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Copositive Plus Matrices Copositive Plus Matrices Willemieke van Vliet Master Thesis in Applied Mathematics October 2011 Copositive Plus Matrices Summary In this report we discuss the set of copositive plus matrices and their properties. We examine certain subsets of copositive plus matrices, copositive plus matrices with small dimensions, and the copositive plus cone and its dual. Furthermore, we consider the Copositive Plus Completion Problem, which is the problem of deciding whether a matrix with unspecified entries can be completed to obtain a copositive plus matrix. The set of copositive plus matrices is important for Lemke's algorithm, which is an al- gorithm for solving the Linear Complementarity Problem (LCP). The LCP is the problem of deciding whether a solution for a specific system of equations exists and finding such a solution. Lemke's algorithm always terminates in a finite number of steps, but for some prob- lems Lemke's algorithm terminates with no solution while the problem does have a solution. However, when the data matrix of the LCP is copositive plus, Lemke's algorithm always gives a solution if such solution exists. Master Thesis in Applied Mathematics Author: Willemieke van Vliet First supervisor: Dr. Mirjam E. D¨ur Second supervisor: Prof. dr. Harry L. Trentelman Date: October 2011 Johann Bernoulli Institute of Mathematics and Computer Science P.O. Box 407 9700 AK Groningen The Netherlands Contents 1 Introduction 1 1.1 Structure . .1 1.2 Notation . .1 2 Copositive Plus Matrices and their Properties 3 2.1 The Class of Copositive Matrices . .3 2.2 Properties of Copositive Matrices . .4 2.3 Properties of Copositive Plus Matrices . .5 2.4 Subsets . .8 2.5 Small Dimensions . 10 2.6 The Copositive Plus Cone and its Dual Cone . 13 2.6.1 The Copositive Plus Cone . 14 2.6.2 The Dual Copositive Plus Cone . 14 2.7 Copositive Plus of Order r ............................. 15 2.8 Copositive Plus Matrices with −1; 0; 1 Entries . 15 3 The Copositive Plus Completion Problem 19 3.1 Unspecified Non-diagonal Elements . 19 3.2 Unspecified Diagonal Entries . 25 4 Lemke's Algorithm 29 4.1 The Linear Complementarity Problem . 29 4.2 Lemke's Algorithm . 31 4.3 Termination and Correctness . 35 4.3.1 Termination for Nondegenerate Problems . 36 4.3.2 Termination for Degenerate Problems . 37 4.3.3 Conditions under which Lemke's Algorithm is Correct . 40 4.4 Applications in Linear- and Quadratic Programming . 40 4.4.1 Linear Programming . 40 4.4.2 Quadratic Programming . 43 4.5 Applications in the Game Theory . 44 4.5.1 Two Person Games . 45 4.5.2 Polymatrix Games . 46 4.6 An Application in Economics . 49 Nomenclature 53 iii iv CONTENTS Index 55 Bibliography 57 Chapter 1 Introduction 1.1 Structure In 1968 Cottle and Dantzig proposed the Linear Complementarity Problem (LCP)[2]. The LCP is the problem of deciding whether a solution for a specific system of equations exists. An algorithm for solving the LCP is Lemke's algorithm which is also called the complementary pivot algorithm. It was proposed by Lemke in 1965 [12] for finding equilibrium points. Lemke's algorithm always terminates in a finite number of steps, but for some problems Lemke's algorithm terminates with no solution while the problem does have a solution. However, when the data matrix of the LCP is Copositive Plus, Lemke's algorithm always gives a solution if such solution exists. In this report we discuss the LCP as well as Lemke's algorithm. Further, we examine the set of copositive plus matrices and their properties. In chapters 2 and 3, we focus on copositive plus matrices. In chapter 2, we discuss some basic properties of the copositive plus matrices. We examine certain subsets of copositive plus matrices, copositive plus matrices with small dimensions, and the copositive plus cone and its dual. Furthermore, we consider matrices which are copositive plus of order r and we consider copositive plus matrices with only −1; 0; 1 entries. In chapter 3, we discuss the Copositive Plus Completion Problem. We consider matrices in which some entries are specified and the remaining entries are unspecified and are free to be chosen, such matrices are called partial matrices. The choice of values for the unspecified entries is a completion of the partial matrix. The Copositive Plus Completion Problem is the problem of deciding which partial matrices have a copositive plus completion. In the first part of this chapter we examine matrices with unspecified non-diagonal entries and in the second part we examine matrices with unspecified diagonal entries. In chapter 4, we discuss the LCP and Lemke's algorithm. We show that Lemke's algorithm always terminates in a finite number of steps. Furthermore, we discuss some applications of the LCP: Linear and Quadratic programming, the problem of finding equilibrium points in two person and polymatrix games, and the problem of finding equilibrium points in economics. 1.2 Notation In this report we will use the following notation. The set of nonnegative matrices is denoted by N and the set of symmetric matrices is denoted by S. 1 2 CHAPTER 1. INTRODUCTION The set R is the set of real numbers. The set of nonnegative real numbers is denoted by n R+. So if a vector v is in R+, then all n entries of the vector v are nonnegative. Further, n n+1 the n-dimensional sphere with radius 1 is defined as the set S = fv 2 R j kvk = 1g. The n n+1 nonnegative quadrant of this sphere is denoted by S+ = fv 2 R+ j kvk = 1g. We denote the ith element of a vector v by vi and the element of the ith row and jth column of a matrix M is denoted by Mij. The vector e is the vector with ones everywhere. The unit vector ei is the vector with at the ith entry an one and zeros everywhere else. Inequality of vectors is always meant entry wise. For example, given a vector v, v ≥ 0 means that every entry of v is nonnegative. T T At last, the inner product of two vectors v1 and v2 is denoted by hv1; v2i = v1 v2(= v2 v1). The norm of a vector v is given by kvk = phv; vi. Furthermore, the infinity norm of a vector v is given by kvk1 = max(jv1j; jv2j;:::; jvnj). Chapter 2 Copositive Plus Matrices and their Properties In the last sixty years, several articles about the properties of the set of copositive matrices are proposed; see for example [3], [4], [17], [16], [6] and [5]. Known is what the cone and the dual cone of these matrices look like and what we can say about this set of matrices for small dimensions. Further, many sufficient and necessary conditions are found for the copositive matrices. Much less is known about the copositive plus matrices, which form a subset of the copositive matrices. These matrices are introduced by C.E. Lemke [12] and the properties of these matrices have been studied by R.W. Cottle, G.J. Habetler, and C.E. Lemke in [3] and [4]; by A.J. Hoffman and F. Pereira in [8]; and by H. V¨aliahoin [17]. In this chapter the most important results of these articles will be presented and we will present some new theorems about copositive plus matrices. 2.1 The Class of Copositive Matrices We will give here the definitions of copositive and copositive plus matrices with respect to symmetric matrices. However, for every non symmetric matrix M, we have that M~ = 1 T ~ 2 (M + M ) is a symmetric matrix. So if a definition of a property holds for M, we say that the corresponding non symmetric matrix M also satisfies this property. We provide the following definitions and notation for the class of copositive matrices. Definition 1. Let M be a real symmetric n×n matrix. The matrix M is said to be copositive, denoted by M 2 C, if T z Mz > 0 for all z > 0: The matrix M is said to be copositive plus, denoted by M 2 C+, if T M 2 C and for z > 0; z Mz = 0 implies Mz = 0: The matrix M is said to be strictly copositive if T z Mz > 0 for all nonzero z > 0: The interior of C is the set of strictly copositive matrices. Therefore, if a matrix M is strictly copositive it will be denoted by M 2 int(C). 3 4 CHAPTER 2. COPOSITIVE PLUS MATRICES AND THEIR PROPERTIES Note that for a non symmetric matrix M and its corresponding symmetric matrix M~ , the quadratic product 1 1 1 zT Mz = zT Mz + zT M T z = zT (M + M T )z = zMz:~ 2 2 2 So above definitions almost holds for non symmetric matrices, the only difference is that for a T T non symmetric copositive plus matrix M, z > 0 with z Mz = 0 implies that (M +M )z = 0. A class of matrices which is close to the class of copositive matrices is the class of positive definite matrices. Definition 2. Let M be a real symmetric n × n matrix. The matrix M is said to be positive semidefinite, denoted by M 2 S+, if T z Mz > 0 for all z: The matrix M is said to be positive definite, denoted by M 2 S++, if zT Mz > 0 for all z 6= 0: Two important properties are the property of inheritance and the property of closure under principal rearrangements. All classes of matrices defined in this section satisfies both properties; see [3].
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