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19
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 69 (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...
Multiple Centrality Corrections in a PrimalDual Method for Linear Programming
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 1995
"... A modification of the (infeasible) primaldual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the efforts to solve and to factoriz ..."
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Cited by 48 (11 self)
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A modification of the (infeasible) primaldual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the efforts to solve and to factorize the KKT systems. For any LP problem, this ratio is determined right after preprocessing the KKT system and prior to the optimization process. The harder the factorization, the more advantageous the higherorder corrections might prove to be. The computational performance of the method is studied on more difficult Netlib problems as well as on tougher and larger reallife LP models arising from applications. The use of multiple centrality corrections gives on the average a 25% to 40% reduction in the number of iterations compared with the widely used secondorder predictorcorrector method. This translates into 20% to 30% savings in CPU time.
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 23 (2 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 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...
On Weighted Centers For Semidefinite Programming
, 1996
"... In this paper, we generalize the notion of weighted centers to semidefinite programming. Our analysis fits in the vspace framework, which is purely based on the symmetric primaldual transformation and does not make use of barriers. Existence and scale invariance properties are proven for the weigh ..."
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Cited by 14 (4 self)
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In this paper, we generalize the notion of weighted centers to semidefinite programming. Our analysis fits in the vspace framework, which is purely based on the symmetric primaldual transformation and does not make use of barriers. Existence and scale invariance properties are proven for the weighted centers. Relations with other primaldual maps are discussed. Key words. semidefinite programming, symmetric primaldual transformation, weighted center. 1 Econometric Institute, Erasmus University Rotterdam, The Netherlands, sturm@few.eur.nl. 2 Econometric Institute, Erasmus University Rotterdam, The Netherlands, zhang@few.eur.nl Sturm and Zhang: On weighted centers for SDP 1 1. Introduction The central path plays a fundamental role in the interior point methodology, both for linear and semidefinite programming. Megiddo [10] showed some highly interesting properties of the central path for linear programming. The fact that ¯centers are the minimizers of the logarithmic barrier f...
Using an Interior Point Method for the Master Problem in a Decomposition Approach
 European Journal of Operational Research
, 1997
"... We addres some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concern the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The parti ..."
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Cited by 11 (7 self)
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We addres some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concern the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems. The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections. Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities). Key words. Convex programming, interior point methods, cutt...
On Adjusting Parameters in Homotopy Methods for Linear Programming
 In Approximation Theory and Optimization, edited by M. Buhmann and A. Iserles
, 1997
"... Several algorithms in optimization can be viewed as following a solution as a parameter or set of parameters is adjusted to a desired value. Examples include homotopy methods in complementarity problems and pathfollowing (infeasible) interiorpoint methods. If we have a metric in solution space th ..."
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Cited by 7 (1 self)
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Several algorithms in optimization can be viewed as following a solution as a parameter or set of parameters is adjusted to a desired value. Examples include homotopy methods in complementarity problems and pathfollowing (infeasible) interiorpoint methods. If we have a metric in solution space that corresponds to the complexity of moving from one solution point to another, there is an induced metric in parameter space, which can be used to guide parameteradjustment schemes. We investigate this viewpoint for feasible and infeasible interiorpoint methods for linear programming. Key words: homotopy methods, pathfollowing methods, interiorpoint algorithms, linear programming, Riemannian metrics Running Header: Adjusting Parameters in Homotopy Methods 1 Introduction This paper is concerned with developing guidelines for optimal adjustment of parameters in homotopy or pathfollowing algorithms in optimization, concentrating on interiorpoint methods for linear programming. The gen...
Target Directions for PrimalDual InteriorPoint Methods for SelfScaled Conic Programming (Extended Abstract)
 1999/NA15, Department of Applied Mathematics and Theoretical Physics, Silver
, 1999
"... ) Raphael Hauser Primaldual interior point methods for convex optimization problems are designed to solve a problem and its dual jointly by making use of convex duality theory. In fact, such algorithms usually aim at reducing the duality gap between primal and dual approximate solutions to zero. Pr ..."
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Cited by 4 (4 self)
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) Raphael Hauser Primaldual interior point methods for convex optimization problems are designed to solve a problem and its dual jointly by making use of convex duality theory. In fact, such algorithms usually aim at reducing the duality gap between primal and dual approximate solutions to zero. Primaldual methods are most conveniently studied in a framework that exhibits the symmetry between the two optimization problems jointly solved by the algorithm. In the linear programming literature such a framework, known as the V space approach, allows for a unified runningtime analysis of several important classes of primaldual interiorpoint algorithms. The unified class of algorithms is known as targetfollowing algorithms (see e.g. [4, 2, 3]). The present article is the second in a series of papers that aim at extending this unified theory to selfscaled conic programming, a class of convex optimization problems which contains linear, semidefinite and convex quadratic programming w...
PrimalDual Symmetric ScaleInvariant SquareRoot Fields for Isotropic SelfScaled Barrier Functionals
, 1999
"... Squareroot fields are differentiable operator fields used in the construction of target direction fields for selfscaled conic programming, a unifying framework for primaldual interiorpoint methods for linear programming, semidefinite programming and secondorder cone programming. In this article ..."
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Cited by 3 (3 self)
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Squareroot fields are differentiable operator fields used in the construction of target direction fields for selfscaled conic programming, a unifying framework for primaldual interiorpoint methods for linear programming, semidefinite programming and secondorder cone programming. In this article we investigate squareroot fields for socalled isotropic selfscaled barrier functionals, i.e. selfscaled barrier functionals that are invariant under rotations of their conic domain of definition. We prove a structure theorem for socalled congruent squareroot fields for isotropic selfscaled barrier functionals in terms of the irreducible decomposition of their domain of definition. Using this structure theorem, we then investigate primaldual symmetry and scaleinvariance of such squareroot fields. In our main theorem we show that these two assumptions together with one additional natural invariance property (socalled canonical reduction) dramatically reduce the degree of freedom in...
NesterovTodd Directions are Newton Directions
, 1999
"... The theory of selfscaled conic programming provides a unified framework for the theories of linear programming, semidefinite programming and convex quadratic programming with convex quadratic constraints. The standard search directions for interiorpoint methods applied to selfscaled conic programm ..."
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Cited by 2 (2 self)
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The theory of selfscaled conic programming provides a unified framework for the theories of linear programming, semidefinite programming and convex quadratic programming with convex quadratic constraints. The standard search directions for interiorpoint methods applied to selfscaled conic programming problems are the socalled NesterovTodd directions. In this article we show that these direction fields are special cases of socalled target directions, a unifying concept for primaldual interior point methods for selfscaled conic programming. In particular, this implies that NesterovTodd directions derive from a Newton system.