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