Linear Programming - Lecture 2 Introduction to Linear...

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Lecture 2: Introduction to Linear programming Contents 1 Definition and geometry A mathematical program is a linear program if it has (i) continuous variables (ii) one linear objective function (iii) all constraints are linear equalities or inequalities A generic linear programming (LP) problem is defined as follows. maximize c T x , subject to a T i x b i , i G, a T i x = b i , i E, a T i x b i , i L, x j 0 , j P, x j 0 , j N. The last two sets of constraints although inequalities are special and are usually treated separately. The vector c is called the objective vector and the numbers b i , i G E L , are called the RHS coefficients. We can, of course, stack up the constraints of each kind and rewrite the above problem in the following form: maximize c T x , subject to A g x b g , A e x = b e , A l x b l , x j 0 , j P, x j 0 , j N. 1.1 Why bother with LPs One part of the answer lies in the geometry of the feasible region of an LP. Since equalities simply state that the problem lies in a lower dimensional space, we will consider LPs with only inequality constraints, i.e. E = . Recall that a linear inequality a T x b divides the entire space into two parts, i.e. the set { x : a T x b } is a halfspace . Thus, the feasible region of an (inequality constrained) LP is an intersection of halfspaces. The set obtained by taking the intersection of halfspaces is called a polyhedron . a T 1 x b 1 a T 2 x b 2 a T 3 x b 3 a T 4 x b 4 a T 5 x b 5 P
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Introduction to Linear Programming 2 Polyhedra are often good first approximations to more complicated feasible sets. F L In the figure above, L is a linear approximation for the nonlinear feasible set F . Therefore, linear programs tend to provide a good approximation to more complicated optimization problems. Second part of the answer to why one should bother with LPs is that LPs can be solved rather efficiently. 1.2 Geometrical solution of LPs Continuing further with this geometrical approach, lets try to examine optimal solutions of LPs. Lets start with a one-dimensional LP maximize cx subject to a 1 x a 2 The optimal solution x of this LP is given by (a) c > 0: x = a 2 (b) c = 0: any x [ a 1 , a 2 ] optimal ... in particular x = a 1 , a 2 (c) c < 0: x = a 1 Moral: The optimal solution is alway at the boundary of the feasible set. Push this geometric approach a little further and consider general (inequality constrained) LPs. maximize c T x , subject to x ∈ P (polytope) For a given scalar z , the set of points x with cost c T x = z is a plane perpendicular to c (a hyperplane ). Thus, an algorithm for solving the LP is to increase z , or equivalently slide the hyperplane in the c direction, until the plane is at the boundary of the feasible region.
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