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17
Global Methods For Nonlinear Complementarity Problems
 MATH. OPER. RES
, 1994
"... Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinea ..."
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Cited by 28 (1 self)
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Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinear least squares problem. Algorithms based on this formulation are applicable to general nonlinear complementarity problems, can be started from any nonnegative starting point, and each iteration only requires the solution of systems of linear equations. Convergence to a solution of the nonlinear complementarity problem is guaranteed under reasonable regularity assumptions. The converge rate is Qlinear, Qsuperlinear, or Qquadratic, depending on the tolerances used to solve the subproblems.
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and compleme ..."
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Cited by 24 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
A General Framework of Continuation Methods for Complementarity Problems
 MATH. OF OPER. RES
, 1994
"... A new class of continuation methods is presented which, in particular, solve linear complementarity problems with copositiveplus and L matrices. Let a# b 2 R be nonnegativevectors. Weembed the complementarity problem with a continuously differentiable mapping f : R in an artificial system o ..."
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Cited by 14 (2 self)
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A new class of continuation methods is presented which, in particular, solve linear complementarity problems with copositiveplus and L matrices. Let a# b 2 R be nonnegativevectors. Weembed the complementarity problem with a continuously differentiable mapping f : R in an artificial system of F (x# y)=(a#ib) and (x# y) 0 # () where F : R is defined by F (x# y)=(x 1 y 1 # ...#x n y n # y ; f(x)) and 0 and i 0 are parameters. A pair (x# y) is a solution of the complementarity problem if and only if it solves ()for = 0 and i = 0. A general idea of continuation methods founded on the system () is as follows.
Degeneracy in Interior Point Methods for Linear Programming
, 1991
"... ... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence ..."
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Cited by 11 (1 self)
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... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence of IPM's, on the trajectories followed by the algorithms, the effect of degeneracy in numerical performance, and on finding basic solutions.
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
Monotone Semidefinite Complementarity Problems
, 1996
"... . In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditi ..."
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Cited by 7 (1 self)
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. In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditions. Especially, for the monotone semidefinite linear complementarity problem, the trajectory converges to an analytic center of the solution set, provided that there exists a strictly complementary solution. Finally, we propose the globally convergent infeasibleinteriorpoint algorithm for the SDCP. Key words Monotone Semidefinite Complementarity Problem, Trajectory, Interior Point Algorithm Research Report B312 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. Let M(n) and S(n) denote the class of n2n real matrices and the class of n2n symmetric real matrices, respectively. Assume that A; B 2 M(n)....
Centers of Monotone Generalized Complementarity Problems
 Math. Oper. Res
, 1996
"... . Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity prob ..."
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Cited by 7 (4 self)
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. Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with C and M: Find (x; y) 2 M " (C 2C 3 ) such that hx; yi = 0. Here C 3 = fy 2 E : hx; yi 0 for all x 2 Cg denotes the dual cone of C. The main result of the paper unifies and extends some results established for monotone complementarity problems in Euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices. Key words Central Trajectory, Path of Centers, Complementarity Problem, Interior Point Algorithm, Linear Program Research Report B303 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. The central trajectory or the path of centers is known ...
A Potential Reduction Newton Method for Constrained Equations
 SIAM Journal on Optimization
, 1997
"... Extending our previous work [11], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A main convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complement ..."
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Cited by 7 (4 self)
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Extending our previous work [11], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A main convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strong convergence results are established in these specializations. 1 Introduction In the paper [11], we have introduced the problem of solving a system of nonlinear equations subject to additional constraints on the variables, i.e., a constrained system of equations. We have demonstrated that constrained equations (CEs) provide a unifying framework for the study of complementarity problems of various types, including the standard nonlinear complementarity problem and the KarushKuhnTucker system of a variational inequality. Postulating a partitioning property of the CE, we have introduced an interior point potential reduction algorithm f...
On A New Homotopy Continuation Trajectory For Nonlinear Complementarity Problems
 Mathematics of Operations Research
, 2001
"... . Most known continuation methods for P 0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and a properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we prop ..."
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Cited by 4 (1 self)
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. Most known continuation methods for P 0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and a properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we propose in this paper a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated. For P 0 complementarity problems, the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods. Particularly, the oftenassumed strictly feasible condition is not required here. When applied to P complementarity problems, the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem, and the entire trajectory converges to the (unique)...
Characterizations of A Homotopy Solution Mapping for Nonlinear Complementarity Problems
 Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences
, 1998
"... . We study several properties of a homotopy solution mapping, based on Zhang and Zhang's homotopy formulation, for continuous nonlinear complementarity problems with such nonmonotone maps as quasimonotone, E 0 , and exceptional regular functions. For these classes of complementarity problems, we e ..."
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Cited by 3 (3 self)
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. We study several properties of a homotopy solution mapping, based on Zhang and Zhang's homotopy formulation, for continuous nonlinear complementarity problems with such nonmonotone maps as quasimonotone, E 0 , and exceptional regular functions. For these classes of complementarity problems, we establish several sufficient conditions to assure the nonemptyness and boundedness of the homotopy solution mapping. Under the P 0 property, all these sufficient conditions also guarantee the uniqueness and continuity of the homotopy solution mapping, and hence the range of this mapping forms a continuous path passing through an arbitrary point in R n ++ \Theta R n ++ to a solution of complementarity problem. Our analysis is very different from Zhang and Zhang's. We use homotopy invariance theorem of degree to develop a general sufficient condition for the nonemptyness of the homotopy solution mapping instead of using the parameterized Sard's theorem. As a result of this, we only require...