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Some Related Problems

Chapter
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Part of the Lecture Notes in Mathematics book series (LNM, volume 2091)

Abstract

This chapter presents some applications of modal intervals to practical problems in different fields. First, the minimax problem, tackled from the definitions of the modal *- and **-semantic extensions of a continuous function. Many real life problems of practical importance can be modelled as continuous minimax optimization problems.

Keywords

Outer Approximation Minimax Problem Modal Interval Interval Vector Proper Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    M.A. Amouzegar, A global optimization method for nonlinear bilevel programming problems. IEEE Trans. Syst. Man Cybern. Part B 29, 771–777 (1999)CrossRefGoogle Scholar
  2. 3.
    J. Armengol, J. Vehí, L. Travé-Massuyès, M.Á. Sainz, Application of modal intervals to the generation of error-bounded envelopes. Reliable Comput. 7(2), 171–185 (2001)CrossRefzbMATHGoogle Scholar
  3. 4.
    J. Armengol, J. Vehí, L. Travé-Massuyès, M.Á. Sainz, Application of multiple sliding time windows to fault detection based on interval models, in 12th International Workshop on Principles of Diagnosis DX 2001. San Sicario, Italy, ed. by Sh. McIlraith, D. Theseider Dupré (2001), pp. 9–16Google Scholar
  4. 5.
    P. Basso, Optimal search for the global maximum of functions with bounded seminorm. SIAM J. Numer. Anal. 9, 888–903 (1985)MathSciNetCrossRefGoogle Scholar
  5. 6.
    E. Baumann, Optimal centered form. BIT 28, 80–87 (1987)MathSciNetCrossRefGoogle Scholar
  6. 7.
    R.J. Bhiwani, B.M. Patre, Solving first order fuzzy equations: A modal interval approach, in 2009 2nd International Conference on Emerging Trends in Engineering and Technology (ICETET) (2009), pp. 953–956Google Scholar
  7. 9.
    J. Bondia, A. Sala, A. Pic, M.A. Sainz, Controller design under fuzzy pole-placement specifications: An interval arithmetic approach. IEEE Trans. Fuzzy Syst. 14(6), 822–836 (2006)CrossRefGoogle Scholar
  8. 10.
    R. Calm, M. García-Jaramillo, J. Bondia, M.A. Sainz, J. Vehí, Comparison of interval and monte carlo simulation for the prediction of postprandial glucose under uncertainty in tipe 1 diabetes mellitus. Comput. Methods Progr. Biomed. 104, 325–332 (2011)CrossRefGoogle Scholar
  9. 12.
    J.M. Danskin, The Theory of Max-Min and Its Applications to Weapons Allocation Problems (Springer, Berlin, 1967)CrossRefGoogle Scholar
  10. 15.
    V.F. Demyanov, V.N. Malozemov, Introduction to Minimax (Dover, New York, 1990)Google Scholar
  11. 17.
    G.D. Erdmann, A new minimax algorithm and its applications to optics problems. Ph.D. thesis, University of Minnesota, USA, 2003Google Scholar
  12. 19.
    M. García-Jaramillo, R. Calm, J. Bondía, J. Vehí, Prediction of postprandial blood glucose under uncertainty and intra-patient variability in type 1 diabetes: a comparative study of three interval models. Comput. Methods Programs Biomed. 108, 325–332 (2012)CrossRefGoogle Scholar
  13. 29.
    A. Goldstein, Modal intervals revisited. Part 1: A generalized interval natural extension. Reliable Comput. 16, 130–183 (2012)Google Scholar
  14. 30.
    A. Goldstein, Modal intervals revisited. Part 2: A generalized interval mean-value extension. Reliable Comput. 16, 184–209 (2012)Google Scholar
  15. 31.
    C. Grandón, G. Chabert, B. Neveu, Generalized interval projection: a new technique for consistent domain extension, in Proceedings of the 20th International Joint Conference on Artifical Intelligence (IJCAI’07), San Francisco, CA (Morgan Kaufmann, Los Altos, 2007), pp. 94–99Google Scholar
  16. 33.
    E. Hansen, Global Optimization Using Interval Analysis (Marcel Dekker, New York, 1992)zbMATHGoogle Scholar
  17. 34.
    E. Hansen, W. Walster, Global Optimization Using Interval Analysis, 2nd edn, revised and expanded (Marcel Dekker, New York, 2004)Google Scholar
  18. 35.
    N. Hayes, System and method to compute narrow bounds on a modal interval spherical projection (Patent Number PCT/US2006/038871), 2007Google Scholar
  19. 36.
    N. Hayes, System and method to compute narrow bounds on a modal interval polynomial function (Patent Number US2008/0256155A1), 2009Google Scholar
  20. 37.
    P. Herrero, Quantified Real Constraint Solving Using Modal Intervals with Applications to Control. Ph.D. thesis, Ph.D. dissertation 1423, University of Girona, Girona (Spain), 2006Google Scholar
  21. 38.
    P. Herrero, R. Calm, J. Vehí, J. Armengol, P. Georgiou, N. Oliver, C. Tomazou, Robust fault detection system for insulin pump therapy using continuous glucose monitoring. J. Diabetes Sci. Technol. 6 (2012)Google Scholar
  22. 41.
    L. Jaulin, Reliable minimax parameter estimation. Reliable Comput. 7(3), 231–246 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 42.
    L. Jaulin, E. Walter, Guaranted bound-error parameter estimation for nonlinear models with uncertain experimental factors. Automatica 35, 849–856 (1993)CrossRefGoogle Scholar
  24. 43.
    L. Jaulin, E. Walter, Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica 29(4), 1053–1064 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 44.
    L. Jaulin, M. Kieffer, O. Didrit, E. Walter, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics (Springer, London, 2001)CrossRefGoogle Scholar
  26. 45.
    S. JongSok, Q. Zhiping, W. Xiaojun, Modal analysis of structures with uncertain-but-bounded parameters via interval analysis. J. Sound Vib. 303, 29–45 (2007)CrossRefGoogle Scholar
  27. 50.
    S. Khodaygan, M.R. Movahhedy, Tolerance analysis of assemblies with asymmetric tolerances by unified uncertaintyaccumulation model based on fuzzy logic. Int. J. Adv. Manuf. Technol. 53, 777–788 (2011)CrossRefGoogle Scholar
  28. 52.
    S. Khodaygan, M.R. Movahhedy, M. Saadat Foumani, Fuzzy-small degrees of freedom representation of linear and angular variations in mechanical assemblies for tolerance analysis and allocation. Mech. Mach. Theory, 46(4), 558–573 (2011)CrossRefzbMATHGoogle Scholar
  29. 53.
    C. Kirjer-Neto, E. Polak, On the conversion of optimization problems with max-min constraints to standard optimization problems. SIAM J. Optim. 8(4), 887–915 (1998)MathSciNetCrossRefGoogle Scholar
  30. 56.
    L. Kupriyanova, Inner estimation of the united solution set of interval algebraic system. Reliable Comput. 1(1), 15–41 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 57.
    L. Kupriyanova, Finding inner estimates of the solution sets to equations with interval coefficients. Ph.D. thesis, Saratov State University, Saratov, Russia, 2000Google Scholar
  32. 58.
    L.S. Lawsdon, A.D. Waren, GRG2 User’s Guide (Prentice hall, Englewood Cliffs, 1982)Google Scholar
  33. 59.
    S. Markov, E. Popova, C. Ullrich, On the solution of linear algebraic equations involving interval coefficients. Iterative Methods Linear Algebra IMACS Ser. Comput. Appl. Math. 3, 216–225 (1996)Google Scholar
  34. 63.
    R.M. Murray, K.J. Åström, S.P. Boyd, R.W. Brockett, G. Stein, Future directions in control in an information-rich world. IEEE Control Syst. Mag. (2003)zbMATHGoogle Scholar
  35. 65.
    A. Neumaier, Interval Methods for Systems of Equations (Cambridge University Press, Cambridge, 1990)zbMATHGoogle Scholar
  36. 67.
    K. Nickel, Optimization using interval mathematics. Freibg. Intervall Ber. 1, 25–47 (1986)Google Scholar
  37. 69.
    P. Herrero, L. Jaulin, J. Vehí, M.A. Sainz Guaranteed set-point computation with application to the control of a sailboat. Int. J. Control Autom. Syst. 8, 1–7 (2010)Google Scholar
  38. 70.
    E. Polak, in Optimization. Algorithms and Consistent Approximations. Applied Mathematical Sciences, vol. 124 (Springer, New York, 1997)Google Scholar
  39. 71.
    S. Ratschan, Applications of quantified constraint solving (2002). http://www.mpi-sb.mpg.de/~ratschan/appqcs.html
  40. 74.
    A. Revert, R. Calm, J. Vehí, J. Bondia, Calculation of the best basal-bolus combination for postprandial glucose control in insulin pump therapy. IEEE Trans. Biomed. Eng. 58, 274–281 (2011)CrossRefGoogle Scholar
  41. 76.
    B. Rustem, M. Howe, Algorithms for Worst-Case Design and Applications to Risk Management (Princeton University Press, Princeton, 2002)zbMATHGoogle Scholar
  42. 77.
    M.Á. Sainz, J.M. Baldasano, Modelo matemático de autodepuración para el bajo Ter (in spanish). Technical report, Junta de Sanejament, Generalitat de Catalunya, 1988Google Scholar
  43. 78.
    M.Á. Sainz, J. Armengol, J. Vehí, Fault diagnosis of the three tanks system using the modal interval analysis. J. Process Control 12(2), 325–338 (2002)CrossRefGoogle Scholar
  44. 79.
    S.P. Shary, Solving the linear interval tolerance problem. Math. Comput. Simul. 39(2), 145–149 (1995)MathSciNetGoogle Scholar
  45. 80.
    S.P. Shary, Algebraic approach to the interval linear static identification, tolerance and control problems, or one more application of kaucher arithmetic. Reliable Comput. 2(1), 3–33 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 84.
    S.P. Shary, Outer estimation of generalized solution sets to interval linear systems. Reliable Comput. 5, 323–335 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 85.
    S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Comput. 8, 321–418 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 91.
    J. Vehí, J. Rodellar, M.Á. Sainz, J. Armengol, Analysis of the robustness of predictive controllers via modal intervals. Reliable Comput. 6(3), 281–301 (2000)CrossRefzbMATHGoogle Scholar
  49. 92.
    Y. Wang, Semantic tolerance modeling based on modal interval, in NSF Workshop on Reliable Engineering Computing, Savannah, Georgia, 2006Google Scholar
  50. 93.
    Y. Wang, Closed-loop analysis in semantic tolerance modeling. J. Mech. Des. 130, 061701–061711 (2008)CrossRefGoogle Scholar
  51. 95.
    Y. Wang, Semantic tolerance modeling with generalized intervals. J. Mech. Des. 130, 081701–081708 (2008)CrossRefGoogle Scholar
  52. 98.
    S. Zuche, A. Neumaier, M.C. Eiermann, Solving minimax problems by interval methods. BIT 30, 742–751 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Informática Matemática Aplicada y Estadística Escola Politecnica SuperiorUniversity of GironaGironaSpain
  2. 2.Enginyeria Elèctrica Electrònica i Automàtica Escola Politecnica SuperiorUniversity of GironaGironaSpain
  3. 3.Imperial College LondonLondonUK
  4. 4.Matemática Económica Financiera y Actuarial Facultad de Economia y EmpresaUniversitat de BarcelonaBarcelonaSpain

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