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PRIMAL-DUALINTERIOR-POINT
MEfHODS
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PRIMA[.-Dui,INTEf ORmPOINT
MEFHODS
Stephen J. WrightArgonne National Laboratory
Argonne, Illinois
Society for Industrial and Applied MathematicsPhiladelphia
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Copyright © 1997 by the Society for Industrial and Applied Mathematics.
10987654
All rights reserved. Printed in the United States of America. No part of this book maybe reproduced, stored, or transmitted in any manner without the written permission ofthe publisher. For information, write to the Society for Industrial and Applied Mathematics,3600 University City Science Center, Philadelphia, PA 19104-2688.
Library of Congress Cataloging-in-Publication Data
Wright, Stephen J., 1960-Primal-dual interior-point methods / Stephen J. Wright.
p. cm.Includes bibliographical references and index.
1. Linear programming. 2. Mathematical optimization. I. Title.T57.74.W75 1997519.72 -- DC20 96-42071
S.LBJÍL is a registered trademark.
ISBN-13: 978-0-898713-82-4 (pbk.)ISBN-10: 0-89871-382-X (pbk.)
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For Nick, Angela, and Charlotte
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Contents
Preface xiii
Notation xvii
1 Introduction 1Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 2Primal-Dual Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4The Central Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7A Primal-Dual Framework . . . . . . . . . . . . . . . . . . . . . . . 8Path-Following Methods . . . . . . . . . . . . . . . . . . . . . . . . 9Potential-Reduction Methods . . . . . . . . . . . . . . . . . . . . . 10Infeasible Starting Points . . . . . . . . . . . . . . . . . . . . . . . 11Superlinear Convergence . . . . . . . . . . . . . . . . . . . . . . . . 12Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Mehrotra's Predictor-Corrector Algorithm . . . . . . . . . . . . . . 14Linear Algebra Issues . . . . . . . . . . . . . . . . . . . . . . . . . 16Karmarkar's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Background: Linear Programming and Interior-PointMethods 21Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Optimality Conditions, Duality, and Solution Sets . . . . . . . . . 23More on Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24The B U N Partition and Strict Complementarity . . . . . . . . . . 27A Strictly Interior Point . . . . . . . . . . . . . . . . . . . . . . . . 29Rank of the Matrix A . . . . . . . . . . . . . . . . . . . . . . . . . 31Bases and Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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vnl
Farkas's Lemma and a Proof of the Goldman-Tucker Result . . . . 34The Central Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Background: Primal Methods . . . . . . . . . . . . . . . . . . . . . 40Primal-Dual Methods: Development of the Fundamental Ideas . . 43Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Complexity Theory 49Polynomial Versus Exponential, Worst Case Versus Average Case . 50Storing the Problem Data: Dimension and Size . . . . . . . . . . . 51The Turing Machine and Rational Arithmetic . . . . . . . . . . . . 52Primal-Dual Methods and Rational Arithmetic . . . . . . . . . . . 52Linear Programming and Rational Numbers . . . . . . . . . . . . . 54Moving to a Solution from an Interior Point . . . . . . . . . . . . . 55Complexity of Simplex, Ellipsoid, and Interior-Point Methods . . . 57Polynomial and Strongly Polynomial Algorithms . . . . . . . . . . 58Beyond the Turing Machine Model . . . . . . . . . . . . . . . . . . 59More on the Real-Number Model and Algebraic Complexity . . . . 60A General Complexity Theorem for Path-Following Methods . . . 61Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Potential-Reduction Methods 65A Primal-Dual Potential-Reduction Algorithm . . . . . . . . . . . 67Reducing ' Forces Convergence . . . . . . . . . . . . . . . . . . . 68A Quadratic Estimate of 4)P Along a Feasible Direction . . . . . . 70Bounding the Coefficients in the Quadratic Approximation . . . . 72An Estimate of the Reduction in I and Polynomial Complexity . 75What About Centrality? . . . . . . . . . . . . . . . . . . . . . . . . 78Choosing p and a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Path-Following Algorithms 83The Short-Step Path-Following Algorithm . . . . . . . . . . . . . . 86Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88The Predictor-Corrector Method . . . . . . . . . . . . . . . . . . . 91A Long-Step Path-Following Algorithm . . . . . . . . . . . . . . . 96Limit Points of the Iteration Sequence . . . . . . . . . . . . . . . . 100Proof of Lemma 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 103Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 104D
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6 Infeasible-Interior-Point Algorithms 107The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Convergence of Algorithm IPF . . . . . . . . . . . . . . . . . . . . 111Technical Results I: Bounds on vkII (xk , sk )I1 . . . . . . . . . . . . . 113Technical Results II: Bounds on (Dk ) -1 Oxk and DkOsk . . . . . . 115Technical Results III: A Uniform Lower Bound on ak . . . . . . . 118Proofs of Theorems 6.1 and 6.2 . . . . . . . . . . . . . . . . . . . . 120Limit Points of the Iteration Sequence . . . . . . . . . . . . . . . . 121
7 Superlinear Convergence and Finite Termination 127Affine-Scaling Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 128An Estimate of (Ox, As): The Feasible Case . . . . . . . . . . . . 129An Estimate of (Ox, Os): The Infeasible Case . . . . . . . . . . . . 133Algorithm PC Is Superlinear . . . . . . . . . . . . . . . . . . . . . 135Nearly Quadratic Methods . . . . . . . . . . . . . . . . . . . . . . 136Convergence of Algorithm LPF+ . . . . . . . . . . . . . . . . . . . 139Convergence of the Iteration Sequence . . . . . . . . . . . . . . . . 144E(A, b, c) and Finite Termination . . . . . . . . . . . . . . . . . . . 145A Finite Termination Strategy . . . . . . . . . . . . . . . . . . . . 146Recovering an Optimal Basis . . . . . . . . . . . . . . . . . . . . . 150More on c(A, b, c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8 Extensions 157The Monotone LCP . . . . . . . . . . . . . . . . . . . . . . . . . . 157Mixed and Horizontal LCP . . . . . . . . . . . . . . . . . . . . . . 160Strict Complementarity and LCP . . . . . . . . . . . . . . . . . . . 162Convex QP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Convex Programming . . . . . . . . . . . . . . . . . . . . . . . . . 164Monotone Nonlinear Complementarity and Variational Inequalities 167Semidefinite Programming . . . . . . . . . . . . . . . . . . . . . . . 168Proof of Theorem 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 171Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 173
9 Detecting Infeasibility 177Self-Duality . . . .. ... . . . .. . .. . .. . . . . .. . .. . . . 178The Simplified HSD Form . . . . . . . . . . . . . . . . . . . . . . . 179The HSD Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Identifying a Solution-Free Region . . . . . . . . . . . . . . . . . . 185D
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Implementations of the HSD Formulations . . . . . . . . . . . . . 188Notes and References'. . . . . . . . . . . . . . . . . . . . . . . . . 190
10 Practical Aspects of Primal-Dual Algorithms 193Motivation for Mehrotra's Algorithm . . . . . . . . . . . . . . . . . 194The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Superquadratic Convergence . . . . . . . . . . . . . . . . . . . . . . 198Second-Order Trajectory-Following Methods . . . . . . . . . . . . . 199Higher-Order Methods . . . . . . . . . . . . . . . . . . . . . . . . . 202Further Enhancements . . . . . . . . . . . . . . . . . . . . . . . . . 204Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 207
11 Implementations 209Three Forms of the Step Equation . . . . . . . . . . . . . . . . . . 210The Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . 211Sparse Cholesky Factorization: Minimum-Degree Orderings . . . . 212Other Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 215Small Pivots in the Cholesky Factorization . . . . . . . . . . . . . 216Dense Columns in A . . . . . . . . . . . . . . . . . . . . . . . . . . 219The Augmented System Formulation . . . . . . . . . . . . . . . . . 220Factoring Symmetric Indefinite Matrices . . . . . . . . . . . . . . . 221Starting Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Alternative Formulations for the Linear Program . . . . . . . . . . 226Free Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Presolving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Primal-Dual Codes . . . . . . . . . . . . . . . . . . . . . . . . . 232Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . 232
A Basic Concepts and Results 239Order Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Convex Sets and Functions . . . . . . . . . . . . . . . . . . . . . . 240KKT Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Convexity and Global Solutions . . . . . . . . . . . . . . . . . . . . 242Self-Duality and a Proof of Theorem 9.1 . . . . . . . . . . . . . . . 243The Natural log Function and a Proof of Lemma 4.1 . . . . . . . . 245Singular Value Decomposition, Matrix Rank, and QR Factorization 246Hoffman's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 248The Stewart—Todd Result and a Proof of Lemma 7.2 . . . . . . . . 249D
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The Sherman—Morrison—Woodbury Formula . . . . . . . . . . . . . 250Asymptotic Convergence . . . . . . . . . . . . . . . . . . . . . . . . 251Storing Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . 253The Turing Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 254
B Software Packages 257BPMPD.................................258CPLEX/Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259HOPDM ..... ............................259LIPS OL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260LOQO..................................261Newton Barrier XPRESS-MP . . . . . . . . . . . . . . . . . . . . . 261OSL/EKKBSLV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262PCx...................................263
Bibliography 265
Index 281
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Preface
Linear programming has been the dominant paradigm in optimization sinceDantzig's development of the simplex method in the 1940s. In 1984, thepublication of a paper by Karmarkar started a wave of research into a newclass of methods known as interior-point methods, and in the decade sincethen, primal-dual algorithms have emerged as the most important and usefulalgorithms from this class.
On the theoretical side, the properties of primal-dual methods for linearprogramming have been quite well understood by researchers since approxi-mately 1994. On the computational side, most interior-point software writ-ten since 1990 has been based on a single primal-dual algorithm: Mehrotra'spredictor-corrector algorithm. The interesting results are, however, widelyscattered in the literature, and most of the relevant papers and reportsassume a significant amount of background knowledge on the part of thereader. In this book, we use a simple, unified framework to describe themajor results, and we provide a straightforward, self-contained account ofthe underlying theory.
Primal-dual methods have excellent theoretical properties, good prac-tical performance, and pleasing relationships to earlier fundamental ideasin mathematical programming. They can be extended to wider classes ofproblems, including convex quadratic programming, monotone linear com-plementarity, and semidefinite programming, without losing their attributesof simplicity and good practical behavior. Extensions to more difficult classesof problems (even to nonlinear programming) are under way, but the issuesbecome more complicated with the loss of linearity and convexity. Theseextensions are keeping many researchers busy.
Chapter 1 presents a broad overview of primal-dual methods and shouldbe read first by all nonspecialist readers. In the next two chapters, we fill insome of the background for primal-dual methods with results from the the-ory of linear programming (Chapter 2) and complexity theory (Chapter 3).
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xiv Preface
Chapter 2 also contains some general historical background on interior-pointmethods. Descriptions and full analyses of the most interesting algorithmsappear in Chapters 4, 5, and 6. An algorithm based on the primal-dualpotential function is the subject of Chapter 4. Chapter 5 describes three al-gorithms of the path-following genre, while Chapter 6 discusses an infeasible-interior-point method that generalizes the long-step path-following methodfrom the preceding chapter. Issues of fast local convergence and finite ter-mination at an optimal solution are discussed in Chapter 7.
Extensions of primal-dual methods beyond linear programming are out-lined in Chapter 8. We discuss monotone linear and nonlinear complemen-tarity problems, convex programming, and semidefinite programming. Tech-niques for reliable detection of infeasible linear programs, including use ofthe homogeneous self-dual formulation, are the subject of Chapter 9.
The last two chapters turn to topics of more immediate practical interest.Most current software is based on Mehrotra's predictor-corrector algorithm,which enhances the algorithms of Chapters 4, 5, and 6 with a higher-ordersearch direction and some effective heuristics. These enhancements are mo-tivated and described in Chapter 10. In Chapter 11, we examine the compu-tational issues involved in implementing primal-dual methods. The agendahere is dominated by large sparse systems of linear equations, which mustbe solved to obtain search directions at each iteration. We also discuss theparticular features of some primal-dual codes that have appeared recentlyin the public domain.
Although this book contains a good deal of analysis, it has a somewhatpractical bias. We tend to focus on algorithms that are closely related topractical methods. Primal-dual affine-scaling algorithms are omitted for thisreason, despite their fascinating theoretical properties. Within each chapter,the intuitive/descriptive sections of the text are well separated from the moretechnical sections; the latter can be skipped safely and left to a rainy day.
This book is intended to fulfill a number of needs. It can be used asa text for a course on interior-point methods for linear programming. Se-lected chapters can be used for a short lecture series within a wider courseon optimization or linear programming. The prerequisites are a basic un-derstanding of real analysis and linear algebra and a familiarity with theelements of duality theory for linear programming.
Optimization practitioners and researchers who are not interior-pointspecialists can use the book as an introduction to some of the major ideasand algorithms. Experts can use it as a reference for the main theoreticalresults and analytical techniques and also as a guide to aspects of the areaD
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Preface xv
with which they are not so familiar.The field of interior-point methods is an active and competitive one, and
the origin of some of the main ideas is sometimes a topic of contention.We have not tried to taxonomize the literature or to trace the developmentof the various ideas, since these issues are of limited interest to studentsand nonspecialists. Still, we have tried to point out the key publications,including papers with comprehensive citation lists. No doubt there are someomissions, for which we apologize.
Two sites on the World Wide Web will interest readers of this book.Interior-Point Methods Online collects new technical reports in the area,together with other relevant information. Its URL is
http://www.mcs.ani.gov/home/otc/InteriorPoint/
The other site is the home page for this book, which contains up-to-dateinformation on software packages, a list of misprints, a feedback column,and so on. The URL is
http://www.siam.org/books/swright/
Other Web sites are mentioned in the text as well. The Web is proving to bea valuable adjunct to research activity in this and other fast-moving areas.
I thank Mihai Anitescu, Uri Ascher, Larry Biegler, Joe Czyzyk, SharonFilipowski, Jean-Pierre Haeberly, Iry Lustig, David Mayne, Michael Over-ton, Florian Potra, Barry Smith, Yin Zhang, and the anonymous refereesof the February 1996 draft for their interest, advice, and extremely valu-able comments, which filled numerous gaps in my knowledge. Gail Pieper'sassiduous proofreading turned up many infelicities of grammar and logic.As always, her help was invaluable. I am also grateful to Paul Plassmannand Jorge Nocedal for their encouragement during the early stages of thisproject. Finally, I thank Susan Ciambrano and the staff at SIAM, who madethe publication process a sheer delight.
STEPHEN J. WRIGHT
ARGONNE, ILLINOIS
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Notation
We summarize the notation and terminology of the book here to avoid con-fusion and save page ruffling.
I1 The space of real n-dimensional vectors.][^+ The nonnegative orthant {x E R I x >_ 0}.R'xn The set of real matrices with dimensions m x n.
II • (I, II • 112 Euclidean norm. For u E IRTh, (lull = (EZ 1 u?)1/2For a matrix M, IIMII = max11u11=1 IIMull.
I I I I 1 The 1-norm. For u E RTh, I I u l l i= ^2 i lul.I.II 'III The oo-norm. For u E lR' , IIull oo = maxi=1,2,...,n I uil.
(Note that IIuIl < IIull 2 < Ilull 1 for any vector u.)e (l,1,...,l)Tlog(.) The natural logarithm: loge
x IRS-vector of primal variables.A IR'-vector of dual variables, that is, Lagrange multipliers
for the equality constraints Ax = b.s Wi-vector of dual slacks, that is, Lagrange multipliers
for the bound constraints x > 0.
A RmXn-coefficient matrix for the linear program.A. ith column of A.problem data
The triple (A, b, c), which completely defines the linearprogramming problem.
L Number of bits required to store the problem data when allcomponents of (A, b, c) are integers or rational numbers.
.F Primal-dual feasible set:
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xviii
Notation
.Fo
duality gap
µrbr,
{(x, A, s) I Ax = b, ATA + s = c, (x, s)>0}.Primal-dual strictly feasible set:
{(x,A,sflAx=b,ATA+s=c,(x, ․)>0).If (x, A, s) is a feasible point, the duality gap is cTx - bT A,
which coincides with xT s.Duality measure, defined as xT s/n.Primal residual, defined as Ax - b.Dual residual, defined as ATA + s - c.
Z* Optimal objective at a solution of the linear program. If(x*, A*, s*) is a primal-dual solution, then cTx* = Z* = bTA*If (x, A, s) is any feasible point, then cTx > Z* > bT A.
I p Set of primal solutions x*.SZD Set of dual solutions (A*, s*).SZ Set of primal-dual solutions (x*, A*, s*); Sl = SIP X 1D•13 Defined by i E 13 C {1, 2,... , n} if xi > 0 for
some primal solution x*.N Defined by i E J1Í C {1, 2, . .. , n} if si > 0 for
some dual slack solution s*.(BUN= {1,2,...,n} and 13í1N= Ó.)
IBI, IA' The number of elements in 8 and JV, respectively.AB, AN Partition of A into columns corresponding to the index
sets 13 and A.xB, xN Subvectors of x that correspond to 13 and N, respectively.
(Similarly for sB, sN, xg, etc.)strictly complementary solution
A solution (x*, A*, s*) such that x; > 0 and sN > 0.€(A, b, c) min(miniEB suPx`ES2p Xi, miniEN sup. ,s*)EQD Si )
M, basis, B A subset of m indices from {1, 2, ... , n} such that thesubmatrix B = [A.j]j EM is nonsingular and B- lb > 0.
k (superscript or subscript)Iteration counter for a sequence, k = 0, 1, 2.....
i, j (subscript)Usual notation for indices of particular components of a
vector or matrix.
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Notation xix
(Xk, ilk , S k) or (x, \, s)Generic iterate of a primal-dual interior-point method.
(Iteration counter k often omitted for clarity.)(Ax, DA, As) Generic primal-dual step, satisfying (1.12) or (1.20),
depending on the context.(x+, A+, s+) Generic "next point"—the new iterate obtained by
taking a step from the current point (x, A, s).
F() A function of (x, A, s) consisting of the equalities from theKarush-Kuhn-Tucker conditions.
J(•) Jacobian of F.
a, ak Step length parameter.unit step A step with a = 1.
(x(a), .\(a), s(a)) (x, A, s) + a(Ax, A,\, As).
(xc(a), \k(a), sk (a)) (xk, %fi k , s 's' ) + a(Oxk , 0Ak , Ask ).
p(a) x(a)TS(a) /n.
µk(a) xI(a)Tsk(a) /n.
X, Xk n x n diagonal matrix constructed from the vector x or xk :
X = diag(xi, X2, ... xn ).
S, Sk n x n diagonal matrix constructed from the vector s or sk :
S = diag(si, S2, ... s om,).D X1/2S-1/2.
AS, AX Diagonal matrices constructed from As and Ax, respectively.
v Vector in R with components (xisi) 1/2
Vmin mini V.V diag(vi,v2,...,vn).r -v + (n/p)µV-le, for some p> n.
affine-scaling directionThe pure Newton direction obtained by setting a = 0 in (1.12
or (1.20).centering direction
The direction obtained by setting a = 1 in (1.12) or (1.20).
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xx Notation
Q Centering parameter for step computation (1.12), (1.20);restricted to the range [0, i].
Range(B) Range space of the matrix B.Null(BT ) Null space of the matrix BT.
(The fundamental theorem of algebra states thatNull(BT ) ® Range(B) = IR, where B has n rows.)
Pu Projection into the subspace U.dist(x, T) Distance from x to the set T, defined as infyET IIx - yll•vertex If T is a polyhedron in RYE, a point x E T is a vertex
if it is the unique minimizer in T of some linear function.
4)P (x, s) = p log xT s - > l log xisi for some p > nThe Tanabe-Todd-Ye potential function.
N2(B) The 2-norm central path neighborhood, defined by{(x, A, s) E .P° I II X Se - peii < Bµ} for given 0 E (0, 1).
N_. (ry)
The oo-norm central path neighborhood, defined by{(x, A, s) E .F° I xisti > yp, VZ} for given 'y E (0, 1).
PL(-y, 3) The infeasible cc-norm neighborhood, defined by
{(x,A,.․) I Ii(rb, rc) ii [II (rv° ,r') Ii/Ao]Rµ, (x, s) > 0 ,x^s >_ -r/t, Vi}
for given ry E (0, 1), /3> 1, and initial point (xo, Ao , so)
LCP Monotone linear complementarity problem.mLCP Mixed monotone linear complementarity problem.hLCP Horizontal monotone linear complementarity problem.
symmetric, skew-symmetricM is symmetric if MT = M, skew-symmetric if MT = -M.
Sn The set of n x n symmetric matrices.D+ The set of diagonal matrices with strictly positive diagonal
elements.
u Unit roundoff error (about 10 -16 for 8-byte floating-pointarithmetic).
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