Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos

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http://dx.doi.org/10.1007/978-1-4615-6103-3_6.

      12 12 Greenberg, H.J. (2000) Simultaneous primal‐dual right‐hand‐side sensitivity analysis from a strictly complementary solution of a linear program. SIAM Journal on Optimization, 10 (2), 427–442, doi: 10.1137/S1052623496310333. URL http://dx.doi.org/10.1137/S1052623496310333.

      13 13 Dantzig, G.B. (1963) Linear programming and extensions, Princeton University Press, Princeton, NJ.

      14 14 Gal, T. (1985) The historical development of parametric programming, in Parametric optimization and approximation, International series of numerical mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d'Analyse numérique, vol. 72 (eds B. Brosowski and F. Deutsch), Birkhäuser Verlag, Basel, pp. 148–165, doi: 10.1007/978‐3‐0348‐6253‐0_10. URL http://dx.doi.org/10.1007/978-3-0348-6253-0_10.

      15 15 Manne, A.S. (1953) Notes on parametric linear programming, RAND Corporation, pp. P–468. URL http://www.rand.org/pubs/papers/P468.

      16 16 Gass, S. and Saaty, T. (1955) The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly, 2 (1–2), 39–45, doi: 10.1002/nav.3800020106. URL http://dx.doi.org/10.1002/nav.3800020106.

      17 17 Orchard‐Hays, W. (1955) The RAND code for the simplex method (SX4): (For the IBM 701 electronic computer), Rand Corporation, Santa Monica, CA.

      18 18 Saaty, T.L. (1959) Coefficient perturbation of a constrained extremum. Operations Research, 7 (3), 294–302, doi: 10.1287/opre.7.3.294. URL http://dx.doi.org/10.1287/opre.7.3.294.

      19 19 Simons, E. (1962) A note on parametric linear programming. Management Science, 8 (3), 355–358, doi: 10.1287/mnsc.8.3.355. URL http://dx.doi.org/10.1287/mnsc.8.3.355.

      20 20 Karabegov, V.K. (1963) A parametric problem in linear programming. USSR Computational Mathematics and Mathematical Physics, 3 (3), 725–741, doi: 10.1016/0041‐5553(63)90297‐0. URL http://www.sciencedirect.com/science/article/pii/0041555363902970.

      21 21 Gal, T. and Davis, G.V. (1978, cop. 1979) Postoptimal analyses, parametric programming and related topics, McGraw‐Hill, London.

      22 22 Gál, T. (1995) Postoptimal analyses, parametric programming, and related topics: degeneracy, multicriteria decision making, redundancy, W. de Gruyter, Berlin and New York, 2nd edn.

      23 23 Acevedo, J. and Pistikopoulos, E.N. (1997) A multiparametric programming approach for linear process engineering problems under uncertainty. Industrial and Engineering Chemistry Research, 36 (3), 717–728, doi: 10.1021/ie960451l.

      24 24 Dua, V. and Pistikopoulos, E.N. (1999) Algorithms for the solution of multiparametric mixed‐integer nonlinear optimization problems. Industrial and Engineering Chemistry Research, 38 (10), 3976–3987, doi: 10.1021/ie980792u.

      25 25 Dinkelbach, W. (1969) Sensitivitätsanalysen und parametrische Programmierung, Ökonometrie und Unternehmensforschung / Econometrics and Operations Research, vol. 12, Springer‐Verlag, Berlin, Heidelberg.

      26 26 Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E.N. (2000) The explicit solution of model predictive control via multiparametric quadratic programming. Proceedings of the American Control Conference, vol. 2, pp. 872–876, doi: 10.1109/ACC.2000.876624.

      27 27 Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E.N. (2002) The explicit linear quadratic regulator for constrained systems. Automatica, 38 (1), 3–20, doi: 10.1016/S0005‐1098(01)00174‐1. URL http://www.sciencedirect.com/science/article/pii/S0005109801001741.

      28 28 Bemporad, A., Borrelli, F., and Morari, M. (2002) Model predictive control based on linear programming ‐ the explicit solution. IEEE Transactions on Automatic Control, 47 (12), 1974–1985, doi: 10.1109/TAC.2002.805688.

      29 29 Bemporad, A., Borrelli, F., and Morari, M. (2000) The explicit solution of constrained LP‐based receding horizon control, in Proceedings of the 39th IEEE Conference on Decision and Control, 2000, vol. 1, pp. 632–637, doi: 10.1109/CDC.2000.912837.

      30 30 Borrelli, F., Bemporad, A., and Morari, M. (2003) Geometric algorithm for multiparametric linear programming. Journal of Optimization Theory and Applications, 118 (3), 515–540, doi: 10.1023/B:JOTA.0000004869.66331.5c. URL URL http://dx.doi.org/10.1023/B%3AJOTA.0000004869.66331.5c.

      31 31 Morari, M., Jones, C.N., Zeilinger, M.N., and Baric, M. (2008) Multiparametric linear programming for control, in CCC 2008. 27th Chinese Control Conference, 2008, pp. 2–4, doi: 10.1109/CHICC.2008.4604876.

      32 32 Jones, C.N., Barić, M., and Morari, M. (2007) Multiparametric linear programming with applications to control. European Journal of Control, 13 (2–3), 152–170, doi: 10.3166/ejc.13.152‐170. URL http://www.sciencedirect.com/science/article/pii/S0947358007708178.

      33 33 Wittmann‐Hohlbein, M. and Pistikopoulos, E.N. (2012) A two‐stage method for the approximate solution of general multiparametric mixed‐integer linear programming problems. Industrial and Engineering Chemistry Research, 51 (23), 8095–8107, doi: 10.1021/ie201408p.

      34 34 Wittmann‐Hohlbein, M. and Pistikopoulos, E.N. (2013) On the global solution of multi‐parametric mixed integer linear programming problems. Journal of Global Optimization, 57 (1), 51–73, doi: 10.1007/s10898‐012‐9895‐2. URL http://dx.doi.org/10.1007/s10898-012-9895-2.

      35 35 Khalilpour, R. and Karimi, I.A. (2014) Parametric optimization with uncertainty on the left hand side of linear programs. Computers and Chemical Engineering, 60, 31–40, doi: 10.1016/j.compchemeng.2013.08.005. URL http://www.sciencedirect.com/science/article/pii/S0098135413002421.

      1 1 If does not have full rank, it is always possible to find an equivalent matrix with a reduced number of rows, which has full rank.

      2 2 Note that this solution can also be directly obtained by solving the set of equations for , which corresponds to the propagation of the solution of the LP at along the parameter space.

      3 3 This does not consider problems arising from scaling and/or round‐off computational errors.

      4 4 Consider Figure 2.4: if the constraint, which only coincides at the single point with the feasible space is chosen as part of the active set, the corresponding parametric solution from Eq. (2.5) will only be valid in that point, based on Eq. (2.6).

      5 5 The geometrical algorithms presented up to that point were limited to at most two parameters [2,25].

      6 6 In his book, Gal also considered the case of left‐hand side uncertainty, however limited to a single parameter and a single row or column [22].

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