1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x

. In this paper we introduce the truncated primal-infeasible dual-feasible 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 state-of-the-art network flow codes. Key Words. Interior point method, linear programming, network flows, primal-infeasible dualfeasible, truncated Newton method, conjugate gradient, maximum flow, experimental test...

A practical warm-start procedure is described for the infeasible primal-dual interior-point method employed to solve the restricted master problem within the cutting-plane 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, interior-point 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, primal-dual algorithm, cutting-plane methods. Supported by the Fonds National de la Recherche Scientifique Su...

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.

The Mizuno-Todd-Ye predictor-corrector 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, predictor-corrector, infeasib...

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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

A predictor-corrector 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, predictor-corrector, infeasible-interiorpoint algorithm, polynomiality, superlinear convergence. Abbreviated Title: An infeasible-interior-point method for LCP. Dep...

An approach to determine primal and dual stepsizes in the infeasible-- interior--point primal--dual method for convex quadratic problems is presented. The approach reduces the primal and dual infeasibilities in each step and allows different stepsizes. The method is derived by investigating the efficient set of a multiobjective optimization problem. Computational results are also given. Keywords: interior point methods, quadratic programming, steplength, efficient set 1 Introduction In the paper we will assume the convex quadratic problem (QP) in the form: min c T x + 1 2 x T Qx# subject to Ax = b# x 0# (1) This work was supported in part by EPSRC grant No. GR/J52655 and Hungarian ResearchFund OTKA T-016413. y H-1518 Budaspest, P.O. BOX63.Hungary 1 where A 2 R m\Thetan is of full row rank, Q 2 R n\Thetan is symmetric positive semidefinite and c# x 2 R n # b 2 R m . The dual of (1) in the Wolfe sense is defined as follows: max b T y ; 1 2 x T Qx# ...

This research studies two computational techniques that improve the practical performance of existing implementations of interior point methods for linear programming. Both are based on the concept of symmetric neighbourhood as the driving tool for the analysis of the good performance of some practical algorithms. The symmetric neighbourhood adds explicit upper bounds on the complementarity