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A PathFollowing InfeasibleInteriorPoint Algorithm for Linear Complementarity Problems
 Optimization Methods and Software
, 1993
"... We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only tha ..."
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Cited by 50 (10 self)
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We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only that a strictly complementary solution exists. 1 Introduction The monotone linear complementarity problem is to find a vector pair (x; y) 2 IR n \Theta IR n such that y = Mx+ h; (x; y) (0; 0); x T y = 0; (1) where h 2 IR n and M is an n \Theta n positive semidefinite matrix. A vector pair (x ; y ) is called a strictly complementary solution of (1) if it satisfies the three conditions in (1) and, in addition, x i + y i ? 0 for each component i = 1; 2; \Delta \Delta \Delta ; n. We denote the solution set for (1) by S and the set of strictly complementary solutions by S c . A number of interior point methods have been proposed for (1). Among recent papers are the predictor...
Local Convergence of InteriorPoint Algorithms for Degenerate Monotone LCP
 Computational Optimization and Applications
, 1993
"... Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is remo ..."
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Cited by 34 (4 self)
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Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed. 1 Introduction In the monotone linear complementarity problem (LCP), we seek a vector pair (x; y) 2 IR n \Theta IR n that satisfies the conditions y = Mx+ q; x 0; y 0; x T y = 0; (1) where q 2 IR n , and M 2 IR n\Thetan is positive semidefinite. We use S to denote the solution set of (1). An assumption that is frequently made in order to prove superlinear convergence of interiorpoint algorithms for (1) is the nondegeneracy assumption: Assumption 1 There is an (x ; y ) 2 S such that x i + y i ? 0 for all i = 1; \Delta \Delta \Delta ; n. In general, we can define three subsets B, N , and J of the index set f1; \Delta \Delta \Delta ; ng by B = fi = 1; \Delta ...
A Computational View of InteriorPoint Methods for Linear Programming
 IN: ADVANCES IN LINEAR AND INTEGER PROGRAMMING
, 1994
"... Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing te ..."
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Cited by 15 (10 self)
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Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical difficulties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of different methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of i...
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...
Linear Algebra for Semidefinite Programming
, 1995
"... Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *subalgebra of M n (IK) over the field IR (or simply a ..."
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Cited by 8 (4 self)
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Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *subalgebra of M n (IK) over the field IR (or simply a *subalgebra) if (i) T forms a subring of M n (IK) with the usual addition A + B and multiplication AB of matrices A; B 2 M n (IK); specifically the zero matrix O and the identity matrix I belong to T . (ii) T is an IRmodule, i.e., a vector space over the field IR; ffA + fiB 2 T for every ff; fi 2 IR and A; B 2 T , (iii) A 3 2 T if A 2 T , where A 3 denotes the conjugate transpose of A 2 M n (IK). The introduction of *subalgebras T provides us with a unified and compact way of handling LPs (linear programs) in IR n , SDPs (semidefinite programs) in M n (IR), M n ( 0 C) and M n (IH), and monotone SDLCPs (semidefinite linear complementarity problems) in those spaces. We can extend t...
Convergence of Infeasible InteriorPoint Algorithms from Arbitrary Starting Points
 SIAM Journal on Optimization
, 1993
"... An important advantage of infeasible interiorpoint methods compared to feasible interiorpoint methods is their ability to be warmstarted from approximate solutions. It is therefore important for the convergence theory of these algorithms not to depend on being able to alter the starting point. In ..."
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Cited by 7 (3 self)
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An important advantage of infeasible interiorpoint methods compared to feasible interiorpoint methods is their ability to be warmstarted from approximate solutions. It is therefore important for the convergence theory of these algorithms not to depend on being able to alter the starting point. In two recent papers, Yin Zhang and Stephen Wright prove convergence results for some infeasible interiorpoint methods. Unfortunately, their analysis places a restriction on the starting point. It is easy to meet the restriction by altering the starting point, but this may take the point farther away from the solution, removing the advantage of warmstarting the algorithms. In this paper we extend Zhang and Wright's results to apply to arbitrary strictly positive starting points. We then present an algorithm for solving the BoxConstrained Linear Complementarity problem and prove its convergence. 1 Introduction Quite often, in using an iterative method to solve a problem, it is possible to u...
A Short Survey on Ten Years Interior Point Methods
, 1995
"... The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to c ..."
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Cited by 3 (0 self)
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The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and outline the major contributions to the field in the last decade. An important class of problems to which interior point methods are applicable is semidefinite optimization, which has recently gained much attention. It has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and can be efficiently solved with interior point methods.
InteriorPoint Methodology for Linear Programming: Duality, Sensitivity Analysis and Computational Aspects
 IN OPTIMIZATION IN PLANNING AND OPERATION OF ELECTRIC POWER SYSTEMS
, 1993
"... In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter. ..."
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Cited by 3 (1 self)
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In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter.
An Affine Scaling Method with an Infeasible Starting Point: Convergence Analysis under Nondegeneracy Assumption
, 1993
"... In this paper, we propose an infeasibleinteriorpoint algorithm for linear programming based on the affine scaling algorithm by Dikin. The search direction of the algorithm is composed of two directions, one for satisfying feasibility and the other for aiming at optimality. Both directions are affi ..."
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Cited by 1 (0 self)
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In this paper, we propose an infeasibleinteriorpoint algorithm for linear programming based on the affine scaling algorithm by Dikin. The search direction of the algorithm is composed of two directions, one for satisfying feasibility and the other for aiming at optimality. Both directions are affine scaling directions of certain linear programming problems. Global convergence of the algorithm is proved under a reasonable nondegeneracy assumption. A summary of analogous global convergence results for general cases obtained in [17] by means of the local potential function is also given. Key Words: Linear Programming, InfeasibleInteriorPoint Methods, Affine Scaling Algorithm, Global Convergence Analysis, Nondegeneracy Assumption Address: y71, Kioicho, Chiyodaku, Tokyo 102 Japan, z467 MinamiAzabu, Minatoku, Tokyo 106 Japan Email address: ymuramatu@hoffman.cc.sophia.ac.jp, ztsuchiya@sun312.ism.ac.jp 1 Introduction In this paper, we propose an infeasibleinteriorpoint algorith...