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A simplified homogeneous and selfdual linear programming algorithm and its implementation
 Annals of Operations Research
, 1996
"... 1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x ..."
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Cited by 62 (5 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
Warm Start of the PrimalDual Method Applied in the CuttingPlane Scheme
 in the Cutting Plane Scheme, Mathematical Programming
, 1997
"... A practical warmstart procedure is described for the infeasible primaldual interiorpoint method employed to solve the restricted master problem within the cuttingplane method. In contrast to the theoretical developments in this field, the approach presented in this paper does not make the unreal ..."
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Cited by 32 (4 self)
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A practical warmstart procedure is described for the infeasible primaldual interiorpoint method employed to solve the restricted master problem within the cuttingplane method. In contrast to the theoretical developments in this field, the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow. Moreover, it treats systematically the case when a large number of cuts are added at one time. The technique proposed in this paper has been implemented in the context of HOPDM, the state of the art, yet public domain, interiorpoint code. Numerical results confirm a high degree of efficiency of this approach: regardless of the number of cuts added at one time (can be thousands in the largest examples) and regardless of the depth of the new cuts, reoptimizations are usually done with a few additional iterations. Key words. Warm start, primaldual algorithm, cuttingplane methods. Supported by the Fonds National de la Recherche Scientifique Su...
A Truncated PrimalInfeasible DualFeasible Network Interior Point Method
, 1994
"... . In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is ..."
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Cited by 29 (3 self)
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. In this paper we introduce the truncated primalinfeasible dualfeasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code, pdnet, is tested on a large set of standard minimum cost network flow test problems. Computational results indicate that the implementation is competitive with stateoftheart network flow codes. Key Words. Interior point method, linear programming, network flows, primalinfeasible dualfeasible, truncated Newton method, conjugate gradient, maximum flow, experimental test...
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many p ..."
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Cited by 23 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
An O(nL) infeasibleinteriorpoint algorithm for LCP with quadratic convergence
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1994
"... The MizunoToddYe predictorcorrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity ..."
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Cited by 18 (10 self)
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The MizunoToddYe predictorcorrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity depends on the quality of the starting point. If the starting points are large enough then the algorithm has O(nL) iteration complexity. If a certain measure of feasibility at the starting point is small enough then the algorithm has O( p nL) iteration complexity. At each iteration both "feasibility' and "optimality" are reduced exactly at the same rate. The algorithm is quadratically convergent for problems having a strictly complementary solution, and therefore its asymptotic efficiency index is p 2. A variant of the algorithm can be used to detect whether solutions with norm less than a given constant exist. . Key Words:linear complementarity problems, predictorcorrector, infeasib...
Combining interiorpoint and pivoting algorithms for
 Linear Programming”, Management Science
, 1996
"... ..."
A New unblocking technique to warmstart interior point methods based on sensitivity analysis
, 2007
"... One of the main drawbacks associated with Interior Point Methods (IPM) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with war ..."
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Cited by 14 (1 self)
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One of the main drawbacks associated with Interior Point Methods (IPM) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with warmstarting for IPM is that an advanced starting point which is close to the boundary of the feasible region, as is typical, might lead to blocking of the search direction. Several techniques have been proposed to address this issue. Most of these aim to lead the iterate back into the interior of the feasible region we classify them as either “modification steps” or “unblocking steps ” depending on whether the modification is taking place before solving the modified problem to prevent future problems, or during the solution if and when problems become apparent. A new “unblocking” strategy is suggested which attempts to directly address the issue of blocking by performing sensitivity analysis on the Newton step with the aim of increasing the size of the step that can be taken. This analysis is used in a new technique to warmstart
On InteriorPoint Warmstarts for Linear and Combinatorial Optimization
, 2008
"... Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we prese ..."
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Cited by 5 (0 self)
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Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we present a new infeasibleinteriorpoint approach to quickly reoptimize an initial problem instance after data perturbations, or a new linear programming relaxation after adding cutting planes for discrete or combinatorial problems. Based on the detailed complexity analysis of the underlying algorithm, we perform a comparative analysis to coldstart initialization schemes and present encouraging computational results with iteration savings around 50 % on average for perturbations of the Netlib linear programs and successive LP relaxations of maxcut and the travelingsalesman problem.
An infeasibleinteriorpoint method for the P*matrix LCP
, 1994
"... A predictorcorrector method for solving the P (k)matrix linear complementarity problems from infeasible starting points is analyzed. Two matrix factorizations and at most three backsolves are to be computed at each iteration. The computational complexity depends on the quality of the starting poi ..."
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Cited by 3 (3 self)
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A predictorcorrector method for solving the P (k)matrix linear complementarity problems from infeasible starting points is analyzed. Two matrix factorizations and at most three backsolves are to be computed at each iteration. The computational complexity depends on the quality of the starting points. If the starting points are large enough then the algorithm has O \Gamma ( + 1) 2 nL \Delta iteration complexity. If a certain measure of feasibility at the starting point is small enough then the algorithm has O (( + 1) p nL) iteration complexity. Both "feasibility' and "optimality" are reduced exactly at the same rate. The algorithm is quadratically convergent for problems having a strictly complementary solution, and therefore its asymptotic efficiency index is p 2 Key Words: Linear complementarity problems, predictorcorrector, infeasibleinteriorpoint algorithm, polynomiality, superlinear convergence. Abbreviated Title: An infeasibleinteriorpoint method for LCP. Dep...
An Algorithm for Perturbed Secondorder Cone Programs
, 2004
"... The secondorder cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Qquadratically to a soluti ..."
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Cited by 2 (2 self)
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The secondorder cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Qquadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples show that the algorithm is good for “warm starting” – for some instances, the solution of a perturbed problem is hit in two iterations.