Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos

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Figure 2.2 A schematic representation of the solution of the mp‐LP problem from Figure 2.1. In (a), the partitioning of the convex, feasible parameter space , as well as the piecewise affine nature of the optimal solution and the convex and piecewise affine nature of the objective function is shown. In (b), the Lagrange multipliers as a function of are shown.

      2.1.1 Local Properties

Consider a fixed, nominal point , which transforms the mp‐LP problem (2.2) into an LP problem of type (2.1). The solution of this LP problem at yields the solution as well as the values of the Lagrange multipliers . Based on these, the indices of the active set are identified:

(2.3a)

      (2.3b)

      Remark 2.3

      In the case where the set from Eq. (2.3) is not unique, the solution is said to be degenerate. The impact of degeneracy on the parametric solution is discussed in Chapter 2.2.

      Together with the equality constraints, which have to be satisfied for any , the following active set matrices and vectors are defined:

(2.4a)

      (2.4b)

      (2.4c)

Note that has to have full rank in order to fulfill the LICQ condition described in Chapter 1. Since the objective function is linear and the constraints are affine, the change of the solution of problem (2.2) based on the basic sensitivity theorem is given by2:

      (2.5a)

      (2.5b)

      (2.5c)

      Based on Eq. (2.5), the following statements regarding the solution around can be made: