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Conjugate of Some Rational Functions and Convex Envelope of Quadratic Functions Over a Polytope Conjugate of some Rational functions and Convex Envelope of Quadratic functions over a Polytope by Deepak Kumar Bachelor of Technology, Indian Institute of Information Technology, Design and Manufacturing, Jabalpur, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Computer Science) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) January 2019 c Deepak Kumar, 2019 The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation en- titled: Conjugate of some Rational functions and Convex Envelope of Quadratic functions over a Polytope submitted by Deepak Kumar in partial fulfilment of the requirements of the degree of Master of Science. Dr. Yves Lucet, I. K. Barber School of Arts & Sciences Supervisor Dr. Heinz Bauschke, I. K. Barber School of Arts & Sciences Supervisory Committee Member Dr. Warren Hare, I. K. Barber School of Arts & Sciences Supervisory Committee Member Dr. Homayoun Najjaran, School of Engineering University Examiner ii Abstract Computing the convex envelope or biconjugate is the core operation that bridges the domain of nonconvex analysis with convex analysis. For a bi- variate PLQ function defined over a polytope, we start with computing the convex envelope of each piece. This convex envelope is characterized by a polyhedral subdivision such that over each member of the subdivision, it has an implicitly defined rational form(square of a linear function over a linear function). Computing this convex envelope involves solving an expo- nential number of subproblems which, in turn, leads to an exponential time algorithm. After that, we compute the conjugate of each such rational function de- fined over a polytope. It is observed that the conjugate has a parabolic subdivision such that over each member of its subdivision, it has an im- plicitly defined fractional form(linear function over square root of a linear function). This computation of the conjugate is performed with a worst-case linear time complexity algorithm. Finally, some directions and insights about computing the maximum of all the conjugates of each piece of a PLQ function and then the conjugate of that to obtain the biconjugate are provided as conjectures for future work. iii Lay Summary Optimization problems occur in a wide variety of fields ranging from Engineering to Mathematical Finance and solving these problems play a crucial role in the process. When the optimization problem is nonconvex it is often difficult to solve. In this thesis, we present a method to convert some class of nonconvex problems to convex problems, which are often easier to solve, and a method to compute their conjugates which can help solving the problem faster than before. iv Table of Contents Abstract . iii Lay Summary . iv Table of Contents . v List of Tables . viii List of Figures . ix List of Notations and Abbreviations . xi Acknowledgements . xii Dedication . xiii Chapter 1: Introduction . 1 1.1 PLQ Functions . 1 1.2 Convex Envelope . 2 1.3 Fenchel conjugate . 4 1.4 Motivation . 5 Chapter 2: Preliminaries and Notations . 8 Chapter 3: Convex Envelope of Bivariate Quadratic Func- tions over a polytope . 13 3.1 Quadratic Reduction . 13 3.2 Problem Formulation[Loc16] . 16 3.2.1 Problem Formulation . 16 3.2.2 Expressions for η(a; b).................. 19 3.2.3 Subproblems . 20 3.3 Exponential subproblems . 22 v TABLE OF CONTENTS 3.4 Optimal Solutions [Loc16] . 25 3.4.1 Solving the subproblems . 25 3.4.2 Functional forms . 26 3.4.3 Solutions . 26 3.5 Convex envelope: the maximum of all subproblems . 29 3.6 Algorithmic Design . 30 3.6.1 Input and Output Data Structures . 30 3.6.2 Main Algorithm . 33 3.6.3 Solving the subproblems . 37 3.7 Map Overlay problem . 40 3.7.1 Data structures for polyhedral region . 41 3.7.2 Sorting vertices in clockwise direction . 42 3.7.3 Map Overlay Algorithm . 45 Chapter 4: Conjugate of a class of Convex Bivariate Rational Functions over a polytope . 49 4.1 Subdifferentials in the interior of the polytope . 50 4.2 Subdifferentials at the Vertices . 55 4.3 Subdifferentials on the edges . 55 4.4 Structure of the conjugate domain . 59 4.5 Conjugate Expressions . 65 4.5.1 Fractional forms . 65 Chapter 5: Algorithmic computation of the Conjugate for a class of Bivariate Rational functions over a poly- tope . 73 5.1 Data Structures . 73 5.1.1 Parabolic region . 74 5.1.2 Output Data Structure . 76 5.2 Algorithm . 79 5.2.1 Main Algorithm . 80 5.2.2 Algorithm for edges . 81 5.3 Example 1 . 83 5.4 Example 2 . 89 Chapter 6: Conclusions and Future work . 96 6.1 Future work . 97 6.1.1 Maximum of all conjugates . 97 6.1.2 Conjugate of bivariate nonconvex PLQ Functions . 98 6.1.3 Convex envelope of bivariate nonconvex PLQ functions 98 vi TABLE OF CONTENTS Bibliography . 100 Appendix . 105 Appendix A: The η(a; b) expressions for the bilinear functions . 106 Appendix B: Solving the subproblems . 108 B.1 Case 1: Quadratic-Quadratic . 108 B.1.1 Solving the equality constraint . 109 B.1.2 Solving the inequalities . 109 B.1.3 Solutions . 111 B.2 Case 2: Quadratic-Linear . 112 B.2.1 Solving the equality constraint . 113 B.2.2 Solving the inequalities . 113 B.2.3 Solutions . 115 B.3 Case 3: Linear-Linear . 116 B.3.1 Solving the equality constraint . 116 B.3.2 Solving the inequalities . 117 B.3.3 Solutions . 118 Appendix C: Conjugate expressions for a rational function over an edge . 119 vii List of Tables Table 3.1 Different cases depending upon the eigenvalues . 14 Table 5.1 Conjugate domain for a rational function . 76 Table 5.2 Conjugate domain for quadratic functions . 80 Table 6.1 Observed intersections for computing the maximum of all conjugates . 97 viii List of Figures Figure 1.1 A univariate convex PLQ Function . 2 Figure 1.2 A non convex univariate PLQ Function . 3 x2 s2 Figure 1.3 f(x) = , its conjugate f ∗(s) = , and the map- 2 2 ping between the two . 5 Figure 1.4 Closed convex envelope (shown in Blue) of each piece of a univariate nonconvex PLQ function . 6 2 Figure 2.1 A Polyhedral subdivision of R . 10 Figure 3.1 η1's polyhedral domain . 22 Figure 3.2 η2(a; b)'s polyhedral domain . 23 Figure 3.3 Nine Subregions formed by overlaying domains of η1 and η2 .......................... 23 Figure 3.4 19 subregions, shown by different colors, formed by intersection of domains of 3 η(a; b)s belonging to a convex edge. 24 Figure 3.5 Input Data Structure . 31 Figure 3.6 Output Data Structure . 32 Figure 3.7 Space division by line y = mx + c . 41 Figure 3.8 Vertex Enumeration Problem . 48 Figure 4.1 Primal and dual mapping for quadratic function in Example 4.4 over lines parallel to x1 + x2 = 0. 52 Figure 4.2 Normal cone at a vertex v 2 P shown as the arrows in the intersection of Red and Green region . 56 Figure 4.3 Polyhedral subdivision for Example 4.14 . 62 Figure 4.4 Parabolic subdivision for r and P in Example 4.22 . 64 Figure 4.5 Parabolic subdivision for r and P from Example 4.5 . 66 Figure 5.1 A subdifferential region . 74 Figure 5.2 Region with only active constraints . 75 ix LIST OF FIGURES Figure 5.3 Proposed subdivision for storing a subdifferential region 75 Figure 5.4 Output Data structure . 77 Figure 5.5 Normal cone division lines for Example 5.3 . 83 Figure 5.6 Conjugate for Example 5.3 . 88 Figure 5.7 Normal cone division lines for Example 5.4 . 90 Figure 5.8 Conjugate for Example 5.4 . 95 x List of Notations and Abbreviations R Real numbers φ Empty set jAj Cardinality of set A dom(f) Effective domain of function f cl(g) Closed envelope of function g convfP Convex envelope of f over a region P conv Closed convex Envelope Sn×n Symmetric Matrix of size n × n int(P ) Interior of the set P ri(P ) Relative interior of set P PLQ Piecewise Linear Quadratic LFT Legendre-Fenchel transform f ∗ Conjugate of function f sup S The supremum of set S inf S The infimum of set S NP (x) Normal cone of set P at x @f(x) Subdifferential of function f at x rf(x) Gradient of function f at x IP Indicator function of set P xi Acknowledgements I would take this opportunity to acknowledge and express my sincere gratitude towards my supervisor Dr. Yves Lucet, for his constant guidance, encouragement, and precious advice throughout my graduate studies. I am very grateful for his endless support, especially for his expertise, patience and insightful suggestions during the course of my research. His attention to details motivated me to pay closer attention to everything I work on, and his positive outlook and confidence in my research inspired me and gave me the right confidence to dig deeper into my work. I would also like to extend my gratitude towards the members of my su- pervisory committee: Dr. Heinz Bauschke and Dr. Warren Hare for spending their valuable time reading my thesis and providing constructive and insight- ful feedback. I am also thankful to all of my committee members for making my defense a wonderful moment of my life. I truly appreciate the support of my family and all of my friends and colleagues of the CCA research group for all they meant to me during the completion of my research.
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