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.
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References
M.A. Amouzegar, A global optimization method for nonlinear bilevel programming problems. IEEE Trans. Syst. Man Cybern. Part B 29, 771–777 (1999)
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)
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. 9–16
P. Basso, Optimal search for the global maximum of functions with bounded seminorm. SIAM J. Numer. Anal. 9, 888–903 (1985)
E. Baumann, Optimal centered form. BIT 28, 80–87 (1987)
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. 953–956
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)
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)
J.M. Danskin, The Theory of Max-Min and Its Applications to Weapons Allocation Problems (Springer, Berlin, 1967)
V.F. Demyanov, V.N. Malozemov, Introduction to Minimax (Dover, New York, 1990)
G.D. Erdmann, A new minimax algorithm and its applications to optics problems. Ph.D. thesis, University of Minnesota, USA, 2003
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)
A. Goldstein, Modal intervals revisited. Part 1: A generalized interval natural extension. Reliable Comput. 16, 130–183 (2012)
A. Goldstein, Modal intervals revisited. Part 2: A generalized interval mean-value extension. Reliable Comput. 16, 184–209 (2012)
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. 94–99
E. Hansen, Global Optimization Using Interval Analysis (Marcel Dekker, New York, 1992)
E. Hansen, W. Walster, Global Optimization Using Interval Analysis, 2nd edn, revised and expanded (Marcel Dekker, New York, 2004)
N. Hayes, System and method to compute narrow bounds on a modal interval spherical projection (Patent Number PCT/US2006/038871), 2007
N. Hayes, System and method to compute narrow bounds on a modal interval polynomial function (Patent Number US2008/0256155A1), 2009
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), 2006
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)
L. Jaulin, Reliable minimax parameter estimation. Reliable Comput. 7(3), 231–246 (2001)
L. Jaulin, E. Walter, Guaranted bound-error parameter estimation for nonlinear models with uncertain experimental factors. Automatica 35, 849–856 (1993)
L. Jaulin, E. Walter, Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica 29(4), 1053–1064 (1993)
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)
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)
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)
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)
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)
L. Kupriyanova, Inner estimation of the united solution set of interval algebraic system. Reliable Comput. 1(1), 15–41 (1995)
L. Kupriyanova, Finding inner estimates of the solution sets to equations with interval coefficients. Ph.D. thesis, Saratov State University, Saratov, Russia, 2000
L.S. Lawsdon, A.D. Waren, GRG2 User’s Guide (Prentice hall, Englewood Cliffs, 1982)
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)
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)
A. Neumaier, Interval Methods for Systems of Equations (Cambridge University Press, Cambridge, 1990)
K. Nickel, Optimization using interval mathematics. Freibg. Intervall Ber. 1, 25–47 (1986)
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)
E. Polak, in Optimization. Algorithms and Consistent Approximations. Applied Mathematical Sciences, vol. 124 (Springer, New York, 1997)
S. Ratschan, Applications of quantified constraint solving (2002). http://www.mpi-sb.mpg.de/~ratschan/appqcs.html
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)
B. Rustem, M. Howe, Algorithms for Worst-Case Design and Applications to Risk Management (Princeton University Press, Princeton, 2002)
M.Á. Sainz, J.M. Baldasano, Modelo matemático de autodepuración para el bajo Ter (in spanish). Technical report, Junta de Sanejament, Generalitat de Catalunya, 1988
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)
S.P. Shary, Solving the linear interval tolerance problem. Math. Comput. Simul. 39(2), 145–149 (1995)
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)
S.P. Shary, Outer estimation of generalized solution sets to interval linear systems. Reliable Comput. 5, 323–335 (1999)
S.P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity. Reliable Comput. 8, 321–418 (2002)
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)
Y. Wang, Semantic tolerance modeling based on modal interval, in NSF Workshop on Reliable Engineering Computing, Savannah, Georgia, 2006
Y. Wang, Closed-loop analysis in semantic tolerance modeling. J. Mech. Des. 130, 061701–061711 (2008)
Y. Wang, Semantic tolerance modeling with generalized intervals. J. Mech. Des. 130, 081701–081708 (2008)
S. Zuche, A. Neumaier, M.C. Eiermann, Solving minimax problems by interval methods. BIT 30, 742–751 (1990)
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Sainz, M.A., Armengol, J., Calm, R., Herrero, P., Jorba, L., Vehi, J. (2014). Some Related Problems. In: Modal Interval Analysis. Lecture Notes in Mathematics, vol 2091. Springer, Cham. https://doi.org/10.1007/978-3-319-01721-1_10
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DOI: https://doi.org/10.1007/978-3-319-01721-1_10
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