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

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Notes

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

2 2 A 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.

2 Multi‐parametric Linear Programming

Consider the following linear programming (LP) problem:

(2.1) equation

where images , images , images , images , images , images is a compact polytope and images 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) equation

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

Remark 2.1

Note that it is possible to add a scalar images to the objective function of an LP problem without influencing the optimal solution. Similarly, it is possible to add an arbitrary scaling function images to an mp‐LP problem without influencing the optimal solution.

Image described by caption.

Figure 2.1 The difference between the solution of an LP and an mp‐LP problem (black dot and line, respectively), where the mp‐LP problem is obtained by treating images as a parameter. In (a), the solution is a single point in one of the vertices of the feasible space, while in (b) the solution is a function of the parameter images .

2.1 Solution Properties

Remark 2.2

Due to the similarities between mp‐LP and multi‐parametric quadratic programming (mp-QP) problems, the different solution strategies available are discussed in detail in Chapter 4.

Figure 2.2 A schematic representation of the solution of the mp‐LP problem from Figure

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Notes

2 Multi‐parametric Linear Programming

Remark 2.1

2.1 Solution Properties Remark 2.2

2.1 Solution Properties

Remark 2.2