Linear Programming
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 3 1 Geometry of Linear Programs
ORIE 6300 Mathematical Programming I September 2, 2014 Lecture 3 Lecturer: David P. Williamson Scribe: Divya Singhvi Last time we discussed how to take dual of an LP in two different ways. Today we will talk about the geometry of linear programs. 1 Geometry of Linear Programs First we need some definitions. Definition 1 A set S ⊆ <n is convex if 8x; y 2 S, λx + (1 − λ)y 2 S, 8λ 2 [0; 1]. Figure 1: Examples of convex and non convex sets Given a set of inequalities we define the feasible region as P = fx 2 <n : Ax ≤ bg. We say that P is a polyhedron. Which points on this figure can have the optimal value? Our intuition from last time is that Figure 2: Example of a polyhedron. \Circled" corners are feasible and \squared" are non feasible optimal solutions to linear programming problems occur at \corners" of the feasible region. What we'd like to do now is to consider formal definitions of the \corners" of the feasible region. 3-1 One idea is that a point in the polyhedron is a corner if there is some objective function that is minimized there uniquely. Definition 2 x 2 P is a vertex of P if 9c 2 <n with cT x < cT y; 8y 6= x; y 2 P . Another idea is that a point x 2 P is a corner if there are no small perturbations of x that are in P . Definition 3 Let P be a convex set in <n. Then x 2 P is an extreme point of P if x cannot be written as λy + (1 − λ)z for y; z 2 P , y; z 6= x, 0 ≤ λ ≤ 1. -
A Lifted Linear Programming Branch-And-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L
A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L. Nemhauser, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332-0205, USA, {[email protected], [email protected], [email protected]} This paper develops a linear programming based branch-and-bound algorithm for mixed in- teger conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algo- rithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations. Key words: nonlinear integer programming; branch and bound; portfolio optimization History: February 2007. 1. Introduction This paper deals with the development of an algorithm for the class of mixed integer non- linear programming (MINLP) problems known as mixed integer conic quadratic program- ming problems. This class of problems arises from adding integrality requirements to conic quadratic programming problems (Lobo et al., 1998), and is used to model several applica- tions from engineering and finance. Conic quadratic programming problems are also known as second order cone programming problems, and together with semidefinite and linear pro- gramming (LP) problems are special cases of the more general conic programming problems (Ben-Tal and Nemirovski, 2001a). -
Linear Programming
Lecture 18 Linear Programming 18.1 Overview In this lecture we describe a very general problem called linear programming that can be used to express a wide variety of different kinds of problems. We can use algorithms for linear program- ming to solve the max-flow problem, solve the min-cost max-flow problem, find minimax-optimal strategies in games, and many other things. We will primarily discuss the setting and how to code up various problems as linear programs At the end, we will briefly describe some of the algorithms for solving linear programming problems. Specific topics include: • The definition of linear programming and simple examples. • Using linear programming to solve max flow and min-cost max flow. • Using linear programming to solve for minimax-optimal strategies in games. • Algorithms for linear programming. 18.2 Introduction In the last two lectures we looked at: — Bipartite matching: given a bipartite graph, find the largest set of edges with no endpoints in common. — Network flow (more general than bipartite matching). — Min-Cost Max-flow (even more general than plain max flow). Today, we’ll look at something even more general that we can solve algorithmically: linear pro- gramming. (Except we won’t necessarily be able to get integer solutions, even when the specifi- cation of the problem is integral). Linear Programming is important because it is so expressive: many, many problems can be coded up as linear programs (LPs). This especially includes problems of allocating resources and business 95 18.3. DEFINITION OF LINEAR PROGRAMMING 96 supply-chain applications. In business schools and Operations Research departments there are entire courses devoted to linear programming. -
Integer Linear Programs
20 ________________________________________________________________________________________________ Integer Linear Programs Many linear programming problems require certain variables to have whole number, or integer, values. Such a requirement arises naturally when the variables represent enti- ties like packages or people that can not be fractionally divided — at least, not in a mean- ingful way for the situation being modeled. Integer variables also play a role in formulat- ing equation systems that model logical conditions, as we will show later in this chapter. In some situations, the optimization techniques described in previous chapters are suf- ficient to find an integer solution. An integer optimal solution is guaranteed for certain network linear programs, as explained in Section 15.5. Even where there is no guarantee, a linear programming solver may happen to find an integer optimal solution for the par- ticular instances of a model in which you are interested. This happened in the solution of the multicommodity transportation model (Figure 4-1) for the particular data that we specified (Figure 4-2). Even if you do not obtain an integer solution from the solver, chances are good that you’ll get a solution in which most of the variables lie at integer values. Specifically, many solvers are able to return an ‘‘extreme’’ solution in which the number of variables not lying at their bounds is at most the number of constraints. If the bounds are integral, all of the variables at their bounds will have integer values; and if the rest of the data is integral, many of the remaining variables may turn out to be integers, too. -
Chapter 11: Basic Linear Programming Concepts
CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately by the mathematical equations of a linear program, the method will find the best solution to the problem. Of course, few complex real-world problems can be expressed perfectly in terms of a set of linear functions. Nevertheless, linear programs can provide reasonably realistic representations of many real-world problems — especially if a little creativity is applied in the mathematical formulation of the problem. The subject of modeling was briefly discussed in the context of regulation. The regulation problems you learned to solve were very simple mathematical representations of reality. This chapter continues this trek down the modeling path. As we progress, the models will become more mathematical — and more complex. The real world is always more complex than a model. Thus, as we try to represent the real world more accurately, the models we build will inevitably become more complex. You should remember the maxim discussed earlier that a model should only be as complex as is necessary in order to represent the real world problem reasonably well. Therefore, the added complexity introduced by using linear programming should be accompanied by some significant gains in our ability to represent the problem and, hence, in the quality of the solutions that can be obtained. You should ask yourself, as you learn more about linear programming, what the benefits of the technique are and whether they outweigh the additional costs. -
Extreme Points and Basic Solutions
EXTREME POINTS AND BASIC SOLUTIONS: In Linear Programming, the feasible region in Rn is defined by P := {x ∈ Rn | Ax = b, x ≥ 0}. The set P , as we have seen, is a convex subset of Rn. It is called a convex polytope. The term convex polyhedron refers to convex polytope which is bounded. Polytopes in two dimensions are often called polygons. Recall that the vertices of a convex polytope are what we called extreme points of that set. Recall that extreme points of a convex set are those which cannot be represented as a proper convex combination of two other (distinct) points of the convex set. It may, or may not be the case that a convex set has any extreme points as shown by the example in R2 of the strip S := {(x, y) ∈ R2 | 0 ≤ x ≤ 1, y ∈ R}. On the other hand, the square defined by the inequalities |x| ≤ 1, |y| ≤ 1 has exactly four extreme points, while the unit disk described by the ineqality x2 + y2 ≤ 1 has infinitely many. These examples raise the question of finding conditions under which a convex set has extreme points. The answer in general vector spaces is answered by one of the “big theorems” called the Krein-Milman Theorem. However, as we will see presently, our study of the linear programming problem actually answers this question for convex polytopes without needing to call on that major result. The algebraic characterization of the vertices of the feasible polytope confirms the obser- vation that we made by following the steps of the Simplex Algorithm in our introductory example. -
The Feasible Region for Consecutive Patterns of Permutations Is a Cycle Polytope
Séminaire Lotharingien de Combinatoire XX (2020) Proceedings of the 32nd Conference on Formal Power Article #YY, 12 pp. Series and Algebraic Combinatorics (Ramat Gan) The feasible region for consecutive patterns of permutations is a cycle polytope Jacopo Borga∗1, and Raul Penaguiaoy 1 1Department of Mathematics, University of Zurich, Switzerland Abstract. We study proportions of consecutive occurrences of permutations of a given size. Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. This allows us to compute the dimension, vertices and faces of the polytope. Finally, we prove that the limits of classical occurrences and consecutive occurrences are independent, in some sense made precise in the extended abstract. As a consequence, the scaling limit of a sequence of permutations induces no constraints on the local limit and vice versa. Keywords: permutation patterns, cycle polytopes, overlap graphs. This is a shorter version of the preprint [11] that is currently submitted to a journal. Many proofs and details omitted here can be found in [11]. 1 Introduction 1.1 Motivations Despite not presenting any probabilistic result here, we give some motivations that come from the study of random permutations. This is a classical topic at the interface of combinatorics and discrete probability theory. There are two main approaches to it: the first concerns the study of statistics on permutations, and the second, more recent, looks arXiv:2003.12661v2 [math.CO] 30 Jun 2020 for the limits of permutations themselves. -
Chapter 7. Linear Programming and Reductions
Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence, and so on. In such cases, we seek a solution that (1) satisfies certain constraints (for instance, the path must use edges of the graph and lead from s to t, the tree must touch all nodes, the subsequence must be increasing); and (2) is the best possible, with respect to some well-defined criterion, among all solutions that satisfy these constraints. Linear programming describes a broad class of optimization tasks in which both the con- straints and the optimization criterion are linear functions. It turns out an enormous number of problems can be expressed in this way. Given the vastness of its topic, this chapter is divided into several parts, which can be read separately subject to the following dependencies. Flows and matchings Introduction to linear programming Duality Games and reductions Simplex 7.1 An introduction to linear programming In a linear programming problem we are given a set of variables, and we want to assign real values to them so as to (1) satisfy a set of linear equations and/or linear inequalities involving these variables and (2) maximize or minimize a given linear objective function. 201 202 Algorithms Figure 7.1 (a) The feasible region for a linear program. (b) Contour lines of the objective function: x1 + 6x2 = c for different values of the profit c. x x (a) 2 (b) 2 400 400 Optimum point Profit = $1900 ¡ ¡ ¡ ¡ ¡ 300 ¢¡¢¡¢¡¢¡¢ 300 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ 200 200 c = 1500 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 1200 ¡ ¡ ¡ ¡ ¡ 100 ¢¡¢¡¢¡¢¡¢ 100 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 600 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ x1 x1 0 100 200 300 400 0 100 200 300 400 7.1.1 Example: profit maximization A boutique chocolatier has two products: its flagship assortment of triangular chocolates, called Pyramide, and the more decadent and deluxe Pyramide Nuit. -
Implementing Customized Pivot Rules in COIN-OR's CLP with Python
Cahier du GERAD G-2012-07 Customizing the Solution Process of COIN-OR's Linear Solvers with Python Mehdi Towhidi1;2 and Dominique Orban1;2 ? 1 Department of Mathematics and Industrial Engineering, Ecole´ Polytechnique, Montr´eal,QC, Canada. 2 GERAD, Montr´eal,QC, Canada. [email protected], [email protected] Abstract. Implementations of the Simplex method differ only in very specific aspects such as the pivot rule. Similarly, most relaxation methods for mixed-integer programming differ only in the type of cuts and the exploration of the search tree. Implementing instances of those frame- works would therefore be more efficient if linear and mixed-integer pro- gramming solvers let users customize such aspects easily. We provide a scripting mechanism to easily implement and experiment with pivot rules for the Simplex method by building upon COIN-OR's open-source linear programming package CLP. Our mechanism enables users to implement pivot rules in the Python scripting language without explicitly interact- ing with the underlying C++ layers of CLP. In the same manner, it allows users to customize the solution process while solving mixed-integer linear programs using the CBC and CGL COIN-OR packages. The Cython pro- gramming language ensures communication between Python and COIN- OR libraries and activates user-defined customizations as callbacks. For illustration, we provide an implementation of a well-known pivot rule as well as the positive edge rule|a new rule that is particularly efficient on degenerate problems, and demonstrate how to customize branch-and-cut node selection in the solution of a mixed-integer program. -
Math 112 Review for Exam Ii (Ws 12 -18)
MATH 112 REVIEW FOR EXAM II (WS 12 -18) I. Derivative Rules • There will be a page or so of derivatives on the exam. You should know how to apply all the derivative rules. (WS 12 and 13) II. Functions of One Variable • Be able to find local optima, and to distinguish between local and global optima. • Be able to find the global maximum and minimum of a function y = f(x) on the interval from x = a to x = b, using the fact that optima may only occur where f(x) has a horizontal tangent line and at the endpoints of the interval. Step 1: Compute the derivative f’(x). Step 2: Find all critical points (values of x at which f’(x) = 0.) Step 3: Plug all the values of x from Step 2 that are in the interval from a to b and the endpoints of the interval into the function f(x). Step 4: Sketch a rough graph of f(x) and pick off the global max and min. • Understand the following application: Maximizing TR(q) starting with a demand curve. (WS 15) • Understand how to use the Second Derivative Test. (WS 16) If a is a critical point for f(x) (that is, f’(a) = 0), and the second derivative is: f ’’(a) > 0, then f(x) has a local min at x = a. f ’’(a) < 0, then f(x) has a local max at x = a. f ’’(a) = 0, then the test tells you nothing. IMPORTANT! For the Second Derivative Test to work, you must have f’(a) = 0 to start with! For example, if f ’’(a) > 0 but f ’(a) ≠ 0, then the graph of f(x) is concave up at x = a but f(x) does not have a local min there. -
A Branch-And-Bound Algorithm for Zero-One Mixed Integer
A BRANCH-AND-BOUND ALGORITHM FOR ZERO- ONE MIXED INTEGER PROGRAMMING PROBLEMS Ronald E. Davis Stanford University, Stanford, California David A. Kendrick University of Texas, Austin, Texas and Martin Weitzman Yale University, New Haven, Connecticut (Received August 7, 1969) This paper presents the results of experimentation on the development of an efficient branch-and-bound algorithm for the solution of zero-one linear mixed integer programming problems. An implicit enumeration is em- ployed using bounds that are obtained from the fractional variables in the associated linear programming problem. The principal mathematical result used in obtaining these bounds is the piecewise linear convexity of the criterion function with respect to changes of a single variable in the interval [0, 11. A comparison with the computational experience obtained with several other algorithms on a number of problems is included. MANY IMPORTANT practical problems of optimization in manage- ment, economics, and engineering can be posed as so-called 'zero- one mixed integer problems,' i.e., as linear programming problems in which a subset of the variables is constrained to take on only the values zero or one. When indivisibilities, economies of scale, or combinatoric constraints are present, formulation in the mixed-integer mode seems natural. Such problems arise frequently in the contexts of industrial scheduling, investment planning, and regional location, but they are by no means limited to these areas. Unfortunately, at the present time the performance of most compre- hensive algorithms on this class of problems has been disappointing. This study was undertaken in hopes of devising a more satisfactory approach. In this effort we have drawn on the computational experience of, and the concepts employed in, the LAND AND DoIGE161 Healy,[13] and DRIEBEEKt' I algorithms. -
Production Models: Maximizing Profits
1 ________________________________________________________________________________________________ Production Models: Maximizing Profits As we saw in the Introduction, mathematical programming is a technique for solving certain kinds of optimization problems Ð maximizing profit, minimizing cost, etc. Ð subject to constraints on resources, capacities, supplies, demands, and the like. AMPL is a language for specifying such mathematical optimization problems. It provides a notation that is very close to the way that you might describe a problem mathematically, so that it is easy to convert from an algebraic description to AMPL. We will concentrate initially on linear programming, which is the best known and eas- iest case; other kinds of mathematical programming are taken up toward the end of the book. This chapter addresses one of the most common applications of linear program- ming: maximizing the profit of some operation, subject to constraints that limit what can be produced. Chapters 2 and 3 are devoted to two other equally common kinds of linear programs, and Chapter 4 shows how linear programming models can be replicated and combined to produce truly large-scale problems. These chapters are written with the beginner in mind, but experienced practitioners of mathematical programming should find them useful as a quick introduction to AMPL. We begin with a linear program (or LP for short) with only two variables, motivated by a mythical steelmaking operation. This will provide a quick review of linear program- ming to refresh your memory if you already have some experience, or to help you get started if you're just learning. We'll show how the same LP can be represented as a gen- eral algebraic model of production, together with specific data.