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of explicit model predictive controllers.

      Part I focuses on the algorithmic developments of multi‐parametric programming. It begins with an overview of the basic sensitivity theorem and progresses to describe solution strategies for linear, quadratic, and mixed‐integer multi‐parametric problems. A chapter for the solution of the aforementioned classes of problems in MATLAB^© is also included. Part I concludes with developments of multi‐parametric programming for the solution of other classes of problems.

      Part II focuses on multi‐parametric model predictive control and its extension for the solution of other control problems. This section concludes with the presentation and applications of PAROC^© framework, a framework for the development and closed loop validation of multi‐parametric model predictive controllers.

      As this book is the outcome of the research work carried out over the last 30 years at the Centre for Process Systems Engineering of Imperial College London and the Texas A&M Energy Institute of Texas A&M University, many colleagues, former and current PhD students, and post‐doctorate/research associates have been involved in the presented work. While a number of them are involved in this project as co‐authors, we would like to take the opportunity to thank in particular our current research team at Texas A&M Energy Institute, particularly Dr. Styliani Avraamidou and Mr. Iosif Pappas.

We would also like to gratefully acknowledge the financial support kindly provided by our many sponsors, EPSRC, NSF, EU/ERC, DOE/CESMII, DOE/RAPID, Shell, Air Products, Eli‐Lilly, and BP.

      Finally, we would like to thank Wiley‐VCH for their enthusiastic support of this effort.

      Richard Oberdieck

      College Station, October 2019

      Efstratios N. Pistikopoulos

      Nikolaos A. Diangelakis

Part I Multi‐parametric Optimization

1 Introduction

In this chapter, the fundamental concepts of mathematical optimization and multi‐parametric programming will be presented. Such concepts will be the foundation towards the development of state‐of‐the‐art multi‐parametric programming strategies and applications, which will appear in this book in the next chapters.

1.1 Concepts of Optimization

      1.1.1 Convex Analysis

      This section presents the idea of convex sets and introduces function convexity. Convexity plays a vital role to establish the required properties which will enable a multi‐parametric solution to hold. In this setting, the following definitions are established.

      Definition 1.1 (Line)

      Consider the points

and

. Then the line that passes through these points is defined as

      (1.1)

      Definition 1.2 (Line Segment)

      The closed line segment joining the points

,

is defined as:

      (1.2)

      Definition 1.3 (Convex Set)

A set

is a convex set, if the closed line segment joining any two points in the set

belongs to the set

for each

such that

.

      1.1.1.1 Properties of Convex Sets

      Let

and

be convex sets defined in

. Then

      1 The intersection of is a convex set.

      2 The summation of two convex sets is a convex set.

      3 Let be a real number. The product is a convex set.

      Examples of convex sets include lines, polytopes and polyhedra, and open and closed halfspaces.

      Definition 1.4 (Convex Function)

      Let

be a convex subset, and the real function

defined in

. The function

is a convex function if for any

,

,

(1.3)

      If strict inequality holds in expression (1.3) for

, then

is a strictly convex function.

      Definition 1.5 (Concave Function)

      Let

be a convex subset, and the real function

defined in

. The function

is a concave function if for any

,

,

(1.4)

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