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Safe and tight linear estimators for global optimization

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Abstract.

Global optimization problems are often approached by branch and bound algorithms which use linear relaxations of the nonlinear constraints computed from the current variable bounds. This paper studies how to derive safe linear relaxations to account for numerical errors arising when computing the linear coefficients. It first proposes two classes of safe linear estimators for univariate functions. Class-1 estimators generalize previously suggested estimators from quadratic to arbitrary functions, while class-2 estimators are novel. When they apply, class-2 estimators are shown to be tighter theoretically (in a certain sense) and almost always tighter numerically. The paper then generalizes these results to multivariate functions. It shows how to derive estimators for multivariate functions by combining univariate estimators derived for each variable independently. Moreover, the combination of tight class-1 safe univariate estimators is shown to be a tight class-1 safe multivariate estimator. Finally, multivariate class-2 estimators are shown to be theoretically tighter (in a certain sense) than multivariate class-1 estimators.

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References

  1. Adjiman, C., Dallwig, S., Floudas, C., Neumaier, A.: A global optimization method, αbb, for general twice-differentiable constrained NLPs - I. theoretical advances. Comput. Chem. Engin. 22, 1137–1158 (1998)

    Article  Google Scholar 

  2. Benhamou, F., McAllester, D., Van Hentenryck, P.: CLP(Intervals) Revisited. In: Proceedings of the International Symposium on Logic Programming (ILPS-94), Ithaca, NY, Nov. 1994, pp. 124–138

  3. Benhamou, F., Older, W.: Applying Interval Arithmetic to Real, Integer and Boolean Constraints. J. Logic Program. 32 (1), 1–24 (1997) July

    Article  MATH  MathSciNet  Google Scholar 

  4. Canny, J.: Some algebraic and geometric computations in pspace. In: Proceedings of the 20th Symposium on the Theory of Computation, 1988, pp. 460–467

  5. Garloff, J., Jansson, C., Smith, A.P.: Lower bound functions for polynomials. J. Comput. Appl. Math. 157, 207–225 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: Nonlinear convex hull relaxation. To appear, 2003

  7. Hentenryck, P.V., McAllister, D., Kapur, D.: Solving Polynomial Systems Using a Branch and Prune Approach. SIAM J. Numer. Anal. 34 (2), (1997)

  8. Jansson, C.: Rigorous error bounds for the optimal value of linear programming problems. In: Proceedings of the First International Workshop on Global Constrained Optimization and Constraint Satisfaction, COCOS 2002, 2003, pp. 59–70

  9. Jansson, C.: A rigorous lower bound for the optimal value of convex optimization problems. J. Global Optim. 28 (1), 121–137 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lebbah, Y., Rueher, M., Michel, C.: A global filtering algorithm for handling systems of quadratic equations and inequations. Lect. Notes Comput. Sci. 2470, 109–123 (2002)

    Article  MathSciNet  Google Scholar 

  11. Lee, S., Grossmann, I.E.: A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems. Comput, Chem. Engin. 25, 1675–1697 (2001)

    Google Scholar 

  12. Lhomme, O.: Consistency Techniques for Numerical Constraint Satisfaction Problems. In: Proceedings of the 1993 International Joint Conference on Artificial Intelligence, Chamberry, France, Aug. 1993

  13. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems. Math. Program. 10, 146–175 (1976)

    MathSciNet  Google Scholar 

  14. Michel, C., Lebbah, Y., Rueher, M.: Safe embedding of the simplex algorithm in a CSP framework. In: Proceedings CPAIOR’03, 2003, pp. 210–220

  15. Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer programming. To appear

  16. Quesada, I., Grossmann, I.E.: A global optimization algorithm for linear fractional and bilinear programs. J. Global Optim. 6, 39–76 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Engin. 19, 551–566 (1995)

    Article  Google Scholar 

  18. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming, vol 65 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, 2002

  19. Van Hentenryck, P., Michel, L., Deville, Y.: Numerica: A Modeling Language for Global Optimization. The MIT Press, Cambridge, MA, USA, 1997

  20. Zamora, J.M., Grossmann, I.E.: A comprehensive global optimization approach for the synthesis of heat exchanger networks with no stream splits. Comp. Chem. Engin. 21 (70), S65–S (1997)

    Google Scholar 

  21. Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. J. Global Optim. 14, 217–249 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Brown University, 1910, Providence, RI 02912, USA

    Glencora Borradaile & Pascal Van Hentenryck

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  1. Glencora Borradaile
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  2. Pascal Van Hentenryck
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Correspondence to Glencora Borradaile.

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Borradaile, G., Hentenryck, P. Safe and tight linear estimators for global optimization. Math. Program. 102, 495–517 (2005). https://doi.org/10.1007/s10107-004-0533-8

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  • DOI: https://doi.org/10.1007/s10107-004-0533-8

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