On-line optimization via off-line optimization! A guided tour to multi-parametric mixed integer and continuous programming
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The last decades have seen the rapid increase in the use of optimization- based techniques for improved design, control and operation of various type of engineering systems. The prime difficulties in applying these types of techniques to real plants arise from the unavoidable presence of variations in the problem parameters such as fluctuations in uncertain inputs and measurements, or variations in inherent system properties and characteristics. These variations readily translate to deviations from the prescribed optimal point, thus, either failing to exploit fully the benefits of the optimization based solution or requiring the repetitive solution of the problem for different values of the problem parameters. Parametric programming is a technique that determines computationally inexpensively the exact mapping of the optimal solution profile in the space of the system parameters. In this way the repetition of the problem solution is avoided, while the optimal solution can readily adapt to the system variability. In our group we have developed algorithms for multi-parametric (mixed integer) linear, quadratic, non-linear and dynamic optimization problems that are commonly encountered in (i) optimization under uncertainty, where the uncertainties are the problem parameters, (ii) multi-objective optimization where the different objectives play the role of the parameters and (iii) on-line control and optimization where the process states correspond to the parameters. In this presentation, we will first give an overview of the mathematical foundations of multi-parametric programming for different classes of mathematical models. We will then discuss its application in the context of model-based optimal control, with emphasis on the control of chemical, biomedical and automotive systems.