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A branch and bound algorithm for quantified quadratic programming

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Abstract

The aim of this paper is to find the global solutions of uncertain optimization problems having a quadratic objective function and quadratic inequality constraints. The bounded epistemic uncertainties in the constraint coefficients are represented using either universal or existential quantified parameters and interval parameter domains. This approach allows to model non-controlled uncertainties by using universally quantified parameters and controlled uncertainties by using existentially quantified ones. While existentially quantified parameters could be equivalently considered as additional variables, keeping them as parameters allows maintaining the quadratic problem structure, which is essential for the proposed algorithm. The branch and bound algorithm presented in the paper handles both universally and existentially quantified parameters in a homogeneous way, without branching on their domains, and uses some dedicated numerical constraint programming techniques for finding a robust, global solution. Several examples clarify the theoretical parts and the tests demonstrate the usefulness of the proposed method.

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Notes

  1. The inconsistency seems to come from the fact that the new constraint after the change of variable involved in Appendix B of [30] contains several occurrences of universally quantified parameters, hence breaking the structure of the problem.

  2. Also called splitting or branching.

  3. Rigorously checking that an interval matrix is full rank can be done e.g. by using the interval Gauss elimination or by preconditioning the matrix using some approximation of generalized inverse of is midpoint.

  4. Necessary for rigorously handling rounding errors.

  5. constrained optimization problems usually have at least one active constraint at a minimizer.

  6. Computed by averaging over the times divided by the number of pruning steps.

  7. The cluster effect is a usual phenomenon when solving global optimization problems, see e.g., [14].

References

  1. Adjiman, C.S., Androulakis, I.P., Maranas, C.D., Floudas, C.A.: A global optimization method \(\alpha \)BB for process design. Comput. Chem. Eng. 20, 419–424 (1996)

    Article  Google Scholar 

  2. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Computer Science and Applied Mathematics (1974)

  3. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization, Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  4. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88(3), 411–424 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben-Tal, A., Nemirovski, A.: On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM J. Optim. 12(3), 811–833 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.F.: Revising hull and box consistency. In: International Conference on Logic Programming, pp. 230–244 (1999)

  8. Blankenship, J.W., Falk, J.E.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19(2), 261–281 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chabert, G., Jaulin, L.: Hull consistency under monotonicity. In: Principle and practices of constraint programming—CP2009, pp. 188–195 (2009)

  10. Clarke, F.H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadephia (1990)

    Book  MATH  Google Scholar 

  11. Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34, 922–938 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Domes, F., Neumaier, A.: Quadratic constraint propagation. Constraints, pp. 404–429 (2010)

  13. Domes, F., Neumaier, A.: Rigorous verification of feasibility. J. Global Optim. 61(2), 255–278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Du, K., Kearfott, R.B.: The cluster problem in multivariate global optimization. J. Global Optim. 5, 253–265 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghaoui, L.E., Lebret, H.: Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 18(4), 1035–1064 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghaoui, L.E., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9(1), 33–52 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goldberg, D.: What every computer scientist should know about floating-point arithmetic. Comput. Surv. 23(1), 5–48 (1991)

    Article  MathSciNet  Google Scholar 

  18. Goldsztejn, A., Domes, F., Chevalier, B.: First order rejection tests for multiple-objective optimization. J. Global Optim. 58(4), 653–672 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goualard, F.: GAOL 3.1.1: Not Just Another Interval Arithmetic Library, 4.0 edn. Laboratoire d’Informatique de Nantes-Atlantique, Nantes (2006)

  20. Hansen, E.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (1992)

    MATH  Google Scholar 

  21. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, Berlin (2001)

    MATH  Google Scholar 

  22. Jeyakumar, V., Li, G.: Robust solutions of quadratic optimization over single quadratic constraint under interval uncertainty. J. Global Optim. 55(2), 209–226 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kearfott, R., Nakao, M., Neumaier, A., Rump, S., Shary, S., van Hentenryck, P.: Standardized notation in interval analysis. In: Proceedings of XIII Baikal International School-seminar “Optimization methods and their applications” (Vol. 4, pp. 106–113). Irkutsk: Institute of Energy Systems, Baikal (2005)

  24. Kearfott, R.B.: On proving existence of feasible points in equality constrained optimization problems. Math. Program. 83(1–3), 89–100 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Kearfott, R.B.: Interval Computations: Introduction, Uses, and Resources. Euromath Bull. 2(1), 95–112 (1996)

    MathSciNet  Google Scholar 

  26. Kearfott, R.B.: Interval computations, rigour and non-rigour in deterministic continuous global optimization. Optim. Methods Softw. 26(2), 259–279 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Knueppel, O.: PROFIL/BIAS: a fast interval library. Computing 53(3–4), 277–287 (1994)

    Article  MathSciNet  Google Scholar 

  28. Lebbah, Y., Michel, C., Rueher, M.: A rigorous global filtering algorithm for quadratic constraints. Constraints 10, 47–65 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lhomme, O.: Consistency techniques for numeric CSPS. IJCAI 1, 232–238 (1993)

    Google Scholar 

  30. Li, M., Gabriel, S., Shim, Y., Azarm, S.: Interval uncertainty-based robust optimization for convex and non-convex quadratic programs with applications in network infrastructure planning. Netw. Spat. Econ. 11, 159–191 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mitsos, A.: Global optimization of semi-infinite programs via restriction of the right-hand side. Optimization 60(10–11), 1291–1308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mitsos, A., Tsoukalas, A.: Global optimization of generalized semi-infinite programs via restriction of the right hand side. J. Global Optim. 61(1), 1–17 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moore, R.: Interval Analysis. Prentice-Hall, Upper Saddle River (1966)

    MATH  Google Scholar 

  34. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press, Cambridge (1990)

    MATH  Google Scholar 

  35. Rump, S.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  36. Schichl, H., Neumaier, A.: Exclusion regions for systems of equations. SIAM J. Numer. Anal. 42(1), 383–408 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sherali, H., Adams, W.: A Reformulation–Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publ., Dordrecht (1999)

    Book  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wolfram Research Inc.: Mathematica 7.0. Wolfram Research Inc., Champaign (2008)

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Vienna, Austria

    F. Domes

  2. CNRS, IRCCYN (UMR 6597), Nantes, France

    A. Goldsztejn

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  1. F. Domes
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  2. A. Goldsztejn
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Correspondence to A. Goldsztejn.

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Domes, F., Goldsztejn, A. A branch and bound algorithm for quantified quadratic programming. J Glob Optim 68, 1–22 (2017). https://doi.org/10.1007/s10898-016-0462-0

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