Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos

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      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.

Notes

      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].