Читать книгу Multi-parametric Optimization and Control - Efstratios N. Pistikopoulos - Страница 51

Consider the following linear programming (LP) problem:

(2.1)

where , , , , , is a compact polytope and is assumed to have full rank.1 This 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.

Figure 2.2 shows some of the properties of the solution of the mpLP problem 2.2.