Journal of Optimization Theory and Applications

, Volume 167, Issue 2, pp 617–643 | Cite as

Continuous Piecewise Linear Delta-Approximations for Univariate Functions: Computing Minimal Breakpoint Systems

Article

Abstract

For univariate functions, we compute optimal breakpoint systems subject to the condition that the piecewise linear approximator, under-, and over-estimator never deviate more than a given \(\delta \)-tolerance from the original function over a given finite interval. The linear approximators, under-, and over-estimators involve shift variables at the breakpoints allowing for the computation of an optimal piecewise linear, continuous approximator, under-, and over-estimator. We develop three non-convex optimization models: two yield the minimal number of breakpoints, and another in which, for a fixed number of breakpoints, the breakpoints are placed such that the maximal deviation is minimized. Alternatively, we use two heuristics which compute the breakpoints subsequently, solving small non-convex problems. We present computational results for 10 univariate functions. Our approach computes breakpoint systems with up to one order of magnitude less breakpoints compared to an equidistant approach.

Keywords

Global optimization Nonlinear programming Mixed-integer nonlinear programming Non-convex optimization 

Mathematics Subject Classification

90C26 

References

  1. 1.
    Kallrath, J.: Combined strategic and operational planning—an MILP success story in chemical industry. OR Spectrum 24(3), 315–341 (2002)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Kallrath, J., Maindl, T.I.: Real Optimization with SAP-APO. Springer (2006)Google Scholar
  3. 3.
    Zheng, Q.P., Rebennack, S., Iliadis, N.A., Pardalos, P.M.: Optimization models in the natural gas industry. In: Rebennack, S., Pardalos, P.M., Pereira, M.V., Iliadis, N.A. (eds.) Handbook of Power Systems I, chap. 6, pp. 121–148. Springer (2010)Google Scholar
  4. 4.
    Frank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey I. Energy Syst. 3(3), 221–258 (2012)CrossRefGoogle Scholar
  5. 5.
    Frank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey II. Energy Syst. 3(3), 259–289 (2012)CrossRefGoogle Scholar
  6. 6.
    Tomlin, J.A.: Special ordered sets and an application to gas supply operating planning. Math. Progr. 45, 69–84 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Beale, E.L.M., Tomlin, J.A.: Special facilities in a general mathematical programming system for nonconvex problem using ordered sets of variables. In: Lawrence, J. (ed.) Proceedings of the Fifth International Conference on Operational Research 1969, pp. 447–454. Tavistock Publishing (1970)Google Scholar
  8. 8.
    Beale, E.L.M.: Two transportation oroblems. In: Proceedings of the Third International Conference on Operational Research 1963, pp. 780–788. Dunod, Paris and English Universities Press (1963)Google Scholar
  9. 9.
    Beale, E.M.L., Forrest, J.J.H.: Global optimization using special ordered sets. Math. Progr. 10, 52–69 (1976)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    de Farias Jr, I.R., Johnson, E.L., Nemhauser, G.L.: A generalized assignment problem with special ordered sets. A polyhedral approach. Math. Progr. 89, 187–203 (2000)CrossRefGoogle Scholar
  11. 11.
    de Farias Jr, I.R., Zhao, M., Zhao, H.: A special ordered set approach for optimizing a discontinuous separable piecewise linear function. Oper. Res. Lett. 36, 234–238 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Leyffer, S., Sartenaer, A., Wanufelle, E.: Branch-and-refine for mixed-integer nonconvex global optimization (2008)Google Scholar
  13. 13.
    Vielma, J.P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 53, 303–315 (2009)MathSciNetGoogle Scholar
  14. 14.
    Vielma, J.P., Nemhauser, G.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Progr. 128, 49–72 (2011)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rosen, J.B., Pardalos, P.M.: Global minimization of large-scale constrained concave quadratic problems by separable programming. Math. Progr. 34, 163–174 (1986)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science. Springer, Berlin (1987)MATHCrossRefGoogle Scholar
  17. 17.
    Geißler, B., Martin, A., Morsi, A., Schewe, L.: Using piecewise linear functions for solving MINLPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 287–314. Springer (2012)Google Scholar
  18. 18.
    Geißler, B.: Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. Dissertation, Universität Erlangen-Nürnberg (2011)Google Scholar
  19. 19.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming. SIAM Rev. 35, 380–429 (1993)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Lopez, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Tsoukalas, A., Rustem, B.: A feasible point adaptation of the blankenship and falk algorithm for semi-infinite programming. Optim. Lett. 5(4), 705–716 (2011)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Blankenship, J.W., Falk, J.E.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19, 268–281 (1976)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rebennack, S., Kallrath, J.: Continuous piecewise linear delta-approximations for bivariate and multivariate functions. J. Optim. Theory Appl. doi:10.1007/s10957-014-0688-2
  24. 24.
    Duistermaat, J., Kol, J.: Multidimensional real analysis I: differentiation. Cambridge Studies in Advanced Mathematics (2004)Google Scholar
  25. 25.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to global optimization, 2nd edn. Kluwer (2000)Google Scholar
  26. 26.
    Kallrath, J., Rebennack, S.: Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate functions. In: Butenko, S., Floudas, C., Rassias, T. (eds.) Optimization in Science and Engineering. Springer (2014)Google Scholar
  27. 27.
    Maranas, C., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4, 135–170 (1994)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Division of Economics and BusinessColorado School of MinesGoldenUSA
  2. 2.Department of AstronomyUniversity of FloridaGainesvilleUSA
  3. 3.BASF SEScientific ComputingLudwigshafenGermany

Personalised recommendations