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27
Implementation of Interior Point Methods for Large Scale Linear Programming
 in Interior Point Methods in Mathematical Programming
, 1996
"... In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on bot ..."
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Cited by 70 (22 self)
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In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasibleprimaldual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasibleprimaldual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...
Presolve Analysis of Linear Programs Prior to Applying an Interior Point Method
 INFORMS Journal on Computing
, 1994
"... Several issues concerning an analysis of large and sparse linear programming problems prior to solving them with an interior point based optimizer are addressed in this paper. Three types of presolve procedures are distinguished. Routines from the first class repeatedly analyze an LP problem formula ..."
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Cited by 34 (6 self)
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Several issues concerning an analysis of large and sparse linear programming problems prior to solving them with an interior point based optimizer are addressed in this paper. Three types of presolve procedures are distinguished. Routines from the first class repeatedly analyze an LP problem formulation: eliminate empty or singleton rows and columns, look for primal and dual forcing or dominated constraints, tighten bounds for variables and shadow prices or just the opposite, relax them to find implied free variables. The second type of analysis aims at reducing a fillin of the Cholesky factor of the normal equations matrix used to compute orthogonal projections and includes a heuristic for increasing the sparsity of the LP constraint matrix and a technique of splitting dense columns in it. Finally, routines from the third class detect, and remove, different linear dependecies of rows and columns in a constraint matrix. Computational results on problems from the Netlib collection, inc...
Convergence of a Class of Inexact InteriorPoint Algorithms for Linear Programs
 Mathematics of Operations Research
, 1996
"... . We present a convergence analysis for a class of inexact infeasibleinteriorpoint methods for solving linear programs. The main feature of inexact methods is that the linear systems defining the search direction at each interiorpoint iteration need not be solved to high accuracy. More precisely, ..."
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Cited by 20 (1 self)
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. We present a convergence analysis for a class of inexact infeasibleinteriorpoint methods for solving linear programs. The main feature of inexact methods is that the linear systems defining the search direction at each interiorpoint iteration need not be solved to high accuracy. More precisely, we allow that these linear systems are only solved to a moderate relative accuracy in the residual , but no assumptions are made on the accuracy of the search direction in the search space. In particular, our analysis does not require that feasibility is maintained even if the initial iterate happened to be a feasible solution of the linear program. AMS 1991 subject classification. Primary: 90C05, Secondary: 65K05, 90C06. Key words. Linear program, infeasibleinteriorpoint method, inexact search direction, linear system, residual, convergence. 1. Introduction Since the publication [6] of Karmarkar's original interiorpoint algorithm for linear programs, numerous variants of the method ...
Row modifications of a sparse Cholesky factorization
 SIAM J. Matrix Anal. Appl
, 2005
"... Abstract. Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDLT, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the modification in the Cholesky factorization associa ..."
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Cited by 15 (4 self)
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Abstract. Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDLT, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the modification in the Cholesky factorization associated with this rank2 modification of C can be computed efficiently using a sparse rank1 technique developed in an earlier paper [SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606627]. We also determine how the solution of a linear system Lx = b changes after changing a row and column of C or after a rankr change in C.
A Computational Study of the Homogeneous Algorithm for LargeScale Convex Optimization
, 1997
"... Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of th ..."
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Cited by 13 (1 self)
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Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for largescale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark. Email: eda@busieco.ou.dk y ...
B.: Towards robust low cost authentication for pervasive devices
 In: IEEE International Conference on Pervasive Computing and Communications (PERCOM’08
, 2008
"... Low cost devices such as RFIDs, sensor network nodes, and smartcards are crucial for building the next generation pervasive and ubiquitous networks. The inherent power and footprint limitations of such networks, prevent us from employing standard cryptographic techniques for authentication which wer ..."
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Cited by 9 (2 self)
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Low cost devices such as RFIDs, sensor network nodes, and smartcards are crucial for building the next generation pervasive and ubiquitous networks. The inherent power and footprint limitations of such networks, prevent us from employing standard cryptographic techniques for authentication which were originally designed to secure high end systems with abundant power. Furthermore, the sharp increase in the number, diversity and strength of physical attacks which directly target the implementation may have devastating consequences in a network setting creating a single point of failure. A compromised node may leak a master key, or may give the attacker an opportunity for injecting faulty messages. In this paper we present a lightweight challenge response authentication scheme based on noisy physical unclonable functions (PUF) that allows for extremely efficient implementations. Furthermore, the inherent properties of PUFs provide cryptographically strong tamper resilience. In a network setting this means that a tampered device will no longer authenticate and in a sense will be isolated from the network. 1
Algorithms and Environments for Complementarity
, 2000
"... Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The ..."
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Cited by 6 (0 self)
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Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The main focus of this thesis is algorithms and environments for solving complementarity problems. Environments, such as AMPL and GAMS, are used by practitioners to easily write large, complex models. Support for these packages is provided by PATH 4.x and SEMI through the customizable solver interface specified in this thesis. The main design feature is the abstraction of core components from the code with implementations tailored to a particular environment supplied either at compile or run time. This solver interface is then used to develop new links to the MATLAB and NEOS tools. Preprocessing techniques are an integral part of linear and mixed integer programming codes and are primarily used to reduce the size and complexity of a model prior to solving it. For example, wasted computation is avoided when an infeasible model is detected.
On Exploiting Problem Structure in a Basis Identification Procedure for Linear Programming
 In: INFORMS Journal on Computing
, 1997
"... During the last decade interiorpoint methods have become an efficient alternative to the simplex algorithm for solution of largescale linear programming (LP) problems. However, in many practical applications of LP, interiorpoint methods have the drawback that they do not generate an optimal basic ..."
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Cited by 5 (0 self)
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During the last decade interiorpoint methods have become an efficient alternative to the simplex algorithm for solution of largescale linear programming (LP) problems. However, in many practical applications of LP, interiorpoint methods have the drawback that they do not generate an optimal basic and nonbasic partition of the variables. This partition is required in the traditional sensitivity analysis and is highly useful when a sequence of related LP problems are solved. Therefore, in this paper we discuss how an optimal basic solution can be generated from the interiorpoint solution. The emphasis of the paper is on how problem structure can be exploited to reduce the computational cost associated with the basis identification. Computational results are presented which indicate that it is highly advantageous to exploit problem structure. Key words: Linear programming, interiorpoint methods, basis identification. 1 Introduction Since the late forties the simplex algorithm has be...
Feasibilitybased bounds tightening via fixed points
"... Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, whi ..."
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Cited by 3 (1 self)
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Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole BranchandBound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial BranchandBound, range reduction, constraint programming. 1