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Piecewise-Linear Approximations of Multidimensional Functions

  • R. Misener
  • C. A. Floudas
Article

Abstract

We develop explicit, piecewise-linear formulations of functions f(x):ℝ n ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.

Approximate optimization Linear interpolation Simplices EPA Complex Emissions Model 

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References

  1. 1.
    Tuy, H.: Convex Analysis and Global Optimization. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1998) zbMATHGoogle Scholar
  2. 2.
    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1999) zbMATHGoogle Scholar
  3. 3.
    Floudas, C.A.: Deterministic Global Optimization: Theory, Methods and Applications. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (2000) Google Scholar
  4. 4.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (2000) zbMATHGoogle Scholar
  5. 5.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Applications, Software, and Applications. Nonconvex Optimization and Its Applications. Kluwer Academic, Norwell (2002) Google Scholar
  6. 6.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (2003) Google Scholar
  7. 7.
    Floudas, C.A., Pardalos, P.M.: State of the art in global optimization: Computational methods and applications—preface. J. Glob. Optim. 7(2), 113 (1995) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Floudas, C.A., Pardalos, P.M. (eds.): State of the Art in Global Optimization: Computational Methods and Applications. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1996) zbMATHGoogle Scholar
  9. 9.
    Floudas, C.A., Pardalos, P.M. (eds.): Frontiers in Global Optimization. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (2004) zbMATHGoogle Scholar
  10. 10.
    Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.: Global optimization in the 21st century: Advances and challenges. Comput. Chem. Eng. 29, 1185–1202 (2005) CrossRefGoogle Scholar
  11. 11.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45, 3–38 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kosmidis, V.D., Perkins, J.D., Pistikopoulos, E.N.: Optimization of well oil rate allocations in petroleum fields. Ind. Eng. Chem. Res. 43(14), 3513–3527 (2004) CrossRefGoogle Scholar
  13. 13.
    Kosmidis, V.D., Perkins, J.D., Pistikopoulos, E.N.: A mixed integer optimization formulation for the well scheduling problem on petroleum fields. Comput. Chem. Eng. 29(7), 1523–1541 (2005) CrossRefGoogle Scholar
  14. 14.
    Buitrago, S., Rodríguez, E., Espin, D.: Global optimization techniques in gas allocation for continuous flow gas lift systems. In: SPE Gas Technology Symposium, Calgary, Alberta, Canada. Society of Petroleum Engineers. SPE 35616 (1996) Google Scholar
  15. 15.
    Misener, R., Gounaris, C.E., Floudas, C.A.: Global optimization of gas lifting operations: A comparative study of piecewise linear formulations. Ind. Eng. Chem. Res. 48(13), 6098–6104 (2009) CrossRefGoogle Scholar
  16. 16.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988) zbMATHGoogle Scholar
  17. 17.
    Floudas, C.A.: Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, New York (1995) zbMATHGoogle Scholar
  18. 18.
    Sherali, H.D.: On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions. Oper. Res. Lett. 28(4), 155–160 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Keha, A.B., de Farias Jr., I.R., Nemhauser, G.L.: Models for representing piecewise linear cost functions. Oper. Res. Lett. 32(1), 44–48 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Williams, H.P.: Model Building in Mathematical Programming. Wiley, Chichester (1978) zbMATHGoogle Scholar
  21. 21.
    Zhang, H., Wang, S.: Linearly constrained global optimization via piecewise-linear approximation. J. Comput. Appl. Math. 214(1), 111–120 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Magnani, A., Boyd, S.P.: Convex piecewise-linear fitting. Optim. Eng. 10, 1–17 (2009) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Rosen, J.B., Pardalos, P.M.: Global minimization of large-scale constrained concave quadratic problems by separable programming. Math. Program. 34(2), 163–174 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science. Springer, Berlin (1987) zbMATHCrossRefGoogle Scholar
  25. 25.
    Meyer, C.A., Floudas, C.A.: Global optimization of a combinatorially complex generalized pooling problem. AIChE J. 52(3), 1027–1037 (2006) CrossRefGoogle Scholar
  26. 26.
    Karuppiah, R., Grossmann, I.E.: Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 30, 650–673 (2006) CrossRefGoogle Scholar
  27. 27.
    Wicaksono, D.S., Karimi, I.A.: Piecewise MILP under-and overestimators for global optimization of bilinear programs. AIChE J. 54(4), 991–1008 (2008) CrossRefGoogle Scholar
  28. 28.
    Gounaris, C.E., Misener, R., Floudas, C.A.: Computational comparison of piecewise-linear relaxations for pooling problems. Ind. Eng. Chem. Res. 48(12), 5742–5766 (2009) CrossRefGoogle Scholar
  29. 29.
    Pham, V., Laird, C., El-Halwagi, M.: Convex hull discretization approach to the global optimization of pooling problems. Ind. Eng. Chem. Res. 48, 1973–1979 (2009) CrossRefGoogle Scholar
  30. 30.
    Mangasarian, O.L., Rosen, J.B., Thompson, M.E.: Global minimization via piecewise-linear underestimation. J. Glob. Optim. 32(1), 1–9 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \({\mathcal{C}}^{2}\)-continuous problems: I. univariate functions. J. Glob. Optim. 42(1), 51–67 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \({{\mathcal{C}}^{2}}\)-continuous problems: II. multivariate functions. J. Glob. Optim. 42(1), 69–89 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Chien, M., Kuh, E.: Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision. IEEE Trans. Circuits Syst. 24(6), 305–317 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Meyer, C.A., Floudas, C.A.: Trilinear monomials with positive or negative domains: Facets of the convex and concave envelopes. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 327–352. Kluwer Academic, Dordrecht (2003) Google Scholar
  35. 35.
    Meyer, C.A., Floudas, C.A.: Trilinear monomials with mixed sign domains: Facets of the convex and concave envelopes. J. Glob. Optim. 29(2), 125–155 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103(2), 207–224 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Hughes, R.B., Anderson, M.R.: Simplexity of the cube. Discrete Math 158(1–3), 99–150 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part 1—convex underestimating problems. Math. Program. 10(1), 147–175 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143–182 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19(4), 403–424 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Carathéodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911) zbMATHCrossRefGoogle Scholar
  43. 43.
    Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In: Lawrence, J. (ed.) Proceedings of the Fifth International Conference on Operational Research. pp. 447–454 (1970) Google Scholar
  44. 44.
    Forrest, J.J.H., Hirst, J.P.H., Tomlin, J.A.: Practical solution of large mixed integer programming problems with umpire. Manage. Sci. 20, 736–773 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    ILOG CPLEX 9.0.2 User’s Manual; ILOG, Mountain View (2005) Google Scholar
  46. 46.
    Brooke, A., Kendrick, D., Meeraus, A.: GAMS: A User’s Guide. GAMS Development Corporation (2005) Google Scholar
  47. 47.
    Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and Its Applications. Kluwer Academic, Dordrecht (1999) zbMATHGoogle Scholar
  48. 48.
    40CFR80.45. Code of federal regulations: complex emissions model, July 2007. http://frwebgate.access.gpo.gov/cgi-bin/get-cfr.cgi
  49. 49.
    40CFR80.41. Code of federal regulations: standards and requirements for compliance, June 2008. http://frwebgate.access.gpo.gov/cgi-bin/get-cfr.cgi
  50. 50.
    Furman, K.C., Androulakis, I.P.: A novel MINLP-based representation of the original complex model for predicting gasoline emissions. Comput. Chem. Eng. 32, 2857–2876 (2008) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Chemical EngineeringPrinceton UniversityPrincetonUSA

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