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Modeling without categorical variables: a mixed-integer nonlinear program for the optimization of thermal insulation systems. (English) Zbl 1273.80009
Summary: Optimal design applications are often modeled by using categorical variables to express discrete design decisions, such as material types. A disadvantage of using categorical variables is the lack of continuous relaxations, which precludes the use of modern integer programming techniques. We show how to express categorical variables with standard integer modeling techniques, and we illustrate this approach on a load-bearing thermal insulation system. The system consists of a number of insulators of different materials and intercepts that minimize the heat flow from a hot surface to a cold surface. Our new model allows us to employ black-box modeling languages and solvers and illustrates the interplay between integer and nonlinear modeling techniques. We present numerical experience that illustrates the advantage of the standard integer model.

80M50 Optimization problems in thermodynamics and heat transfer
90C11 Mixed integer programming
90C30 Nonlinear programming
Full Text: DOI
[1] Abhishek K, Leyffer S, Linderoth JT (2006) FilMINT: an outer-approximation-based solver for nonlinear mixed integer programs. Preprint ANL/MCS-P1374-0906, Mathematics and Computer Science Division, Argonne National Lab · Zbl 1243.90142
[2] Abramson MA (2004) Mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm. Optim Eng 5:157–177 · Zbl 1085.90033 · doi:10.1023/B:OPTE.0000033373.79886.54
[3] Abramson MA, Audet C, Dennis JE (2007) Filter pattern search algorithms for mixed variable constrained optimization problems. Pac J Optim 3(3):477–500 · Zbl 1138.65043
[4] Audet C, Dennis J (2004) A pattern search filter method for nonlinear programming without derivatives. SIAM J Optim 14(4):980–1010 · Zbl 1073.90066 · doi:10.1137/S105262340138983X
[5] Audet C, Dennis JE (2000) Pattern search algorithms for mixed variable programming. SIAM J Optim 11(3):573–594 · Zbl 1035.90048 · doi:10.1137/S1052623499352024
[6] Beal JM, Shukla A, Brezhneva OA, Abramson MA (2008) Optimal sensor placement for enhancing sensitivity to change in stiffness for structural health monitoring. Optim Eng 9:119–142 · Zbl 1175.74059 · doi:10.1007/s11081-007-9023-1
[7] Bonami P, Biegler LT, Conn AR, Cornuéjols G, Grossmann IE, Laird CD, Lee J, Lodi A, Margot F, Sawaya N, Wächter A (2008) An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim 5:186–204 · Zbl 1151.90028 · doi:10.1016/j.disopt.2006.10.011
[8] Dakin RJ (1965) A tree search algorithm for mixed programming problems. Comput J 8:250–255 · Zbl 0154.42004 · doi:10.1093/comjnl/8.3.250
[9] Fletcher R, Leyffer S, Toint P (2002) On the global convergence of a Filter-SQP algorithm. SIAM J Optim 13:44–59 · Zbl 1029.65063 · doi:10.1137/S105262340038081X
[10] Fourer R, Gay DM, Kernighan BW (2003) AMPL: a modelling language for mathematical programming, 2nd edn. Books/Cole–Thomson Learning, Florence
[11] Grossmann IE (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3:227–252 · Zbl 1035.90050 · doi:10.1023/A:1021039126272
[12] Gupta OK, Ravindran A (1985) Branch and bound experiments in convex nonlinear integer programming. Manag Sci 31:1533–1546 · Zbl 0591.90065 · doi:10.1287/mnsc.31.12.1533
[13] Kokkolaras M, Audet C, Dennis JE (2001) Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system. Optim Eng 2:5–29 · Zbl 1078.90595 · doi:10.1023/A:1011860702585
[14] Leyffer S (1998a) User manual for MINLP. University of Dundee, Dundee
[15] Leyffer S (1998b) User manual for MINLP-BB. University of Dundee, Dundee
[16] Nemhauser GL, Savelsbergh MWP, Sigismondi GC (1994) MINTO, a mixed INTeger optimizer. Oper Res Lett 15:47–58 · Zbl 0806.90095 · doi:10.1016/0167-6377(94)90013-2
[17] Quesada I, Grossmann IE (1992) An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput Chem Eng 16:937–947 · doi:10.1016/0098-1354(92)80028-8
[18] Zhao Z, Meza J, van Hove M (2006) Using pattern search methods for surface structure determination of nanostructures. J Phys, Condens Matter 18:8693–8706 · doi:10.1088/0953-8984/18/39/002
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