Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

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1 Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations. Outline of Simplex Method. Runtimes for Linear Program Solvers. Readings: Read text section 11.6, and sections 1 and 2 of Tom Ferguson s notes (see course homepage). A factory makes (thousand) widgets of type 1 and of type 2. Total profit for making x = (, ) T is: profit = + 2 Due to a limited resource (e.g. time) we require: + 4 Two waste products from making widget 1 are required for widget 2. So we need to make enough of widget 1 to supply the construction of widget 2. These constraints are: Finally, both and must be non-negative. How many widgets of each type should be made to maximize profit? 1 2 Linear Programming: Standard Form Widget Factory Example: Continued. Consider a real-valued, unknown, n-vector x = (,,, x n ) T. A linear programming problem in standard form (A, b, c) has the three components: Constants: A an m x n matrix, b an m vector, Objective Function: We wish to choose x to maximize: c an n vector. c T x = c 1 + c 2 + c n x n Linear function of x with x subject to the following constraints: Pose Widget problem as a linear program in Standard Form. Need to specify constants, (A, b, c). Unknowns: x = (, ) T number (in thousands) of the two widget types. Objective function (profit): c T x = c 1 + c 2 = + 2, so c T = (c 1, c 2 ) = (1, 2). A x b For an m x n matrix A, and an m vector b: A x b so Non-negativity Constraints: x 0 Linear inequality constraints on x Notation: For two K-vectors x and y, x y iff x k y k for each k = 1, 2,, K. Other inequalities (, etc.) defined similarly. Non-negativity Constraints: x 0 3 4

2 Graphing the Widget Factory Example Graphing the Widget Factory Example Linear Program specified by (A, b, c). Example: x = (, ) T. Linear Program specified by (A, b, c). Ax b, c T x = 4 c T x increases in direction c Mixtures of widgets with the same profit c T x = 2 Non-negativity: x Graphing the Widget Factory Example Graphing the Widget Factory Example Example: x = (, ) T. Linear Program specified by (A, b, c). Example: x = (, ) T. Linear Program specified by (A, b, c). Ax b, Ax b, Max profit: c T x Non-negativity: x 0. Direction of increasing profit. Non-negativity: x

3 Graphing the Widget Factory Example Linear Programming: Example Applications Example: x = (, ) T. Linear Program specified by (A, b, c). Linear programming is quite a general framework. For example: Ax b, Network flow problems can be written as linear programs (LPs) for the unknown flow f(e). The constraints are: 0 f(e) c(e). flow conservation at each v є V\{s,t}. f ( e) f ( e) 0 e in to v e out of v Non-negativity: x 0. Optimal Solution: x T = (25, 27)/13 which can be rewritten as LHS 0, and LHS 0. The objective function is max sum{ f(e) e = (s, v), v є V } Integer Linear Programming. Weighted scheduling problems, the Knapsack problem, etc. can also be written as LPs, although for these we seek integer valued solutions Linear Programming Theory: Example Given the constants (A, b, c), consider the linear program: Objective Function: Maximize c T x, where x = (,,, x n ) T. A x b Non-negativity Constraints: x Define the feasible set F = { x Ax b and x 0 }. F could be empty (no solutions to all the constraints). F is a convex polytope. (A region of [R n defined by the intersection of finitely many half-spaces, e.g., a T k x b k and x j 0.) Convexity of F: Let u, v є F, and let s є [0, 1]. Then su + (1-s)v є F Pf: u, v є F implies Au b and Av b. So A [su + (1-s)v ] = s Au + (1-s)Av sb + (1-s)b = b, for s є [0, 1]. A similar argument shows su + (1-s)v 0. Therefore su + (1-s)v є F

4 Example Example unbounded Linear Programming Theory: Characterization of a Vertex Three Dimensional Example: Vertices What s a vertex of the feasible set? Let P be the (m+n) x n matrix and p the (m+n)-vector which represents both the problem and non-negativity constraints: Vertex: Choose n = 3 problem or non-negativity constraints, solve equalities for v: x 3 + 3x x x Let s = {s 1, s 2,, s n } be a selection of n row numbers, 1 s i m+n. Define Q(s) to be the n x n matrix formed from the s-rows of P, and q(s) the n-vector formed from the same rows of p. Defn: A point v є [R n is a vertex of the feasible set F iff there exists an s such that: Q(s) is nonsingular, v = [Q(s)] -1 q(s), i.e., v satisfies the n equalities selected by s, and v є F, i.e., v is feasible. See 2D examples above, and 3D example next. Feasible set F is this closed polytope. Vertex is a soln of: + 3x 3 = x 3 = 400 = 0 Vertex is a soln of: + 3x 3 = x 3 = x 3 =

5 Linear Programming Theory: Characterization of a Solution Solution at Vertex: Example 1 Given the constants (A, b, c), consider the linear program: Objective Function: Maximize c T x, where x = (,,, x n ) T, c 0. A x b Non-negativity Constraints: x Define: Feasible set F = { x A x b and x 0 } Theorem: The linear program above either: 1. has no solution, in which case either the feasible set F is empty, or the objective c T x unbounded (and F is unbounded), 2. has a solution x*. In case 2, x* can be taken to be a vertex of the polytope F (there may also be non-vertex solutions x є F with c T x = c T x*) + 4 Optimal solution at vertex See examples Solution at Vertex: Example 2 Solution along Closed Segment: Example Optimal solution at vertex. + 4 Optimal solutions at vertices and along segment between c must be special for such a case to occur

6 Three Dimensional Example Variations in the Formulation of Linear Programs Colours indicate value of c T x. Larger c T x hotter colours. Feasible set F is this closed polytope. x 3 + 3x x x Maximize c T x, c T = (2, 3, 4), Multiple solns iff c is perpendicular to an edge or face of F. A given LP can be expressed in many equivalent forms. Given an LP in standard form, with constants (A, b, c). Some alternatives: minimize c T x equivalent to maximizing c T x. Constraint a T x b equivalent to -a T x -b Constraint a T x = b iff a T x b and a T x -b. Ax b iff Ax + z = b and z 0, with slack variables z = (z 1,, z m ) T. min e can be rewritten as min(e + + e -) with the linear constraint e = e + -e - and the non-negativity constraint e +, e - 0. These alternatives are useful for posing problems as LPs in standard form, or reposing LPs in alternative forms useful for computation A Sketch of the Simplex Method Three Dimensional Example: Revisited Simplex method: Given an LP in standard form (A, b, c). Let P and p be: Let v be a feasible vertex. So v = v(s), where s = {s 1,, s n } denotes a set of n selected rows of P v p, such that Q(s) v = q(s) and Q(s) is nonsingular (see the defn. of a vertex, above). while true * Consider each neighbour s of s (i.e., s and s differ only in one element.). Choose an edge v(s) to v(s ) s.t. the objective increases. If there is no such edge, v(s) is a solution. Stop. If there is an edge leaving v(s) on which the objective is unbounded, then there is no solution to this LP. Stop. Set s s, v v (s ) end * Modulo non-cycling conditions Feasible set F is this closed polytope. x 3 + 3x x x Maximize c T x, c T = (2, 3, 4), c T x gives shading. Multiple solns iff c is perpendicular to an edge or face of F

7 Pivoting Obtaining an Initial Vertex The step from v(s) to v(s ) is called pivoting. One row of Pv p is dropped from s, and it is replaced by another row to form s. The selection of a pivot is guaranteed not to decrease the objective function. We need an initial feasible vertex to start the Simplex Algorithm. Given the LP constants (A, b, c), consider the start-up LP: Objective Function: Maximize -z -z m, where z = (z 1, z 2,, z m ) T. Equivalent to minimizing z 1 + +z m A x - z b, Non-negativity Constraints: x, z 0 If some care is taken to avoid cycling, the Simplex Algorithm is guaranteed to converge to a solution after finitely many pivot steps. In an efficient implementation, each pivot step costs O((m+n)n) real number operations. Unfortunately, simplex may visit exponentially many vertices in contrived cases. E.g., number of choices for s, (n+m) choose n. For this start-up LP we have the initial guess, x = 0, z = b - where b k - =-bk if b k < 0 and 0 otherwise. This start-up LP has a solution (x 0, 0) (i.e., with z = 0 and the objective function equal to 0) iff x 0 is a feasible solution of the original LP. Simplex will return a feasible vertex x 0 on this start-up LP, so long as the original feasible set F is not empty Runtime for Simplex Algorithm Runtimes for Linear Programming Solvers Worst case runtime is exponential. The Simplex Algorithm might visit exponentially many vertices as m and n grow. In practice: the method is highly efficient, typically requires a number of steps which is just a small multiple of the number of variables. LPs with thousands or even millions of variables are routinely solved using the simplex method on modern computers. efficient, highly sophisticated implementations are available. Interior point methods provided the first polynomial time algorithms known for LP. These iterate through the interior of the feasible set F. Ellipsoid algorithm, Khachiyan, Interior point projective method, Karmarkar, Interior point methods are now generally considered competitive with the simplex method in most, though not all, applications. Sophisticated software packages are available. Integer Linear Programming (ILP): An LP problem but with the added constraint that the solution vector x must be integer valued. ILP is NP-hard

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