput Chem Eng 28(8):1169–1192
Borchers B, Mitchell JE (1994) An improved branch and bound algorithm for mixed integer nonlinear programs. Comput Oper Res 21:359–367
Chelouah R, Siarry P (2000) Tabu search applied to global optimization. Eur J Oper Res 123:256–270
Chen CS, Sarin S, Ram B (1993) A mixed-integer programming model for a class of assortment problems. Eur J Oper Res 63:362–367
Dolan WB, Cummings PT, Levan MD (1989) Process optimization via simulated annealing: application to network design. AIChE J 35:725–736
Duran M, Grossmann IE (1986) An outer-approximation algorithm for a class of mixed integer nonlinear programs. Math Program 36:307–339
Fletcher R, Leyffer S (1994) Solving mixed integer nonlinear programs by outer approximation. Math Program 66:327–349
Floudas CA (1995) Nonlinear and mixed integer optimization: fundamentals and applications. Oxford University Press, New York
Floudas CA (1999a) Global optimization in design and control of chemical process systems. J Process Control 10:125–134
Floudas CA (1999b) Recent advances in global optimization for process synthesis, design and control: Enclosure of all solutions. Comput Chem Eng 23:S963–S973
Floudas CA (2000) Deterministic global optimization: theory, methods and application. Kluwer Academic, Boston
Floudas CA, Akrotirianakis IG, Caratzoulas S, Meyer CA, Kallrath J (2005) Global optimization in the 21st century: advances and challenges. Comput Chem Eng 29:1185–1202
Floudas CA, Gounaris CE (2009) A review of recent advances in global optimization. J Glob Optim 45(1):3–38
Floudas CA, Pardalos PM (1996) State of the art in global optimization: computational methods and applications. Kluwer Academic, Boston
Floudas CA, Pardalos PM, Adjiman CS, Esposito WR, Gumus Z, Harding ST, Klepeis JL, Meyer CA, Schweiger CA (1999) Handbook of test problems for local and global optimization. Kluwer Academic, Dordrecht
Geoffrion AM (1972) Generalized Benders decomposition. J Optim Theory Appl 10:237–260
GroßB, Hammel U, Maldaner P, Meyer A, Roosen P, Schütz M (1996) Optimization of heat exchanger networks by means of evolution strategies. Lect Notes Comput Sci 1141:1002–1011
Grossmann IE (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3:227–252
Grossmann IE, Biegler LT (2004) Part II. Future perspective on optimization. Comput Chem Eng 28(8):1193–1218
Lee S, Grossmann IE (2000) New algorithms for nonlinear generalized disjunctive programming. Comput Chem Eng 24:2125–2142
Lejeune MA (2003) Heuristic optimization of experimental designs. Eur J Oper Res 147:484–498
Lewin R (1998) A generalized method for HEN synthesis using stochastic optimization. II. Synthesis of cost-optimal networks. Comput Chem Eng 22:1387–1405
Leyffer S (2001) Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput Optim Appl 18:295–309
Lin B, Miller DC (2004) Solving heat exchanger network synthesis problems with tabu search. Comput Chem Eng 28:1451–1464
Lindo (2004) LINGO, Release 11. Lindo System Inc., Chicago
Lundell A, Westerlund T (2009) Convex underestimation strategies for signomial functions. Optim Methods Softw 24(4–5):505–522
Lundell A, Westerlund J, Westerlund T (2009) Some transformation techniques with applications in global optimization. J Glob Optim 43(2–3):391–405
Luo Y, Yuan X, Liu Y (2007) An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Comput Chem Eng 31:153–162
Maranas CD, Floudas CA (1995) Finding all solutions of nonlinearly constrained systems of equations. J Glob Optim 7:143–182
Maranas CD, Floudas CA (1997) Global optimization in generalized geometric programming. Comput Chem Eng 21:351–370
Pedamallu CS, Ozdamar L (2008) Investigating a hybrid simulated annealing and local search algorithm for constrained optimization. Eur J Oper Res 185:1230–1245
Pörn R, Harjunkoski I, Westerlund T (1999) Convexification of different classes of non-convex MINLP problems. Comput Chem Eng 23:439–448
Quesada I, Grossmann IE (1992) An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput Chem Eng 16:937–947
Salcedo RL (1992) Solving nonconvex nonlinear programming problems with adaptive random search. Ind Eng Chem Res 31:262–273
Shopova EG, Vaklieva-Bancheva NG (2006) BASIC—a genetic algorithm for engineering problems solution. Comput Chem Eng 30:1293–1309
Stubbs RA, Mehrotra S (1999) A branch-and-cut method for 0-1 mixed convex programming. Math Program 86:515–532
Tawarmalani M, Sahinidis NV (2002) Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications. Kluwer Academic, Dordrecht
Tsai JF (2005) Global optimization for nonlinear fractional programming problems in engineering design. Eng Optim 37:399–409
Tsai JF, Li HL, Hu NZ (2002) Global optimization for signomial discrete programming problems in engineering design. Eng Optim 34:613–622
Tsai JF, Lin MH (2006) An optimization approach for solving signomial discrete programming problems with free variables. Comput Chem Eng 30:1256–1263
Tsai JF, Lin MH (2008) Global optimization of signomial mixed-integer nonlinear programming problems with free variables. J Glob Optim 42:39–49
Tsai JF, Lin MH, Hu YC (2008) Finding multiple solutions to general integer linear programs. Eur J Oper Res 184:802–809
Westerlund T (2007) Some transformation techniques in global optimization. In: Liberti L, Maculan N (eds) Global optimization: from theory to implementation. Springer, New York, pp 45–74
Westerlund T, Pettersson F (1995) An extended cutting plane method for solving convex MINLP problems. Comput Chem Eng 19:S131–S136
Young CT, Zheng Y, Yeh CW, Jang SS (2007) Information-guided genetic algorithm approach to the solution of MINLP problems. Ind Eng Chem Res 46(5):1527–1537
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, MH., Tsai, JF. Finding multiple optimal solutions of signomial discrete programming problems with free variables. Optim Eng 12, 425–443 (2011). https://doi.org/10.1007/s11081-011-9137-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11081-011-9137-3