DOCSLIB.ORG

Abstracted Primal-Dual Affine Programming

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2013 Abstracted primal-dual affine ogrpr amming Tien Chih Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Chih, Tien, "Abstracted primal-dual affine ogrpr amming" (2013). Graduate Student Theses, Dissertations, & Professional Papers. 10765. https://scholarworks.umt.edu/etd/10765 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. ABSTRACTED PRIMAL-DUAL AFFINE PROGRAMMING By Tien Chih B.A. University of Hawaii at Hilo, 2007 M.A. The University of Montana, USA, 2009 Dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics The University of Montana Missoula, MT December 2013 Approved by: Sandy Ross, Associate Dean of the Graduate School Graduate School Dr. George McRae, Chair Mathematical Sciences Dr. Kelly McKinnie Mathematical Sciences Dr. Jennifer McNulty Mathematical Sciences Dr. Thomas Tonev Mathematical Sciences Dr. Ronald Premuroso Accounting UMI Number: 3624605 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3624605 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Chih, Tien Ph.D., December 2013 Mathematics Abstracted Primal-Dual Affine Programming Committee Chair: George McRae, Ph.D. The classical study of linear (affine) programs, pioneered by George Dantzig and Albert Tucker, studies both the theory, and methods of solutions for a linear (affine) primal-dual maximization-minimization program, which may be described as follows: \Given A 2 m×n;~b 2 m;~c 2 n; d 2 , find ~x 2 n such that A~x ≤ ~b, and ~x ≥ 0, that R R R R R ? m maximizes the affine functional f(~x) := ~c · ~x − d; and find ~y 2 R such that A ~y ≥ ~c, and ~y ≥ 0, that minimizes the affine functional g(~y) := ~y ·~b − d." In this classical setting, there are several canonical results dealing with the primal-dual as- pect of affine programming. These include: I: Tucker's Key Equation, II: Weak Duality Theo- rem, III: Convexity of Solutions, IV: Fundamental Theorem of Linear (Affine) Programming, V: Farkas' Lemma,VI: Complementary Slackness Theorem, VII: Strong Duality Theorem, VIII: Existence-Duality Theorem, IX: Simplex Algorithm. We note that although the classical setting involves finite dimensional real vector spaces, moreover the classical viewpoint of these problems, the key results, and the solutions are extremely coordinate and basis dependent. However, these problems may be stated in much greater generality. We can define a function-theoretic, rather than coordinate-centric, view of these problem statements. Moreover, we may change the underlying ring, or abstract to a potentially infinite dimensional setting. Integer programming is a well known example of such a generalization. It is natural to ask then, which of the classical facts hold in a general setting, and under what hypothesis would they hold? We describe the various ways that one may generalize the statement of an affine program. Beginning with the most general case, we prove these facts using as few hypotheses as possible. Given each additional hypothesis, we prove all facts that may be proved in this setting, and provide counterexamples to the remaining facts, until we have successfully established all of our classical results. ii Acknowledgements I would like to give special thanks for my advisor, Dr. George McRae. His perspective, knowledge and experience has forever changed the way I think of and do Mathematics. His patience, guidance and support, coupled with his insight and expertise, has made completing this project possible, although his influence throughout my PhD work extends far beyond that. I would also like to thank my committee: Dr. Kelly McKinnie, Dr. Jenny McNulty, Dr. Thomas Tonev, and Dr. Ron Premuroso, for going above and beyond to continuously sup- porting me, providing me with help and advice, and for their making available their knowledge and expertise. Their continued support throughout this process has been invaluable. I would also like to thank the Department of Mathematical Sciences at the University of Montana. In particular I would like to thank my masters advisor Dr. Nikolaus Vonesson for continuing to be someone I could run to for help, even after the conclusion of our work, and Dr. Eric Chesebro for his advice and help critiquing this document. I would also like to thank the Associate Chair of Graduate Studies, Dr. Emily Stone for her continued diligence and support during this entire process. I would also like to thank all of the graduate students for sharing my passion of mathe- matics, and giving me their unique insights into a shared field we all love. I would like to give special thanks to all the graduate students who assisted in editing this document. Addition- ally, I would like to thank former graduate student Dr. Demitri Plessas has being especially willing to act as a sounding board during my entire time as a graduate student. I would also like to give special thanks for my family for their personal support through these years and all years prior. Finally, I would like to thank Tara Ashley Clayton for her ongoing and continuous sup- port through the entire schooling process, even as she attends to her own schooling as well. Her ability to understand and cope with the odd and sometimes aggravating lifestyle of a PhD student is immeasurable, and her emotional support during the more trying times of this process is invaluable. To her I give my love and everlasting gratitude. iii Notation and Conventions R a ring, typically a (ordered) (division ring) . 57 (58) (29) X a left R module (vector space), space of solutions . 57 (29) Y a left R module (vector space), space of constraints . 57 (29) HomR(X; Y ) collection of left homomorphisms from X to Y .................... 57 ∗ ∗ X ;Y HomR(X; R); HomR(Y; R) ........................................ 57 A an element of HomR(X; Y ) ....................................... 57 ~b element of Y , upper bound of primal problem . 57 c element of Homr(X; R), primal objective . 57 ⊕ the non negatives of ............................................ 67 X^ collection of spanning maps when X a vector space . 84 Y^ collection of spanning maps when Y a vector space . 84 X the image of 1R under the induced inclusions (basis of X) . 84 Y the image of 1R under the induced inclusions (basis of Y ) . 84 αi the row-like projection mapy ^i ◦ A; y^i 2 Y^ ......................... 85 A collection of all row-like projection maps fαig ..................... 85 M an oriented matroid defined on a set E . 131 α^i the map defined on X ⊕ R to simulate bi − αi in OM program . 155 X theα ^i that forms a vector(circuit) . 155 Y theα ^j induced byAnX 155 BB : Y ! f+; −; 0g, records coefficient of f forα ^j . 155 CC : X ! f+; −; 0g, records coefficient ofα ^i as summand of g . 155 AA : X × Y ! f+; −; 0g, records coefficient ofα ^i as summand ofα ^j 155 iv Contents 1 Introduction: Primal-Dual Affine Programs, Classical and Abstract1 1.1 The Goal.......................................1 1.2 Classical Affine Primal-Dual Programs.......................2 1.2.1 An Example of a Classical Affine Primal-Dual Program.........2 1.2.2 Introduction to Classical Affine Programming...............6 1.2.3 Basic facts.................................. 10 1.2.4 Results regarding Order........................... 11 1.2.5 Facts using R ................................. 19 1.2.6 Simplex Algorithm.............................. 24 1.3 Generalizations.................................... 42 1.3.1 Generalizing the Ring of Scalars...................... 42 1.3.2 Generalizing the Dimension or Rank.................... 46 v 1.3.3 Generalizing the Cones........................... 47 1.4 General Framework.................................. 48 1.5 Results......................................... 49 1.5.1 Results about affine maps.......................... 49 1.5.2 Results about Duality............................ 50 1.5.3 Results Classifying Solutions........................ 50 1.5.4 Results about Structure: Tucker Tableaux and Oriented Matroids... 51 1.5.5 Results about Optimal Solutions: The Simplex Algorithm........ 53 1.6 Potential Difficulty in an Abstract Situation.................... 53 1.7 Summary of Results................................. 54 2 Ordered Rings and Modules 56 2.1 Introduction...................................... 56 2.2 General Rings..................................... 57 2.3 Some Properties of Ordered Rings and Modules.................. 59 2.3.1 Properties of Ordered Rings......................... 59 2.3.2 Modules and Cones over Ordered Rings.................. 65 2.4
Recommended publications
  • Revised Primal Simplex Method Katta G

    Revised Primal Simplex Method Katta G

    6.1 Revised Primal Simplex method Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor First put LP in standard form. This involves floowing steps. • If a variable has only a lower bound restriction, or only an upper bound restriction, replace it by the corresponding non- negative slack variable. • If a variable has both a lower bound and an upper bound restriction, transform lower bound to zero, and list upper bound restriction as a constraint (for this version of algorithm only. In bounded variable simplex method both lower and upper bound restrictions are treated as restrictions, and not as constraints). • Convert all inequality constraints as equations by introducing appropriate nonnegative slack for each. • If there are any unrestricted variables, eliminate each of them one by one by performing a pivot step. Each of these reduces no. of variables by one, and no. of constraints by one. This 128 is equivalent to having them as permanent basic variables in the tableau. • Write obj. in min form, and introduce it as bottom row of original tableau. • Make all RHS constants in remaining constraints nonnega- tive. 0 Example: Max z = x1 − x2 + x3 + x5 subject to x1 − x2 − x4 − x5 ≥ 2 x2 − x3 + x5 + x6 ≤ 11 x1 + x2 + x3 − x5 =14 −x1 + x4 =6 x1 ≥ 1;x2 ≤ 1; x3;x4 ≥ 0; x5;x6 unrestricted. 129 Revised Primal Simplex Algorithm With Explicit Basis Inverse. INPUT NEEDED: Problem in standard form, original tableau, and a primal feasible basic vector. Original Tableau x1 ::: xj ::: xn −z a11 ::: a1j ::: a1n 0 b1 . am1 ::: amj ::: amn 0 bm c1 ::: cj ::: cn 1 α Initial Setup: Let xB be primal feasible basic vector and Bm×m be associated basis.
  • Chapter 6: Gradient-Free Optimization

    Chapter 6: Gradient-Free Optimization

    AA222: MDO 133 Monday 30th April, 2012 at 16:25 Chapter 6 Gradient-Free Optimization 6.1 Introduction Using optimization in the solution of practical applications we often encounter one or more of the following challenges: • non-differentiable functions and/or constraints • disconnected and/or non-convex feasible space • discrete feasible space • mixed variables (discrete, continuous, permutation) • large dimensionality • multiple local minima (multi-modal) • multiple objectives Gradient-based optimizers are efficient at finding local minima for high-dimensional, nonlinearly- constrained, convex problems; however, most gradient-based optimizers have problems dealing with noisy and discontinuous functions, and they are not designed to handle multi-modal problems or discrete and mixed discrete-continuous design variables. Consider, for example, the Griewank function: n n x2 f(x) = P i − Q cos pxi + 1 4000 i i=1 i=1 (6.1) −600 ≤ xi ≤ 600 Mixed (Integer-Continuous) Figure 6.1: Graphs illustrating the various types of functions that are problematic for gradient- based optimization algorithms AA222: MDO 134 Monday 30th April, 2012 at 16:25 Figure 6.2: The Griewank function looks deceptively smooth when plotted in a large domain (left), but when you zoom in, you can see that the design space has multiple local minima (center) although the function is still smooth (right) How we could find the best solution for this example? • Multiple point restarts of gradient (local) based optimizer • Systematically search the design space • Use gradient-free optimizers Many gradient-free methods mimic mechanisms observed in nature or use heuristics. Unlike gradient-based methods in a convex search space, gradient-free methods are not necessarily guar- anteed to find the true global optimal solutions, but they are able to find many good solutions (the mathematician's answer vs.
  • Combinatorial Characterizations of K-Matrices

    Combinatorial Characterizations of K-Matrices

    Combinatorial Characterizations of K-matricesa Jan Foniokbc Komei Fukudabde Lorenz Klausbf October 29, 2018 We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Pt´ak on linear-algebraic character- izations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies the original proof substantially by exploiting the duality of oriented matroids. As an application, we show that a simple principal pivot method applied to the linear complementarity problems with K-matrices converges very quickly, by a purely combinatorial argument. Key words: P-matrix, K-matrix, oriented matroid, linear complementarity 2010 MSC: 15B48, 52C40, 90C33 1 Introduction A matrix M ∈ Rn×n is a P-matrix if all its principal minors (determinants of principal submatrices) are positive; it is a Z-matrix if all its off-diagonal elements are non-positive; and it is a K-matrix if it is both a P-matrix and a Z-matrix. Z- and K-matrices often occur in a wide variety of areas such as input–output produc- tion and growth models in economics, finite difference methods for partial differential equations, Markov processes in probability and statistics, and linear complementarity problems in operations research [2]. In 1962, Fiedler and Pt´ak [9] listed thirteen equivalent conditions for a Z-matrix to be a K-matrix. Some of them concern the sign structure of vectors: arXiv:0911.2171v3 [math.OC] 3 Aug 2010 a This research was partially supported by the project ‘A Fresh Look at the Complexity of Pivoting in Linear Complementarity’ no.
  • Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10

    Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10

    CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex.
  • The Simplex Algorithm in Dimension Three1

    The Simplex Algorithm in Dimension Three1

    The Simplex Algorithm in Dimension Three1 Volker Kaibel2 Rafael Mechtel3 Micha Sharir4 G¨unter M. Ziegler3 Abstract We investigate the worst-case behavior of the simplex algorithm on linear pro- grams with 3 variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the “random edge” rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is known to be subexponential without dimension restriction, as well as Zadeh’s de- terministic history-dependent rule, for which no non-polynomial instances in general dimensions have been found so far. 1 Introduction The simplex algorithm is a fascinating method for at least three reasons: For computa- tional purposes it is still the most efficient general tool for solving linear programs, from a complexity point of view it is the most promising candidate for a strongly polynomial time linear programming algorithm, and last but not least, geometers are pleased by its inherent use of the structure of convex polytopes. The essence of the method can be described geometrically: Given a convex polytope P by means of inequalities, a linear functional ϕ “in general position,” and some vertex vstart, 1Work on this paper by Micha Sharir was supported by NSF Grants CCR-97-32101 and CCR-00- 98246, by a grant from the U.S.-Israeli Binational Science Foundation, by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.
  • Complete Enumeration of Small Realizable Oriented Matroids

    Complete Enumeration of Small Realizable Oriented Matroids

    Complete enumeration of small realizable oriented matroids Komei Fukuda ∗ Hiroyuki Miyata † Institute for Operations Research and Graduate School of Information Institute of Theoretical Computer Science, Sciences, ETH Zurich, Switzerland Tohoku University, Japan [email protected] [email protected] Sonoko Moriyama ‡ Graduate School of Information Sciences, Tohoku University, Japan [email protected] September 26, 2012 Abstract Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e., non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points and 5- dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices. 1 Introduction Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures or types is often more important than their metric structures. The combinatorial type of a point configuration is defined by all possible partitions of the points by a hyperplane (the definition given in (2.1)), and encodes various important information such as the convexity, the face lattice of the convex hull and all possible triangulations.
  • 11.1 Algorithms for Linear Programming

    11.1 Algorithms for Linear Programming

    6.854 Advanced Algorithms Lecture 11: 10/18/2004 Lecturer: David Karger Scribes: David Schultz 11.1 Algorithms for Linear Programming 11.1.1 Last Time: The Ellipsoid Algorithm Last time, we touched on the ellipsoid method, which was the first polynomial-time algorithm for linear programming. A neat property of the ellipsoid method is that you don’t have to be able to write down all of the constraints in a polynomial amount of space in order to use it. You only need one violated constraint in every iteration, so the algorithm still works if you only have a separation oracle that gives you a separating hyperplane in a polynomial amount of time. This makes the ellipsoid algorithm powerful for theoretical purposes, but isn’t so great in practice; when you work out the details of the implementation, the running time winds up being O(n6 log nU). 11.1.2 Interior Point Algorithms We will finish our discussion of algorithms for linear programming with a class of polynomial-time algorithms known as interior point algorithms. These have begun to be used in practice recently; some of the available LP solvers now allow you to use them instead of Simplex. The idea behind interior point algorithms is to avoid turning corners, since this was what led to combinatorial complexity in the Simplex algorithm. They do this by staying in the interior of the polytope, rather than walking along its edges. A given constrained linear optimization problem is transformed into an unconstrained gradient descent problem. To avoid venturing outside of the polytope when descending the gradient, these algorithms use a potential function that is small at the optimal value and huge outside of the feasible region.
  • Chapter 1 Introduction

    Chapter 1 Introduction

    Chapter 1 Introduction 1.1 Motivation The simplex method was invented by George B. Dantzig in 1947 for solving linear programming problems. Although, this method is still a viable tool for solving linear programming problems and has undergone substantial revisions and sophistication in implementation, it is still confronted with the exponential worst-case computational effort. The question of the existence of a polynomial time algorithm was answered affirmatively by Khachian [16] in 1979. While the theoretical aspects of this procedure were appealing, computational effort illustrated that the new approach was not going to be a competitive alternative to the simplex method. However, when Karmarkar [15] introduced a new low-order polynomial time interior point algorithm for solving linear programs in 1984, and demonstrated this method to be computationally superior to the simplex algorithm for large, sparse problems, an explosion of research effort in this area took place. Today, a variety of highly effective variants of interior point algorithms exist (see Terlaky [32]). On the other hand, while exterior penalty function approaches have been widely used in the context of nonlinear programming, their application to solve linear programming problems has been comparatively less explored. The motivation of this research effort is to study how several variants of exterior penalty function methods, suitably modified and fine-tuned, perform in comparison with each other, and to provide insights into their viability for solving linear programming problems. Many experimental investigations have been conducted for studying in detail the simplex method and several interior point algorithms. Whereas these methods generally perform well and are sufficiently adequate in practice, it has been demonstrated that solving linear programming problems using a simplex based algorithm, or even an interior-point type of procedure, can be inadequately slow in the presence of complicating constraints, dense coefficient matrices, and ill- conditioning.
  • A Branch-And-Bound Algorithm for Zero-One Mixed Integer

    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.
  • Coms: Complexes of Oriented Matroids Hans-Jürgen Bandelt, Victor Chepoi, Kolja Knauer

    Coms: Complexes of Oriented Matroids Hans-Jürgen Bandelt, Victor Chepoi, Kolja Knauer

    COMs: Complexes of Oriented Matroids Hans-Jürgen Bandelt, Victor Chepoi, Kolja Knauer To cite this version: Hans-Jürgen Bandelt, Victor Chepoi, Kolja Knauer. COMs: Complexes of Oriented Matroids. Journal of Combinatorial Theory, Series A, Elsevier, 2018, 156, pp.195–237. hal-01457780 HAL Id: hal-01457780 https://hal.archives-ouvertes.fr/hal-01457780 Submitted on 6 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMs: Complexes of Oriented Matroids Hans-J¨urgenBandelt1, Victor Chepoi2, and Kolja Knauer2 1Fachbereich Mathematik, Universit¨atHamburg, Bundesstr. 55, 20146 Hamburg, Germany, [email protected] 2Laboratoire d'Informatique Fondamentale, Aix-Marseille Universit´eand CNRS, Facult´edes Sciences de Luminy, F-13288 Marseille Cedex 9, France victor.chepoi, kolja.knauer @lif.univ-mrs.fr f g Abstract. In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asym- metric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of \conditional oriented matroid" (abbreviated COM).
  • Arxiv:1706.05795V4 [Math.OC] 4 Nov 2018

    Arxiv:1706.05795V4 [Math.OC] 4 Nov 2018

    BCOL RESEARCH REPORT 17.02 Industrial Engineering & Operations Research University of California, Berkeley, CA 94720{1777 SIMPLEX QP-BASED METHODS FOR MINIMIZING A CONIC QUADRATIC OBJECTIVE OVER POLYHEDRA ALPER ATAMTURK¨ AND ANDRES´ GOMEZ´ Abstract. We consider minimizing a conic quadratic objective over a polyhe- dron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic qua- dratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem us- ing the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution meth- ods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve con- vex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed al- gorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they pro- vide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP- based branch-and-bound algorithm with extended formulations.
  • ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes

    ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes

    ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow­ ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera­ lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) .