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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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of the projection are the following:

       Solving a multi‐parametric linear programming (mp‐LP) problem (see e.g. [8])

       Performing a Fourier–Motzkin (FM) elimination (see, e.g. [9])

      In addition, the concept of a hybrid projection is introduced:

      Definition 1.12 (Hybrid Projection)

      Consider the set images. Then, the hybrid projection images of images onto images is defined as

      (1.34)equation

      By inspection it is clear that (i) images is obtained by performing at most images projections, one for each combination of the binary variables and consequently (ii) images is generally a union of at most images possibly overlapping polytopes.

      A hybrid projection can thereby be performed by solving a multi‐parametric mixed‐integer programming problem purely based on feasibility requirements.

      1.3.3 Modeling of the Union of Polytopes

      The aim is to represent a union of polytopes images as a single set of linear inequality constraints in order to seamlessly include them within multi‐parametric programming problems. However, in order to address the possible non‐convexity within unions of polytopes, the introduction of suitable binary variables is required. First, consider that a point images if and only if there exists at least one images such that images. Thus, one binary variable images is defined such that

      (1.35a)equation

      (1.35b)equation

      (1.36b)equation

      (1.37a)equation

      (1.37b)equation

      (1.37c)equation

      (1.37d)equation

      where images, images. Thus, the final formulation of the union as a set of linear inequality constraints featuring binary variables is given as

      (1.38)equation

      1 1 Fiacco, A.V. (1983) Introduction to sensitivity and stability analysis in nonlinear programming, Mathematics in science and engineering, vol. v, 165, Academic Press, New York.

      2 2 Boyd, S.P. and Vandenberghe, L. (2004) Convex optimization, Cambridge University Press, Cambridge, UK and New York.

      3 3 Telgen, J. (1981) Redundancy and linear programs, Mathematisch Centrum, Amsterdam.

      4 4 Karwan, M.H., Lotfi, V., Telgen, J., and Zionts, S. (1983) Redundancy in mathematical programming: a state‐of‐the‐art survey, Lecture notes in economics and mathematical systems, vol. 206, Springer‐Verlag, Berlin and New York.

      5 5 Brearley, A.L., Mitra, G., and Williams, H.P. (1975) Analysis of mathematical programming problems prior to applying the simplex algorithm. Mathematical Programming, 8 (1), 54–83, doi: 10.1007/BF01580428.

      6 6 Suard, R., Lofberg, J., Grieder, P., Kvasnica, M., and Morari, M. (2004) Efficient computation of controller partitions in multi‐parametric programming, in 2004 43rd IEEE Conference on Decision and Control (CDC), vol. 4, pp. 3643–3648, doi: 10.1109/CDC.2004.1429297.

      7 7 Jones,


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