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Multi-parametric Optimization and Control. Efstratios N. PistikopoulosЧитать онлайн книгу.

Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos


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2.1. In (a), the partitioning of the convex, feasible parameter space images, as well as the piecewise affine nature of the optimal solution images and the convex and piecewise affine nature of the objective function images is shown. In (b), the Lagrange multipliers images as a function of images are shown.

      2.1.1 Local Properties

      (2.3b)equation

      Remark 2.3

      In the case where the set images 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 images, the following active set matrices and vectors are defined:

      (2.4b)equation

      (2.4c)equation

      (2.5a)equation

      (2.5b)equation

      (2.5c)equation

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

       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 Chapter 1 have to hold.

       In order for Eq. (2.5) to remain the optimal solution around a nominal point , it needs to be feasible, i.e.(2.6a) (2.6b) Note that since the values of the Lagrange multipliers do not change as a function of , the optimality requirement from the Karush‐Kuhn‐Tucker conditions can be omitted from the construction of the feasible region.

      Lemma 2.1

      Every critical region is uniquely defined by its active set.

       Proof

      By inspection of Eq. (2.7), it is clear that the differences between any two critical regions are the values of images, images, and images, respectively, which only depend on the active set images. Thus, the set of active constraints images uniquely defines the critical region images, which completes the proof.

      Lemma 2.2

      The maximum number of critical regions, images, for problem 2.2 is given by

      (2.8)equation

      Proof

      Consider images. Then an optimal solution of the resulting LP problem is guaranteed to lie in a vertex, thus featuring images active constraints. However, as the equality constraints have to be fulfilled for all images,


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