Luenberger/ LINEAR and NONLINEAR PROGRAMMING, 2Nd Ed
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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. -
Oracles, Ellipsoid Method and Their Uses in Convex Optimization Lecturer: Matt Weinberg Scribe:Sanjeev Arora
princeton univ. F'17 cos 521: Advanced Algorithm Design Lecture 17: Oracles, Ellipsoid method and their uses in convex optimization Lecturer: Matt Weinberg Scribe:Sanjeev Arora Oracle: A person or agency considered to give wise counsel or prophetic pre- dictions or precognition of the future, inspired by the gods. Recall that Linear Programming is the following problem: maximize cT x Ax ≤ b x ≥ 0 where A is a m × n real constraint matrix and x; c 2 Rn. Recall that if the number of bits to represent the input is L, a polynomial time solution to the problem is allowed to have a running time of poly(n; m; L). The Ellipsoid algorithm for linear programming is a specific application of the ellipsoid method developed by Soviet mathematicians Shor(1970), Yudin and Nemirovskii(1975). Khachiyan(1979) applied the ellipsoid method to derive the first polynomial time algorithm for linear programming. Although the algorithm is theoretically better than the Simplex algorithm, which has an exponential running time in the worst case, it is very slow practically and not competitive with Simplex. Nevertheless, it is a very important theoretical tool for developing polynomial time algorithms for a large class of convex optimization problems, which are much more general than linear programming. In fact we can use it to solve convex optimization problems that are even too large to write down. 1 Linear programs too big to write down Often we want to solve linear programs that are too large to even write down (or for that matter, too big to fit into all the hard drives of the world). -
Nonlinear Optimization Using the Generalized Reduced Gradient Method Revue Française D’Automatique, Informatique, Recherche Opé- Rationnelle
REVUE FRANÇAISE D’AUTOMATIQUE, INFORMATIQUE, RECHERCHE OPÉRATIONNELLE.RECHERCHE OPÉRATIONNELLE LEON S. LASDON RICHARD L. FOX MARGERY W. RATNER Nonlinear optimization using the generalized reduced gradient method Revue française d’automatique, informatique, recherche opé- rationnelle. Recherche opérationnelle, tome 8, no V3 (1974), p. 73-103 <http://www.numdam.org/item?id=RO_1974__8_3_73_0> © AFCET, 1974, tous droits réservés. L’accès aux archives de la revue « Revue française d’automatique, in- formatique, recherche opérationnelle. Recherche opérationnelle » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.LR.O. (8* année, novembre 1974, V-3, p. 73 à 104) NONLINEAR OPTIMIZATION USING THE-GENERALIZED REDUCED GRADIENT METHOD (*) by Léon S. LÀSDON, Richard L. Fox and Margery W. RATNER Abstract. — This paper describes the principles and logic o f a System of computer programs for solving nonlinear optimization problems using a Generalized Reduced Gradient Algorithm, The work is based on earlier work of Âbadie (2). Since this paper was written, many changes have been made in the logic, and significant computational expérience has been obtained* We hope to report on this in a later paper. 1. INTRODUCTION Generalized Reduced Gradient methods are algorithms for solving non- linear programs of gênerai structure. This paper discusses the basic principles of GRG, and constructs a spécifie GRG algorithm. The logic of a computer program implementing this algorithm is presented by means of flow charts and discussion. -
Combinatorial Structures in Nonlinear Programming
Combinatorial Structures in Nonlinear Programming Stefan Scholtes¤ April 2002 Abstract Non-smoothness and non-convexity in optimization problems often arise because a combinatorial structure is imposed on smooth or convex data. The combinatorial aspect can be explicit, e.g. through the use of ”max”, ”min”, or ”if” statements in a model, or implicit as in the case of bilevel optimization where the combinatorial structure arises from the possible choices of active constraints in the lower level problem. In analyzing such problems, it is desirable to decouple the combinatorial from the nonlinear aspect and deal with them separately. This paper suggests a problem formulation which explicitly decouples the two aspects. We show that such combinatorial nonlinear programs, despite their inherent non-convexity, allow for a convex first order local optimality condition which is generic and tight. The stationarity condition can be phrased in terms of Lagrange multipliers which allows an extension of the popular sequential quadratic programming (SQP) approach to solve these problems. We show that the favorable local convergence properties of SQP are retained in this setting. The computational effectiveness of the method depends on our ability to solve the subproblems efficiently which, in turn, depends on the representation of the governing combinatorial structure. We illustrate the potential of the approach by applying it to optimization problems with max-min constraints which arise, for example, in robust optimization. 1 Introduction Nonlinear programming is nowadays regarded as a mature field. A combination of important algorithmic developments and increased computing power over the past decades have advanced the field to a stage where the majority of practical prob- lems can be solved efficiently by commercial software. -
Nonlinear Integer Programming ∗
Nonlinear Integer Programming ∗ Raymond Hemmecke, Matthias Koppe,¨ Jon Lee and Robert Weismantel Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Nu- merous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considera- tions and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research. Raymond Hemmecke Otto-von-Guericke-Universitat¨ Magdeburg, FMA/IMO, Universitatsplatz¨ 2, 39106 Magdeburg, Germany, e-mail: [email protected] Matthias Koppe¨ University of California, Davis, Dept. of Mathematics, One Shields Avenue, Davis, CA, 95616, USA, e-mail: [email protected] Jon Lee IBM T.J. -
12. Coordinate Descent Methods
EE 546, Univ of Washington, Spring 2014 12. Coordinate descent methods theoretical justifications • randomized coordinate descent method • minimizing composite objectives • accelerated coordinate descent method • Coordinate descent methods 12–1 Notations consider smooth unconstrained minimization problem: minimize f(x) x RN ∈ n coordinate blocks: x =(x ,...,x ) with x RNi and N = N • 1 n i ∈ i=1 i more generally, partition with a permutation matrix: U =[PU1 Un] • ··· n T xi = Ui x, x = Uixi i=1 X blocks of gradient: • f(x) = U T f(x) ∇i i ∇ coordinate update: • x+ = x tU f(x) − i∇i Coordinate descent methods 12–2 (Block) coordinate descent choose x(0) Rn, and iterate for k =0, 1, 2,... ∈ 1. choose coordinate i(k) 2. update x(k+1) = x(k) t U f(x(k)) − k ik∇ik among the first schemes for solving smooth unconstrained problems • cyclic or round-Robin: difficult to analyze convergence • mostly local convergence results for particular classes of problems • does it really work (better than full gradient method)? • Coordinate descent methods 12–3 Steepest coordinate descent choose x(0) Rn, and iterate for k =0, 1, 2,... ∈ (k) 1. choose i(k) = argmax if(x ) 2 i 1,...,n k∇ k ∈{ } 2. update x(k+1) = x(k) t U f(x(k)) − k i(k)∇i(k) assumptions f(x) is block-wise Lipschitz continuous • ∇ f(x + U v) f(x) L v , i =1,...,n k∇i i −∇i k2 ≤ ik k2 f has bounded sub-level set, in particular, define • ⋆ R(x) = max max y x 2 : f(y) f(x) y x⋆ X⋆ k − k ≤ ∈ Coordinate descent methods 12–4 Analysis for constant step size quadratic upper bound due to block coordinate-wise -
Process Optimization
Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007 Process Optimization • Typical Industrial Problems • Mathematical Programming Software • Mathematical Basis for Optimization • Lagrange Multipliers and the Simplex Algorithm • Generalized Reduced Gradient Algorithm • On-Line Optimization • Mixed Integer Programming and the Branch and Bound Algorithm • Chemical Production Complex Optimization New Results • Using one computer language to write and run a program in another language • Cumulative probability distribution instead of an optimal point using Monte Carlo simulation for a multi-criteria, mixed integer nonlinear programming problem • Global optimization Design vs. Operations • Optimal Design −Uses flowsheet simulators and SQP – Heuristics for a design, a superstructure, an optimal design • Optimal Operations – On-line optimization – Plant optimal scheduling – Corporate supply chain optimization Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100 Optimization Programming Languages • GAMS - General Algebraic Modeling System • LINDO - Widely used in business applications -
A Generalized Dual Phase-2 Simplex Algorithm1
A Generalized Dual Phase-2 Simplex Algorithm1 Istvan´ Maros Department of Computing, Imperial College, London Email: [email protected] Departmental Technical Report 2001/2 ISSN 1469–4174 January 2001, revised December 2002 1This research was supported in part by EPSRC grant GR/M41124. I. Maros Phase-2 of Dual Simplex i Contents 1 Introduction 1 2 Problem statement 2 2.1 The primal problem . 2 2.2 The dual problem . 3 3 Dual feasibility 5 4 Dual simplex methods 6 4.1 Traditional algorithm . 7 4.2 Dual algorithm with type-1 variables present . 8 4.3 Dual algorithm with all types of variables . 9 5 Bound swap in dual 11 6 Bound Swapping Dual (BSD) algorithm 14 6.1 Step-by-step description of BSD . 14 6.2 Work per iteration . 16 6.3 Degeneracy . 17 6.4 Implementation . 17 6.5 Features . 18 7 Two examples for the algorithmic step of BSD 19 8 Computational experience 22 9 Summary 24 10 Acknowledgements 25 Abstract Recently, the dual simplex method has attracted considerable interest. This is mostly due to its important role in solving mixed integer linear programming problems where dual feasible bases are readily available. Even more than that, it is a realistic alternative to the primal simplex method in many cases. Real world linear programming problems include all types of variables and constraints. This necessitates a version of the dual simplex algorithm that can handle all types of variables efficiently. The paper presents increasingly more capable dual algorithms that evolve into one which is based on the piecewise linear nature of the dual objective function. -
The Gradient Sampling Methodology
The Gradient Sampling Methodology James V. Burke∗ Frank E. Curtisy Adrian S. Lewisz Michael L. Overtonx January 14, 2019 1 Introduction in particular, this leads to calling the negative gra- dient, namely, −∇f(x), the direction of steepest de- The principal methodology for minimizing a smooth scent for f at x. However, when f is not differentiable function is the steepest descent (gradient) method. near x, one finds that following the negative gradient One way to extend this methodology to the minimiza- direction might offer only a small amount of decrease tion of a nonsmooth function involves approximating in f; indeed, obtaining decrease from x along −∇f(x) subdifferentials through the random sampling of gra- may be possible only with a very small stepsize. The dients. This approach, known as gradient sampling GS methodology is based on the idea of stabilizing (GS), gained a solid theoretical foundation about a this definition of steepest descent by instead finding decade ago [BLO05, Kiw07], and has developed into a a direction to approximately solve comprehensive methodology for handling nonsmooth, potentially nonconvex functions in the context of op- min max gT d; (2) kdk2≤1 g2@¯ f(x) timization algorithms. In this article, we summarize the foundations of the GS methodology, provide an ¯ where @f(x) is the -subdifferential of f at x [Gol77]. overview of the enhancements and extensions to it To understand the context of this idea, recall that that have developed over the past decade, and high- the (Clarke) subdifferential of a locally Lipschitz f light some interesting open questions related to GS. -
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. -
Matroidal Subdivisions, Dressians and Tropical Grassmannians
Matroidal subdivisions, Dressians and tropical Grassmannians vorgelegt von Diplom-Mathematiker Benjamin Frederik Schröter geboren in Frankfurt am Main Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Wilhelm Stannat Gutachter: Prof. Dr. Michael Joswig Prof. Dr. Hannah Markwig Senior Lecturer Ph.D. Alex Fink Tag der wissenschaftlichen Aussprache: 17. November 2017 Berlin 2018 Zusammenfassung In dieser Arbeit untersuchen wir verschiedene Aspekte von tropischen linearen Räumen und deren Modulräumen, den tropischen Grassmannschen und Dressschen. Tropische lineare Räume sind dual zu Matroidunterteilungen. Motiviert durch das Konzept der Splits, dem einfachsten Fall einer polytopalen Unterteilung, wird eine neue Klasse von Matroiden eingeführt, die mit Techniken der polyedrischen Geometrie untersucht werden kann. Diese Klasse ist sehr groß, da sie alle Paving-Matroide und weitere Matroide enthält. Die strukturellen Eigenschaften von Split-Matroiden können genutzt werden, um neue Ergebnisse in der tropischen Geometrie zu erzielen. Vor allem verwenden wir diese, um Strahlen der tropischen Grassmannschen zu konstruieren und die Dimension der Dressschen zu bestimmen. Dazu wird die Beziehung zwischen der Realisierbarkeit von Matroiden und der von tropischen linearen Räumen weiter entwickelt. Die Strahlen einer Dressschen entsprechen den Facetten des Sekundärpolytops eines Hypersimplexes. Eine besondere Klasse von Facetten bildet die Verallgemeinerung von Splits, die wir Multi-Splits nennen und die Herrmann ursprünglich als k-Splits bezeichnet hat. Wir geben eine explizite kombinatorische Beschreibung aller Multi-Splits eines Hypersimplexes. Diese korrespondieren mit Nested-Matroiden. Über die tropische Stiefelabbildung erhalten wir eine Beschreibung aller Multi-Splits für Produkte von Simplexen.