introduction-lp-duality1 - Chapter 9 Linear programming The...

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Chapter 9 Linear programming The nature of the programmes a computer scientist has to conceive often requires some knowl- edge in a specific domain of application, for example corporate management, network proto- cols, sound and video for multimedia streaming,. . . Linear programming is one of the necessary knowledges to handle optimization problems. These problems come from varied domains as production management, economics, transportation network planning, . . . For example, one can mention the composition of train wagons, the electricity production, or the flight planning by airplane companies. Most of these optimization problems do not admit an optimal solution that can be computed in a reasonable time, that is in polynomial time (See Chapter 3). However, we know how to ef- ficiently solve some particular problems and to provide an optimal solution (or at least quantify the difference between the provided solution and the optimal value) by using techniques from linear programming. In fact, in 1947, G.B. Dantzig conceived the Simplex Method to solve military planning problems asked by the US Air Force that were written as a linear programme, that is a system of linear equations. In this course, we introduce the basic concepts of linear programming. We then present the Simplex Method, following the book of V. Chv´atal [2]. If you want to read more about linear programming, some good references are [6, 1]. The objective is to show the reader how to model a problem with a linear programme when it is possible, to present him different methods used to solve it or at least provide a good ap- proximation of the solution. To this end, we present the theory of duality which provide ways of finding good bounds on specific solutions. We also discuss the practical side of linear programming: there exist very efficient tools to solve linear programmes, e.g. CPLEX [3] and GLPK [4]. We present the different steps leading to the solution of a practical problem expressed as a linear programme. 9.1 Introduction A linear programme is a problem consisting in maximizing or minimizing a linear function while satisfying a finite set of linear constraints. 129
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130 CHAPTER 9. LINEAR PROGRAMMING Linear programmes can be written under the standard form : Maximize n j = 1 c j x j Subject to: n j = 1 a i j x j b i for all 1 i m x j 0 for all 1 j n . (9.1) All constraints are inequalities (and not equations) and all variables are non-negative. The variables x j are referred to as decision variables . The function that has to be maximized is called the problem objective function . Observe that a constraint of the form n j = 1 a i j x j b i may be rewritten as n j = 1 ( a i j ) x j b i . Similarly, a minimization problem may be transformed into a maximization problem: minimizing n j = 1 c j x j is equivalent to maximizing n j = 1 ( c j ) x j . Hence, every maximization or minimization problem subject to linear constraints can be reformulated in the standard form (See Exercices 9.1 and 9.2.).
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