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X.: Implementation of interior point methods for large scale linear programming
 Interior Point Methods in Mathematical Programming. Kluwer Acad Pub
, 1996
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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 16 (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...
Mixed Integer Optimization in the Chemical Process Industry
 Experience, Potential and Future. Trans. I. Chem. E., 78, Part A
, 2000
"... roper organization, planning and design of production, storage locations, transportation and scheduling are vital to retain the competitive edge of companies in the global economy.Typical additional problems in the chemical industry suitable for optimization are process design, process synthesis and ..."
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Cited by 11 (1 self)
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roper organization, planning and design of production, storage locations, transportation and scheduling are vital to retain the competitive edge of companies in the global economy.Typical additional problems in the chemical industry suitable for optimization are process design, process synthesis and multicomponent blended¯ow problems leading to nonlinear or even mixed integer nonlinear models. Mixed integer optimization (MIP) determines optimal solutions of such complex problems; the development of new algorithms, software and hardware allow the solution of larger problems in acceptable times. This tutorial paper addresses two groups. The focus towards the ®rst group (managers and senior executives) and readers who are not very familiar with mathematical programming is to create some awareness regarding the potential bene®ts of MIP, to transmit a sense of what kind of problems can be tackled, and to increase the acceptance of MIP. The second group (a more technical audience with some background in mathematical optimization), might rather appreciate the stateoftheart view on goodmodelling practice, algorithms and an outlook into global optimization. Some realworld MIP problems solved by BASF’s mathematical consultants are brie¯y described, among them discrete blending and multistage production planning and distribution with several sites, products and periods. Finally, there is a focus on future perspectives and sources of MIP support are indicated from academia, software providers and consulting ®rms.
A Strongly Polynomial Rounding Procedure Yielding a Maximally Complementary Solution for P*(κ) Linear Complementarity Problems
, 1998
"... We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the var ..."
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Cited by 7 (3 self)
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We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the variables on, or close to the central path fall apart in three classes in which these variables are O(1); O() and O( p ), respectively. The constants hidden in these bounds are expressed in, or bounded by, the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and close enough to the central path, the rounding procedure produces a maximally complementary solution in at most O(n³) arithmetic operations. The result implies that Interior Point Methods (IPMs) not only converge to a complementary solution of P () LCPs but, when furnished with our rounding procedure, they can produce a max...
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...
Basis and Tripartition Identification for Quadratic Programming and Linear Complementarity Problems  From an interior solution to an optimal basis and viceversa
, 1996
"... Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexb ..."
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Cited by 3 (1 self)
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Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal complementary solutions. Maximal complementary solutions can be characterized by optimal (tri)partitions. On the other hand, the solutions provided by simplexbased pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A tripartition identification algorithm is an algorithm which generates a maximal complementary solution (and its corresponding tripartition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal...
Effective Finite Termination Procedures in InteriorPoint Methods for Linear Programming

, 1998
"... Due to the structure of the solution set an exact solution to... bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of ..."
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Cited by 3 (1 self)
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Due to the structure of the solution set an exact solution to... bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of projection attempts needed to find an exact solution. When early termination is the goal, projection from a pure composite Newton step is advocated. However, the cost may exceed the benefits due to the average need of more than one projection attempt to find an exact solution. Variants of Mehrotra's predictorcorrector primaldual interiorpoint algorithm provide the foundation for most practical interiorpoint codes. To take advantage of all available algorithmic information, we investigate the behavior of the Tapia predictorcorrector indicator, which incorporates the corrector step, to identify the optimal partition. Globally, the Tapia predictorcorrector indicator behaves poorly as do all indicators, but locally exhibits fast convergence.
A Stable PrimalDual Approach for Linear Programming
"... This paper studies a primaldual interior/exteriorpoint pathfollowing approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primaldual optimality equations Fu(x, y, z) = 0. Under nonde ..."
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This paper studies a primaldual interior/exteriorpoint pathfollowing approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primaldual optimality equations Fu(x, y, z) = 0. Under nondegeneracy assumptions, this nonlinear system is wellposed,i.e. it has a nonsingular Jacobian at optimality and is not necessarily illconditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate boththe primal and dual feasibility equations. This results in a single bilinear equation that maintains the wellposedness property. We then apply both a direct solution techniqueas well as a preconditioned conjugate gradient method (PCG), within an inexact Newton framework, directly on the linearized equations. This is done without forming the usualnormal equations, NEQ, or augmented system. Sparsity is maintained. The work of aniteration for the PCG approach consists almost entirely in the (approximate) solution of this wellposed linearized system. Therefore, improvements depend on efficient preconditioning.
EXPERIMENTS WITH A HYBRID INTERIOR POINT/COMBINATORIAL APPROACH FOR NETWORK FLOW PROBLEMS
"... Interior Point (IP) algorithms for Min Cost Flow (MCF) problems have been shown to be competitive with combinatorial approaches, at least on some problem classes and for very large instances. This is in part due to availability of specialized crossover routines for MCF; these allow early termination ..."
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Cited by 2 (2 self)
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Interior Point (IP) algorithms for Min Cost Flow (MCF) problems have been shown to be competitive with combinatorial approaches, at least on some problem classes and for very large instances. This is in part due to availability of specialized crossover routines for MCF; these allow early termination of the IP approach, sparing it with the final iterations where the KKT systems become more difficult to solve. As the crossover procedures are nothing but combinatorial approaches to MCF that are only allowed to perform few iterations, the IP algorithm can be seen as a complex “multiple crash start ” routine for the combinatorial ones. We report our experiments about allowing one primaldual combinatorial algorithm to MCF to perform as many iterations as required to solve the problem after being warmstarted by an IP approach. Our results show that the efficiency of the combined approach critically depends on the accurate selection of a set of parameters among very many possible ones, for which designing accurate guidelines appears not to be an easy task; however, they also show that the combined approach can be competitive with the original combinatorial algorithm, at least on some “difficult” instances.
Numerical Stability in Linear Programming and Semidefinite Programming
, 2006
"... We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms. ..."
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Cited by 1 (1 self)
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We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.