Efstratios N. Pistikopoulos - Multi-parametric Optimization and Control

(1.36a)

(1.36b)

(1.36c)

Based on [10,11], Eqs. ( 1.36a)–( 1.36c) are reformulated as

(1.37a)

(1.37b)

(1.37c)

(1.37d)

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

(1.38)

The remainder of this book is organized in two parts. In the first part, the theoretical and algorithmic essentials of multi‐parametric programming problems will be established. These include algorithms for the solution of multi‐parametric linear programming (mp‐LP), multi‐parametric quadratic programming (mp‐QP), multi‐parametric mixed‐integer linear programming (mp‐MILP), and multi‐parametric mixed‐integer quadratic programming (mp‐MIQP) problems. On the other hand, the latter of these parts is focused on the applications of multi‐parametric programming and specifically on its utilization to provide solutions to receding horizon optimization problems such as model predictive control.

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8 8 Kouramas, K.I., Panos, C., Faísca, N.P., and Pistikopoulos, E.N. (2013) An algorithm for robust explicit/multi‐parametric model predictive control. Automatica, 49 (2), 381–389, doi: 10.1016/j.automatica.2012.11.035. URL http://www.sciencedirect.com/science/article/pii/S0005109812005717.

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11 11 Williams, H.P. (2013) Model building in mathematical programming, Wiley, Hoboken, NJ, 5th edn.

1 1A function is called pseudo‐convex if for all feasible where we have .

2 2A function is called quasi‐convex if for all feasible and we have . Note that a quasi‐concave function is a function whose negative is quasi‐convex.

Consider the following linear programming (LP) problem:

(2.1)

where , , , , , is a compact polytope and is assumed to have full rank. 1This problem formulation requires the deterministic knowledge of all elements of problem ( 2.1). However, in many applications values such as prices, demand, and risk might change over time and have great impact on the solution. In order to take this into account, the uncertainty can be considered explicitly by formulating a multi‐parametric linear programming (mp‐LP) problem:

(2.2)

where , , , and is a compact polytope. The difference between problems ( 2.1) and ( 2.2) is the presence of the bounded uncertain parameters in the constraints. As a result, the solution , the Lagrange multipliers and , and the optimal objective function of problem ( 2.2) are obtained as a function of , i.e. , , , and , respectively. A schematic representation of the difference between problem ( 2.1) and ( 2.2) is shown in Figure 2.1.