Multi-parametric Optimization and Control. Efstratios N. Pistikopoulos
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Figure 2.2 shows some of the properties of the solution of the mpLP problem 2.2.
Remark 2.1
Note that it is possible to add a scalar
Figure 2.1 The difference between the solution of an LP and an mp‐LP problem (black dot and line, respectively), where the mp‐LP problem is obtained by treating
2.1 Solution Properties
Remark 2.2
Due to the similarities between mp‐LP and multi‐parametric quadratic programming (mp-QP) problems, the different solution strategies available are discussed in detail in Chapter 4.
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
2.1.1 Local Properties
Consider a fixed, nominal point
(2.3a)
(2.3b)
Remark 2.3
In the case where the set
Together with the equality constraints, which have to be satisfied for any
(2.4a)
(2.4b)
(2.4c)
Note that
(2.5a)
(2.5b)
(2.5c)
Based on Eq. (2.5), the following statements regarding the solution around
The optimization variables are affine functions of the parameter .
In the case of mp‐LP problems, the values of the Lagrange multipliers and do not change as a function of around a nominal point .
The square matrix is invertible since the SCS and LICQ conditions of