DOCSLIB.ORG

Matroidal Subdivisions, Dressians and Tropical Grassmannians

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Außerdem präsentieren wir Berechnungen für explizite untere Schranken der Anzahl der Facetten einiger Sekundärpolytope von Hypersimplexen. Berechnungen und Algorithmen spielen auch im Weiteren eine wichtige Rolle. Wir führen eine neue Methode zum Berechnen von tropischen linearen Räumen und sogar allgemeiner von dualen Komplexen von polyedrischen Unterteilungen ein. Diese Methode basiert auf einem Algorithmus von Ganter (1984) für endliche Hüllensysteme. Außerdem beschreiben wir die Implementierung eines algebraischen Teilkörpers der formalen Puiseux-Reihen. Dieser kann eingesetzt werden zum Lösen von linearen Programmen und konvexen Hüllenproblemen, die jeweils von einem reelen Parameter abhängen. Darüber hinaus ist dieses Werkzeug sowohl für tropische Konvexgeometrie als auch tropische algebraische Geometrie wertvoll. Tropische Varietäten, wie zum Beispiel tropische lineare Räume oder tropische Grass- mannsche, sind der gemeinsame Schnitt von endlich vielen tropischen Hyperflächen. Die Menge der zu den Hyperflächen gehörenden Polynome bildet eine tropische Basis. Für den allgemeinen Fall geben wir eine explizite obere Schranke für den Grad an, den die Polynome in einer tropischen Basis benötigen. Als Anwendung berechnen wir f-Vektoren von tropischen Varietäten und veranschaulichen die Unterschiede zwischen Gröbnerbasen und tropischen Basen. iii Abstract In this thesis we study various aspects of tropical linear spaces and their moduli spaces, the tropical Grassmannians and Dressians. Tropical linear spaces are dual to matroid subdivisions. Motivated by the concept of splits, the simplest case of a subdivision, a new class of matroids is introduced, which can be studied via techniques from polyhedral geometry. This class is very large as it strictly contains all paving matroids. The structural properties of these split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians and the dimension of the Dressian. In particular, a relation between matroid realizability and certain tropical linear spaces is elaborated. The rays of a Dressian correspond to facets of the secondary polytope of a hypersimplex. A special class of facets is obtained by a generalization of splits, called multi-splits or originally, in Herrmann’s work, k-splits. We give an explicit combinatorial description of all multi-splits of the hypersimplex. These are in correspondence to nested matroids and, via the tropical Stiefel map, also to multi-splits of products of simplices. Hence, we derive a description for all multi-splits of a product of simplices. Moreover, a computational result leads to explicit lower bounds on the total number of facets of secondary polytopes of hypersimplices. Other computational aspects are also part of our research: A new method for computing tropical linear spaces and more general duals of polyhedral subdivisions is developed and implemented in the software polymake. This is based on Ganter’s algorithm (1984) for finite closure systems. Additionally, we describe the implementation of a subfield of the field of formal Puiseux series. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is useful for computations in convex and algebraic tropical geometry. Tropical varieties, as for example tropical linear spaces or tropical Grassmannians, are intersections of finitely many tropical hypersurfaces. The set of corresponding polynomials is a tropical basis. We give an explicit upper bound for the degree of a general tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner bases and tropical bases. v Acknowledgments First of all, I would like to express my gratitude to my advisor Michael Joswig for his continuous support during the work on my thesis and the possibilities he gave me. I thank him, Hannah Markwig, Alex Fink and Wilhelm Stannat for refereeing my thesis. Further thanks go to my other co-authors Simon Hampe, Georg Loho and Benjamin Lorenz. I am indebted to Alex Fink, Takayuki Hibi, Hiroshi Hirai and Raman Sanyal for their hospitality. Furthermore, I would like to thank all of them and many others for fruitful and enriching conversations and discussions that I had during the time of writing this thesis. Especially, I would like to mention here Kazuo Murota, Thorsten Theobald and Timo de Wolff. Special thanks go to the members and former members of my group “Discrete Mathematics / Geometry” at TU Berlin, Benjamin Assarf, Simon Hampe, Robert Loewe, Georg Loho, Benjamin Lorenz, Marta Panizzut, Antje Schulz, Kristin Shaw and André Wagner for all their support and discussions. I also thank them and additionally Jan Hofmann and Olivier Sète for proof-reading parts of the manuscript. Moreover, I would like to thank my family, my mother, father and sister as well as my friends and colleagues for the encouragement throughout the last years. vii Contents Zusammenfassung iii Abstract v Acknowledgments vii 1 Introduction to tropical linear spaces 1 1.1 Matroids and subdivisions . ......................... 1 1.2 The Dressian .................................. 3 1.3 The tropical Grassmannian . ........................ 4 1.4Aboutthisthesis................................ 6 2 Matroids from hypersimplex splits 9 2.1 Introduction ................................... 9 2.2 Matroid polytopes and their facets . ................... 11 2.3 Matroid subdivisions and tropical linear spaces ............... 22 2.4 Rays of the Dressian .............................. 28 2.5 Remarks and open questions ......................... 34 2.6 Some matroid statistics ............................ 38 3 Multi-splits of hypersimplices 41 3.1 Introduction ................................... 41 3.2 Multi-splits of the hypersimplex ........................ 43 3.3 Matroid subdivisions and multi-splits ..................... 52 3.4 Coarsest matroid subdivisions ......................... 61 4 Algorithms for tropical linear spaces 67 4.1 Introduction ................................... 67 viii Contents 4.2 Closure systems, lower sets and matroids . .................. 68 4.3 Extended tight spans .............................. 71 4.4Tropicallinearspaces.............................. 73 4.4.1 Performance comparison ........................ 75 4.5 A case study on f-vectors of tropical linear spaces ............. 76 4.5.1 The Dressian Dr(3, 8)......................... 77 4.6Outlook..................................... 80 4.6.1 Higher Dressians ............................ 80 4.6.2 Further optimization .......................... 81 5 The degree of a tropical basis 83 5.1 Introduction ................................... 83 5.2 Degree bounds ................................. 84 5.3 Examples .................................... 86 5.4 Non-constant coefficients ............................ 88 5.5 The f-vector of a tropical variety ....................... 89 5.6 Open questions ................................. 92 6 Polyhedral computations over Puiseux fractions 93 6.1 Introduction ................................... 93 6.2 Ordered fields and rational functions ..................... 94 6.3 Parameterized polyhedra . ........................... 96 6.4Tropicaldualconvexhulls...........................102 6.5 Implementation .................................104 6.6 Computations ..................................105 6.6.1 Using polymake ............................105 6.6.2
Load more

Recommended publications
  • Discover Linear Algebra Incomplete Preliminary Draft
    Discover Linear Algebra Incomplete Preliminary Draft Date: November 28, 2017 L´aszl´oBabai in collaboration with Noah Halford All rights reserved. Approved for instructional use only. Commercial distribution prohibited. c 2016 L´aszl´oBabai. Last updated: November 10, 2016 Preface TO BE WRITTEN. Babai: Discover Linear Algebra. ii This chapter last updated August 21, 2016 c 2016 L´aszl´oBabai. Contents Notation ix I Matrix Theory 1 Introduction to Part I 2 1 (F, R) Column Vectors 3 1.1 (F) Column vector basics . 3 1.1.1 The domain of scalars . 3 1.2 (F) Subspaces and span . 6 1.3 (F) Linear independence and the First Miracle of Linear Algebra . 8 1.4 (F) Dot product . 12 1.5 (R) Dot product over R ................................. 14 1.6 (F) Additional exercises . 14 2 (F) Matrices 15 2.1 Matrix basics . 15 2.2 Matrix multiplication . 18 2.3 Arithmetic of diagonal and triangular matrices . 22 2.4 Permutation Matrices . 24 2.5 Additional exercises . 26 3 (F) Matrix Rank 28 3.1 Column and row rank . 28 iii iv CONTENTS 3.2 Elementary operations and Gaussian elimination . 29 3.3 Invariance of column and row rank, the Second Miracle of Linear Algebra . 31 3.4 Matrix rank and invertibility . 33 3.5 Codimension (optional) . 34 3.6 Additional exercises . 35 4 (F) Theory of Systems of Linear Equations I: Qualitative Theory 38 4.1 Homogeneous systems of linear equations . 38 4.2 General systems of linear equations . 40 5 (F, R) Affine and Convex Combinations (optional) 42 5.1 (F) Affine combinations .
    [Show full text]
  • 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
    LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set.
    [Show full text]
  • A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems
    ' : .v ' : V. • ' '^!V'> r" '%': ' W ' ::r'' NISTIR 4900 Computing and Applied Mathematics Laboratory A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems P.T. Boggs, J.W. Tolle, A.J. Kearsley August 1992 Technology Administration U.S. DEPARTMENT OF COMMERCE National Institute of Standards and Technology Gaithersburg, MD 20899 “QC— 100 .U56 4900 1992 Jc. NISTIR 4900 A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems P. T. Boggs J. W. Tone A. J. Kearsley U.S. DEPARTMENT OF COMMERCE Technology Administration National Institute of Standards and Technology Computing and Applied Mathematics Laboratory Applied and Computational Mathematics Division Gaithersburg, MD 20899 August 1992 U.S. DEPARTMENT OF COMMERCE Barbara Hackman Franklin, Secretary TECHNOLOGY ADMINISTRATION Robert M. White, Under Secretary for Technology NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY John W. Lyons, Director si.- MT-- ’•,•• /'k /-.<• -Vj* #' ^ ( "> M !f>' J ' • ''f,i sgsHv.w,^. ^ , .!n'><^.'". tr 'V 10 i^ailCin'i.-'J/ > ' it i>1l • •»' W " %'. ' ' -fft ?" . , - '>! .-' .S A Truncated SQP Algorithm for Large Scale Nonlinear Programming Problems * Paul T. Boggs ^ Jon W. ToUe ^ Anthony J. Kearsley § August 4, 1992 Abstract We consider the inequality constrained nonlinear programming problem and an SQP algorithm for its solution. We are primarily concerned with two aspects of the general procedure, namely, the approximate solution of the quadratic program, and the need for an appropriate merit function. We first describe an (iterative) interior-point method for the quadratic programming subproblem that, no matter when it its terminated, yields a descent direction for a suggested new merit function. An algorithm based on ideas from trust-region and truncated Newton methods, is suggested and some of our preliminary numerical results are discussed.
    [Show full text]
  • A Bound Strengthening Method for Optimal Transmission Switching in Power Systems
    1 A Bound Strengthening Method for Optimal Transmission Switching in Power Systems Salar Fattahi∗, Javad Lavaei∗;+, and Alper Atamturk¨ ∗ ∗ Industrial Engineering and Operations Research, University of California, Berkeley + Tsinghua-Berkeley Shenzhen Institute, University of California, Berkeley. Abstract—This paper studies the optimal transmission switch- transmission switches) in the system. Furthermore, The Energy ing (OTS) problem for power systems, where certain lines are Policy Act of 2005 explicitly addresses the “difficulties of fixed (uncontrollable) and the remaining ones are controllable siting major new transmission facilities” and calls for the via on/off switches. The goal is to identify a topology of the power grid that minimizes the cost of the system operation while utilization of better transmission technologies [2]. satisfying the physical and operational constraints. Most of the Unlike in the classical network flows, removing a line from existing methods for the problem are based on first converting a power network may improve the efficiency of the network the OTS into a mixed-integer linear program (MILP) or mixed- due to physical laws. This phenomenon has been observed integer quadratic program (MIQP), and then iteratively solving and harnessed to improve the power system performance by a series of its convex relaxations. The performance of these methods depends heavily on the strength of the MILP or MIQP many authors. The notion of optimally switching the lines of formulations. In this paper, it is shown that finding the strongest a transmission network was introduced by O’Neill et al. [3]. variable upper and lower bounds to be used in an MILP or Later on, it has been shown in a series of papers that the MIQP formulation of the OTS based on the big-M or McCormick incorporation of controllable transmission switches in a grid inequalities is NP-hard.
    [Show full text]
  • Linear Algebra (VII)
    Linear Algebra (VII) Yijia Chen 1. Review Basis and Dimension. We fix a vector space V. Lemma 1.1. Let A, B ⊆ V be two finite sets of vectors in V, possibly empty. If A is linearly indepen- dent and can be represented by B. Then jAj 6 jBj. Theorem 1.2. Let S ⊆ V and A, B ⊆ S be both maximally linearly independent in S. Then jAj = jBj. Definition 1.3. Let e1,..., en 2 V. Assume that – e1,..., en are linearly independent, – and every v 2 V can be represented by e1,..., en. Equivalently, fe1,..., eng is maximally linearly independent in V. Then fe1,..., eng is a basis of V. Note that n = 0 is allowed, and in that case, it is easy to see that V = f0g. By Theorem 1.2: 0 0 Lemma 1.4. If fe1,..., eng and fe1,..., emg be both bases of V with pairwise distinct ei’s and with 0 pairwise distinct ei, then n = m. Definition 1.5. Let fe1,..., eng be a basis of V with pairwise distinct ei’s. Then the dimension of V, denoted by dim(V), is n. Equivalently, if rank(V) is defined, then dim(V) := rank(V). 1 Theorem 1.6. Assume dim(V) = n and let u1,..., un 2 V. (1) If u1,..., un are linearly independent, then fu1,..., ung is a basis. (2) If every v 2 V can be represented by u1,..., un, then fu1,..., ung is a basis. Steinitz exchange lemma. Theorem 1.7. Assume that dim(V) = n and v1,..., vm 2 V with 1 6 m 6 n are linearly indepen- dent.
    [Show full text]
  • A Scenario-Based Branch-And-Bound Approach for MES Scheduling in Urban Buildings Mainak Dan, Student Member, Seshadhri Srinivasan, Sr
    1 A Scenario-based Branch-and-Bound Approach for MES Scheduling in Urban Buildings Mainak Dan, Student Member, Seshadhri Srinivasan, Sr. Member, Suresh Sundaram, Sr. Member, Arvind Easwaran, Sr. Member and Luigi Glielmo, Sr. Member Abstract—This paper presents a novel solution technique for Storage Units scheduling multi-energy system (MES) in a commercial urban PESS Power input to the electrical storage [kW]; building to perform price-based demand response and reduce SoCESS State of charge of the electrical storage [kW h]; energy costs. The MES scheduling problem is formulated as a δ mixed integer nonlinear program (MINLP), a non-convex NP- ESS charging/discharging status of the electrical stor- hard problem with uncertainties due to renewable generation age; and demand. A model predictive control approach is used PTES Cooling energy input to the thermal storage [kW]; to handle the uncertainties and price variations. This in-turn SoCTES State of charge of the thermal storage [kW h]; requires solving a time-coupled multi-time step MINLP dur- δTES charging/discharging status of the thermal storage; ing each time-epoch which is computationally intensive. This T ◦C investigation proposes an approach called the Scenario-Based TES Temperature inside the TES [ ]; Branch-and-Bound (SB3), a light-weight solver to reduce the Chiller Bank computational complexity. It combines the simplicity of convex −1 programs with the ability of meta-heuristic techniques to handle m_ C;j Cool water mass-flow rate of chiller j [kg h ]; complex nonlinear problems. The performance of the SB3 solver γC;j Binary ON-OFF status of chiller j; is validated in the Cleantech building, Singapore and the results PC Power consumed in the chiller bank [kW]; demonstrate that the proposed algorithm reduces energy cost by QC;j Thermal energy supplied by chiller j [kW]; about 17.26% and 22.46% as against solving a multi-time step heuristic optimization model.
    [Show full text]
  • The Big-M Method with the Numerical Infinite M
    Optimization Letters https://doi.org/10.1007/s11590-020-01644-6 ORIGINAL PAPER The Big-M method with the numerical infinite M Marco Cococcioni1 · Lorenzo Fiaschi1 Received: 30 March 2020 / Accepted: 8 September 2020 © The Author(s) 2020 Abstract Linear programming is a very well known and deeply applied field of optimization theory. One of its most famous and used algorithms is the so called Simplex algorithm, independently proposed by Kantoroviˇc and Dantzig, between the end of the 30s and the end of the 40s. Even if extremely powerful, the Simplex algorithm suffers of one initialization issue: its starting point must be a feasible basic solution of the problem to solve. To overcome it, two approaches may be used: the two-phases method and the Big-M method, both presenting positive and negative aspects. In this work we aim to propose a non-Archimedean and non-parametric variant of the Big-M method, able to overcome the drawbacks of its classical counterpart (mainly, the difficulty in setting the right value for the constant M). We realized such extension by means of the novel computational methodology proposed by Sergeyev, known as Grossone Methodol- ogy. We have validated the new algorithm by testing it on three linear programming problems. Keywords Big-M method · Grossone methodology · Infinity computer · Linear programming · Non-Archimedean numerical computing 1 Introduction Linear Programming (LP) is a branch of optimization theory which studies the min- imization (or maximization) of a linear objective function, subject to linear equality and/or linear inequality constraints. LP has found a lot of successful applications both in theoretical and real world contexts, especially in countless engineering applications.
    [Show full text]
  • Two Dimensional Search Algorithms for Linear Programming
    University of Nebraska at Omaha DigitalCommons@UNO Mathematics Faculty Publications Department of Mathematics 8-2019 Two dimensional search algorithms for linear programming Fabio Torres Vitor Follow this and additional works at: https://digitalcommons.unomaha.edu/mathfacpub Part of the Mathematics Commons Two dimensional search algorithms for linear programming by Fabio Torres Vitor B.S., Maua Institute of Technology, Brazil, 2013 M.S., Kansas State University, 2015 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Industrial and Manufacturing Systems Engineering Carl R. Ice College of Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 2019 Abstract Linear programming is one of the most important classes of optimization problems. These mathematical models have been used by academics and practitioners to solve numerous real world applications. Quickly solving linear programs impacts decision makers from both the public and private sectors. Substantial research has been performed to solve this class of problems faster, and the vast majority of the solution techniques can be categorized as one dimensional search algorithms. That is, these methods successively move from one solution to another solution by solving a one dimensional subspace linear program at each iteration. This dissertation proposes novel algorithms that move between solutions by repeatedly solving a two dimensional subspace linear program. Computational experiments demonstrate the potential of these newly developed algorithms and show an average improvement of nearly 25% in solution time when compared to the corresponding one dimensional search version. This dissertation's research creates the core concept of these two dimensional search algorithms, which is a fast technique to determine an optimal basis and an optimal solution to linear programs with only two variables.
    [Show full text]
  • G. Gutin, Computational Optimisation, Royal Holloway 2008
    Computational Optimisation Gregory Gutin April 4, 2008 Contents 1 Introduction to Computational Optimisation 1 1.1 Introduction.................................... 1 1.2 Algorithm efficiency and problem complexity . ...... 3 1.3 Optimality and practicality . .... 4 2 Introduction to Linear Programming 6 2.1 Linearprogramming(LP)model . 6 2.2 FormulatingproblemsasLPproblems . .... 6 2.3 GraphicalsolutionofLPproblems . .... 9 2.4 Questions ..................................... 13 3 Simplex Method 15 3.1 StandardForm .................................. 15 3.2 SolutionsofLinearSystems . 17 3.3 TheSimplexMethod............................... 18 3.4 Questions ..................................... 22 4 Linear Programming Approaches 25 4.1 Artificial variables and Big-M Method . ..... 25 4.2 Two-PhaseMethod................................ 26 4.3 Shadowprices................................... 28 4.4 Dualproblem................................... 29 4.5 DecompositionofLPproblems . 32 1 CONTENTS 2 4.6 LPsoftware.................................... 34 4.7 Questions ..................................... 34 5 Integer Programming Modeling 36 5.1 Integer Programming vs. Linear Programming . ...... 36 5.2 IPproblems.................................... 37 5.2.1 Travelling salesman problem . 37 5.2.2 KnapsackProblem ............................ 38 5.2.3 Binpackingproblem.. .. .. .. .. .. .. .. 38 5.2.4 Set partitioning/covering/packing problems . ........ 39 5.2.5 Assignment problem and generalized assignment problem ...... 40 5.3 Questions ....................................
    [Show full text]
  • A One-Phase Interior Point Method for Nonconvex Optimization
    Stanford University A one-phase interior point method for nonconvex optimization Oliver Hinder, Yinyu Ye January 12, 2018 Abstract The work of W¨achter and Biegler [40] suggests that infeasible-start interior point methods (IPMs) developed for linear programming cannot be adapted to nonlinear optimization without significant mod- ification, i.e., using a two-phase or penalty method. We propose an IPM that, by careful initialization and updates of the slack variables, is guaranteed to find a first-order certificate of local infeasibility, local optimality or unboundedness of the (shifted) feasible region. Our proposed algorithm differs from other IPM methods for nonconvex programming because we reduce primal feasibility at the same rate as the barrier parameter. This gives an algorithm with more robust convergence properties and closely resem- bles successful algorithms from linear programming. We implement the algorithm and compare with IPOPT on a subset of CUTEst problems. Our algorithm requires a similar median number of iterations, but fails on only 9% of the problems compared with 16% for IPOPT. Experiments on infeasible variants of the CUTEst problems indicate superior performance for detecting infeasibility. The code for our implementation can be found at https://github.com/ohinder/OnePhase. 1 Introduction Consider the problem min f(x) (1a) n x2R a(x) ≤ 0; (1b) where the functions a : Rn ! Rm and f : Rn ! R are twice differentiable and might be nonconvex. Examples of real-world problems in this framework include truss design, robot control, aircraft control, and aircraft design, e.g., the problems TRO11X3, ROBOT, AIRCRAFTA, AVION2 in the CUTEst test set [16].
    [Show full text]
  • Math 121: Linear Algebra and Applications
    Math 121: Linear Algebra and Applications Course Taught by Elden Elmanto Notes Taken by Forrest Flesher Fall 2019 Welcome to Math 121: Linear Algebra and Applications. I’m Forrest, your course assistant for the semester. Some important information: • The course webpage is: https://scholar.harvard.edu/elden/classes/math-121-fall-2019 • Office hour times, course times and more information is located in the syllabus, which is at: https://scholar.harvard.edu/files/elden/files/syllabus-121_fall_2019-final. pdf Office hours are at the following times in Science Center 231: – Sundays: 2:00-3:00 pm – Thursdays: 2:00-3:00 pm • I will hold two office hours per week. The first Friday office hour willbean interactive LaTeX introduction session. My office hours are at the following times: – Mondays: 8:00-10:00 pm, Leverett Dining hall – Fridays: 3:00-4:15 pm, Science Center 309A • The text for the course is Axler’s “Linear Algebra Done Right.” • I will by typing up notes in class, since there will be times when the course deviates from the book. • My email is [email protected]. Email with any questions, comments, or concerns, especially if you find a mistake in these notes. • We will use the Canvas site for submitting/grading problem sets. • For submitting your problem sets, please use LATEX. This is strongly encouraged for the first few psets, and required after the fifth pset. If you are newtexand need help getting started, come to office hours. I’d recommend using overleaf: https://www.overleaf.com 1 Math 121: Linear Algebra and Applications Fall 2019 Contents 1 September 3, 2019 4 1.1 Introduction ....................................
    [Show full text]
  • DRAFT Syllabus and Synopses for the Preliminary Examination in Mathematics 2014–2015 for Examination in 2015
    DRAFT Syllabus and Synopses for the Preliminary Examination in Mathematics 2014{2015 for examination in 2015 Contents 1 Foreword 3 2 Syllabus 5 2.1 Mathematics I . .5 2.2 Mathematics II . .7 2.3 Mathematics III . .8 2.4 Mathematics IV . 10 2.5 Mathematics V . 11 3 Mathematics I 12 3.1 Introductory Courses . 12 3.1.1 Introduction to University Level Mathematics | Dr. Neumann | 8 MT .................................... 12 3.1.2 Introduction to Complex Numbers | Prof. Kirwan | 2 MT . 13 3.2 Linear Algebra I | Prof. Kirwan | 14 MT . 14 3.3 Linear Algebra II | Prof. Lauder | 8HT . 16 3.4 Groups and Group Actions | Dr. Neale | 8 HT and 8 TT . 17 4 Mathematics II 19 4.1 Analysis I: Sequences and Series | Prof. Priestley | 14 MT . 19 4.2 Analysis II: Continuity and Differentiability | Prof. Heath-Brown | 16 HT 20 4.3 Analysis III: Integration | Prof. Capdeboscq | 8 TT . 22 5 Mathematics III 24 1 5.1 Introductory Calculus | Dr. Wilkins | 16 MT . 24 5.2 Probability | Prof. Goldschmidt | 16 MT . 25 5.3 Statistics | Dr. Laws | 8 HT . 26 6 Mathematics IV 28 6.1 Geometry | Prof. Lauder | 7 MT . 28 6.2 Dynamics | Dr Earl | 8 MT and 8 HT . 29 6.3 Optimisation | Prof. McDiarmid | 8 TT . 30 6.4 Constructive Mathematics | Prof. Hauser | 8 TT . 30 7 Mathematics V 32 7.1 Multivariable Calculus | Prof. Gaffney | 16 HT . 32 7.2 Fourier Series and PDEs | Prof. Baker | 16 HT . 33 7.3 Applications | Prof. Mason | 8 TT . 34 8 Computational Mathematics 35 8.1 Computational Mathematics | Dr.
    [Show full text]