Downloads & Free Reading Options - Results

We consider the problem of finding an n-agent joint-policy for the optimal finite-horizon control of a decentralized Pomdp (Dec-Pomdp). This is a problem of very high complexity (NEXP-hard in n >= 2). In this paper, we propose a new mathematical programming approach for the problem. Our approach is based on two ideas: First, we represent each agent's policy in the sequence-form and not in the tree-form, thereby obtaining a very compact representation of the set of joint-policies. Second, using this compact representation, we solve this problem as an instance of combinatorial optimization for which we formulate a mixed integer linear program (MILP). The optimal solution of the MILP directly yields an optimal joint-policy for the Dec-Pomdp. Computational experience shows that formulating and solving the MILP requires significantly less time to solve benchmark Dec-Pomdp problems than existing algorithms. For example, the multi-agent tiger problem for horizon 4 is solved in 72 secs with the MILP whereas existing algorithms require several hours to solve it.

We propose a novel approach to automatically tracking cell populations in time-lapse images. To account for cell occlusions and overlaps, we introduce a robust method that generates an over-complete set of competing detection hypotheses. We then perform detection and tracking simultaneously on these hypotheses by solving to optimality an integer program with only one type of flow variables. This eliminates the need for heuristics to handle missed detections due to occlusions and complex morphology. We demonstrate the effectiveness of our approach on a range of challenging sequences consisting of clumped cells and show that it outperforms state-of-the-art techniques.

We introduce a general technique to create an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by creating a new binary variable for each generated value. Initial experiments show that the extended formulation can have a more compact complete description than the original formulation. We prove that, using this reformulation technique, the facet description decomposes into one ``linking polyhedron'' per block and the ``aggregated polyhedron''. Each of these polyhedra can be analyzed separately. For the case of identical coefficients in a block, we provide a complete description of the linking polyhedron and a polynomial-time separation algorithm. Applied to the knapsack with a fixed number of distinct coefficients, this theorem provides a complete description in an extended space with a polynomial number of variables.

This article is from PLoS ONE , volume 9 . Abstract In this paper, we consider the Minimum Reaction Insertion (MRI) problem for finding the minimum number of additional reactions from a reference metabolic network to a host metabolic network so that a target compound becomes producible in the revised host metabolic network in a Boolean model. Although a similar problem for larger networks is solvable in a flux balance analysis (FBA)-based model, the solution of the FBA-based model tends to include more reactions than that of the Boolean model. However, solving MRI using the Boolean model is computationally more expensive than using the FBA-based model since the Boolean model needs more integer variables. Therefore, in this study, to solve MRI for larger networks in the Boolean model, we have developed an efficient Integer Programming formalization method in which the number of integer variables is reduced by the notion of feedback vertex set and minimal valid assignment. As a result of computer experiments conducted using the data of metabolic networks of E. coli and reference networks downloaded from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, we have found that the developed method can appropriately solve MRI in the Boolean model and is applicable to large scale-networks for which an exhaustive search does not work. We have also compared the developed method with the existing connectivity-based methods and FBA-based methods, and show the difference between the solutions of our method and the existing methods. A theoretical analysis of MRI is also conducted, and the NP-completeness of MRI is proved in the Boolean model. Our developed software is available at “http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/minRect/minRect.html.”

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones.

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

Load disaggregation based on aided linear integer programming (ALIP) is proposed. We start with a conventional linear integer programming (IP) based disaggregation and enhance it in several ways. The enhancements include additional constraints, correction based on a state diagram, median filtering, and linear programming-based refinement. With the aid of these enhancements, the performance of IP-based disaggregation is significantly improved. The proposed ALIP system relies only on the instantaneous load samples instead of waveform signatures, and hence does not crucially depend on high sampling frequency. Experimental results show that the proposed ALIP system performs better than the conventional IP-based load disaggregation system.

In the Integer Quadratic Programming problem input is an n*n integer matrix Q, an m*n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x in Z^n, minimizing x^TQx, subject to Ax

This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of \"Ozlen and Azizo\u{g}lu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.

We transform join ordering into a mixed integer linear program (MILP). This allows to address query optimization by mature MILP solver implementations that have evolved over decades and steadily improved their performance. They offer features such as anytime optimization and parallel search that are highly relevant for query optimization. We present a MILP formulation for searching left-deep query plans. We use sets of binary variables to represent join operands and intermediate results, operator implementation choices or the presence of interesting orders. Linear constraints restrict value assignments to the ones representing valid query plans. We approximate the cost of scan and join operations via linear functions, allowing to increase approximation precision up to arbitrary degrees. Our experimental results are encouraging: we are able to find optimal plans for joins between 60 tables; a query size that is beyond the capabilities of prior exhaustive query optimization methods.

In this paper, an integer programming approach is introduced to construct Unequal Error Protection (UEP) codes for multiuser broadcast channels. We show that the optimal codes can be constructed that satisfy the integer programming bound. Based on the bound, we compute asymptotic code rate and perform throughput analysis for the degraded broadcast channel.

Answering science questions posed in natural language is an important AI challenge. Answering such questions often requires non-trivial inference and knowledge that goes beyond factoid retrieval. Yet, most systems for this task are based on relatively shallow Information Retrieval (IR) and statistical correlation techniques operating on large unstructured corpora. We propose a structured inference system for this task, formulated as an Integer Linear Program (ILP), that answers natural language questions using a semi-structured knowledge base derived from text, including questions requiring multi-step inference and a combination of multiple facts. On a dataset of real, unseen science questions, our system significantly outperforms (+14%) the best previous attempt at structured reasoning for this task, which used Markov Logic Networks (MLNs). It also improves upon a previous ILP formulation by 17.7%. When combined with unstructured inference methods, the ILP system significantly boosts overall performance (+10%). Finally, we show our approach is substantially more robust to a simple answer perturbation compared to statistical correlation methods.

Answering science questions posed in natural language is an important AI challenge. Answering such questions often requires non-trivial inference and knowledge that goes beyond factoid retrieval. Yet, most systems for this task are based on relatively shallow Information Retrieval (IR) and statistical correlation techniques operating on large unstructured corpora. We propose a structured inference system for this task, formulated as an Integer Linear Program (ILP), that answers natural language questions using a semi-structured knowledge base derived from text, including questions requiring multi-step inference and a combination of multiple facts. On a dataset of real, unseen science questions, our system significantly outperforms (+14%) the best previous attempt at structured reasoning for this task, which used Markov Logic Networks (MLNs). It also improves upon a previous ILP formulation by 17.7%. When combined with unstructured inference methods, the ILP system significantly boosts overall performance (+10%). Finally, we show our approach is substantially more robust to a simple answer perturbation compared to statistical correlation methods.

Considering the intermittency of renewable energy systems, a sizing and scheduling model is proposed for a finite number of static electric loads. The model objective is to maximize solar energy utilization with and without storage. For the application of optimal load size selection, the energy production of a solar photovoltaic is assumed to be consumed by a finite number of discrete loads in an off-grid system using mixed-integer linear programming. Additional constraints are battery charge and discharge limitations and minimum uptime and downtime for each unit. For a certain solar power profile the model outputs optimal unit size as well as the optimal scheduling for both units and battery charge and discharge (if applicable). The impact of different solar power profiles and minimum up and down time constraints on the optimal unit and battery sizes are studied. The battery size required to achieve full solar energy utilization decreases with the number of units and with increased flexibility of the units (shorter on and off-time). A novel formulation is introduced to model quasi-dynamic units that gradually start and stop and the quasi-dynamic units increase solar energy utilization. The model can also be applied to search for the optimal number of units for a given cost function.

Considering the intermittency of renewable energy systems, a sizing and scheduling model is proposed for a finite number of static electric loads. The model objective is to maximize solar energy utilization with and without storage. For the application of optimal load size selection, the energy production of a solar photovoltaic is assumed to be consumed by a finite number of discrete loads in an off-grid system using mixed-integer linear programming. Additional constraints are battery charge and discharge limitations and minimum uptime and downtime for each unit. For a certain solar power profile the model outputs optimal unit size as well as the optimal scheduling for both units and battery charge and discharge (if applicable). The impact of different solar power profiles and minimum up and down time constraints on the optimal unit and battery sizes are studied. The battery size required to achieve full solar energy utilization decreases with the number of units and with increased flexibility of the units (shorter on and off-time). A novel formulation is introduced to model quasi-dynamic units that gradually start and stop and the quasi-dynamic units increase solar energy utilization. The model can also be applied to search for the optimal number of units for a given cost function.

"NPS-55-82-009"--Cover

"NPS-55-82-009"--Cover

Journal of Research of the National Bureau of Standards

Bibliography: leaves 40-42

Bibliography: pt. 1, p. 46-47; pt. 2, p. 48

Bibliography: pt. 1, p. 46-47; pt. 2, p. 48

The Integer Programming Problem (IP) for a polytope P \subseteq R^n is to find an integer point in P or decide that P is integer free. We give an algorithm for an approximate version of this problem, which correctly decides whether P contains an integer point or whether a (1+\eps) scaling of P around its barycenter is integer free in time O(1/\eps^2)^n. We reduce this approximate IP question to an approximate Closest Vector Problem (CVP) in a "near-symmetric" semi-norm, which we solve via a sieving technique first developed by Ajtai, Kumar, and Sivakumar (STOC 2001). Our main technical contribution is an extension of the AKS sieving technique which works for any near-symmetric semi-norm. Our results also extend to general convex bodies and lattices.

In many common scenarios, programmers need to implement functionality that is already provided by some third party library. This paper presents a tool called Hunter that facilitates code reuse by finding relevant methods in large code bases and automatically synthesizing any necessary wrapper code. The key technical idea underlying our approach is to use types to both improve search results and guide synthesis. Specifically, our method computes similarity metrics between types and uses this information to solve an integer linear programming (ILP) problem in which the objective is to minimize the cost of synthesis. We have implemented Hunter as an Eclipse plug-in and evaluate it by (a) comparing it against S6, a state-of-the-art code reuse tool, and (b) performing a user study. Our evaluation shows that Hunter compares favorably with S6 and significantly increases programmer productivity.

The Lovasz Local Lemma due to Erdos and Lovasz is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson to derive good approximation algorithms for such problems. We use our extension of the Local Lemma to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension/degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan, to obtain constructive (algorithmic) versions of our results for covering integer programs.

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path- width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.

Voting problems are central in the area of social choice. In this article, we investigate various voting systems and types of control of elections. We present integer linear programming (ILP) formulations for a wide range of NP-hard control problems. Our ILP formulations are flexible in the sense that they can work with an arbitrary number of candidates and voters. Using the off-the-shelf solver Cplex, we show that our approaches can manipulate elections with a large number of voters and candidates efficiently.

We give a survey of work on the number of vertices of the convex hull of integer points defined by the system of linear inequalities. Also, we present our improvement of some of these.

A Multiregression Dynamic Model (MDM) is a class of multivariate time series that represents various dynamic causal processes in a graphical way. One of the advantages of this class is that, in contrast to many other Dynamic Bayesian Networks, the hypothesised relationships accommodate conditional conjugate inference. We demonstrate for the first time how straightforward it is to search over all possible connectivity networks with dynamically changing intensity of transmission to find the Maximum a Posteriori Probability (MAP) model within this class. This search method is made feasible by using a novel application of an Integer Programming algorithm. The efficacy of applying this particular class of dynamic models to this domain is shown and more specifically the computational efficiency of a corresponding search of 11-node Directed Acyclic Graph (DAG) model space. We proceed to show how diagnostic methods, analogous to those defined for static Bayesian Networks, can be used to suggest embellishment of the model class to extend the process of model selection. All methods are illustrated using simulated and real resting-state functional Magnetic Resonance Imaging (fMRI) data.

This document demonstrates that the efficient approach for diagnosis of Petri nets via integer linear programming may be unable to detect a fault even if the system is diagnosable.

ADA704480

ADA704480

This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of these models are around $0.3\%$ of the optimal policy cost and computational times are reasonable.

Lecture from a Business Mathematics course (2005 - 6) by KIT

In the minimum common string partition (MCSP) problem two related input strings are given. "Related" refers to the property that both strings consist of the same set of letters appearing the same number of times in each of the two strings. The MCSP seeks a minimum cardinality partitioning of one string into non-overlapping substrings that is also a valid partitioning for the second string. This problem has applications in bioinformatics e.g. in analyzing related DNA or protein sequences. For strings with lengths less than about 1000 letters, a previously published integer linear programming (ILP) formulation yields, when solved with a state-of-the-art solver such as CPLEX, satisfactory results. In this work, we propose a new, alternative ILP model that is compared to the former one. While a polyhedral study shows the linear programming relaxations of the two models to be equally strong, a comprehensive experimental comparison using real-world as well as artificially created benchmark instances indicates substantial computational advantages of the new formulation.

In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We give several examples for which an approximate solution should be found very quickly, such as management of a hybrid-electric vehicle drivetrain and control of switched-mode power converters. Our numerical experiments suggest that our method is very effective in finding a feasible point with small objective value; indeed, we find that in many cases, it finds the global solution.

This paper presents Multi-Objective Integer Non-Linear Programming (MOINLP) involving Negative Total Float (NTF) for improving the basic model of Multi-Objective Programming (MOP) in case the optimization of the additional cost for Project Scheduling Compression (PSC). Using the basic MOP to solve the more complex problems is a challenging task. We suspect that Negative Total Float (NTF) having an indication to make the basic MOP to solve the more general case, both simple and complex of PSC. The purpose of this research is identifying the conflicting objectives in PSC problem using NTF and improving MOINLP by involving the NTF parameter to solve the PSC problem. The Solver Application, which is an add-in of MS Excel, is used to perform optimization process to the model developed. The results show that NTF has an important role to identify the conflicting objectives in PSC. We define NTF is an automatic maximum value of the activity duration reduction to achieve due date of PSC. Furthermore, the use of NTF as a constraint in MOINLP can solve the more general case for both simple and complex PSC problem. Base on the condition, we state that the basic MOP is still significant to solve the PSC complex problems using MOINLP as a sophisticated MOP technique. 

This article is from PLoS ONE , volume 8 . Abstract Identification of communities in complex networks is an important topic and issue in many fields such as sociology, biology, and computer science. Communities are often defined as groups of related nodes or links that correspond to functional subunits in the corresponding complex systems. While most conventional approaches have focused on discovering communities of nodes, some recent studies start partitioning links to find overlapping communities straightforwardly. In this paper, we propose a new quantity function for link community identification in complex networks. Based on this quantity function we formulate the link community partition problem into an integer programming model which allows us to partition a complex network into overlapping communities. We further propose a genetic algorithm for link community detection which can partition a network into overlapping communities without knowing the number of communities. We test our model and algorithm on both artificial networks and real-world networks. The results demonstrate that the model and algorithm are efficient in detecting overlapping community structure in complex networks.

This article is from PLoS ONE , volume 8 . Abstract Identification of communities in complex networks is an important topic and issue in many fields such as sociology, biology, and computer science. Communities are often defined as groups of related nodes or links that correspond to functional subunits in the corresponding complex systems. While most conventional approaches have focused on discovering communities of nodes, some recent studies start partitioning links to find overlapping communities straightforwardly. In this paper, we propose a new quantity function for link community identification in complex networks. Based on this quantity function we formulate the link community partition problem into an integer programming model which allows us to partition a complex network into overlapping communities. We further propose a genetic algorithm for link community detection which can partition a network into overlapping communities without knowing the number of communities. We test our model and algorithm on both artificial networks and real-world networks. The results demonstrate that the model and algorithm are efficient in detecting overlapping community structure in complex networks.

Thesis advisor, Richard E. Rosenthal

Projection from a Business Mathematics course (2005 - 6) by KIT

In this paper we consider the use of extended formulations in LP-based algorithms for mixed integer conic quadratic programming (MICQP). Extended formulations have been used by Vielma, Ahmed and Nemhauser (2008) and Hijazi, Bonami and Ouorou (2013) to construct algorithms for MICQP that can provide a significant computational advantage. The first approach is based on an extended or lifted polyhedral relaxation of the Lorentz cone by Ben-Tal and Nemirovski (2001) that is extremely economical, but whose approximation quality cannot be iteratively improved. The second is based on a lifted polyhedral relaxation of the euclidean ball that can be constructed using techniques introduced by Tawarmalani and Sahinidis (2005). This relaxation is less economical, but its approximation quality can be iteratively improved. Unfortunately, while the approach of Vielma, Ahmed and Nemhauser is applicable for general MICQP problems, the approach of Hijazi, Bonami and Ouorou can only be used for MICQP problems with convex quadratic constraints. In this paper we show how a homogenization procedure can be combined with the technique by Tawarmalani and Sahinidis to adapt the extended formulation used by Hijazi, Bonami and Ouorou to a class of conic mixed integer programming problems that include general MICQP problems. We then compare the effectiveness of this new extended formulation against traditional and extended formulation-based algorithms for MICQP. We find that this new formulation can be used to improve various LP-based algorithms. In particular, the formulation provides an easy-to-implement procedure that, in our benchmarks, significantly improved the performance of commercial MICQP solvers.

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed {\em n-fold integer programming problems} are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the $d$-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting stock problem.

To obtain a better understanding of the trade-offs between various objectives, Bi-Objective Integer Programming (BOIP) algorithms calculate the set of all non-dominated vectors and present these as the solution to a BOIP problem. Historically, these algorithms have been compared in terms of the number of single-objective IPs solved and total CPU time taken to produce the solution to a problem. This is equitable, as researchers can often have access to widely differing amounts of computing power. However, the real world has recently seen a large uptake of multi-core processors in computers, laptops, tablets and even mobile phones. With this in mind, we look at how to best utilise parallel processing to improve the elapsed time of optimisation algorithms. We present two methods of parallelising the recursive algorithm presented by Ozlen, Burton and MacRae. Both new methods utilise two threads and improve running times. One of the new methods, the Meeting algorithm, halves running time to achieve near-perfect parallelisation. The results are compared with the efficiency of parallelisation within the commercial IP solver IBM ILOG CPLEX, and the new methods are both shown to perform better.

We present an integer programming model for the ferry scheduling problem, improving existing models in various ways. In particular, our model has reduced size in terms of the number of variables and constraints compared to existing models by a factor of approximately O(n), where n being the number of ports. The model also handles efficiently load/unload time constraints, crew scheduling and passenger transfers. Experiments using real world data produced high quality solutions in 12 hours using CPLEX 12.4 with a performance guarantee of within 15% of optimality, on average. This establishes that using a general purpose integer programming solver is a viable alternative in solving the ferry scheduling problem of moderate size.

We present an integer programming model for the ferry scheduling problem, improving existing models in various ways. In particular, our model has reduced size in terms of the number of variables and constraints compared to existing models by a factor of approximately O(n), where n being the number of ports. The model also handles efficiently load/unload time constraints, crew scheduling and passenger transfers. Experiments using real world data produced high quality solutions in 12 hours using CPLEX 12.4 with a performance guarantee of within 15% of optimality, on average. This establishes that using a general purpose integer programming solver is a viable alternative in solving the ferry scheduling problem of moderate size.

We present an integer programming model for the ferry scheduling problem, improving existing models in various ways. In particular, our model has reduced size in terms of the number of variables and constraints compared to existing models by a factor of approximately O(n), where n being the number of ports. The model also handles efficiently load/unload time constraints, crew scheduling and passenger transfers. Experiments using real world data produced high quality solutions in 12 hours using CPLEX 12.4 with a performance guarantee of within 15% of optimality, on average. This establishes that using a general purpose integer programming solver is a viable alternative in solving the ferry scheduling problem of moderate size.

A group theoretic algorithm for the integer program has been computer programmed and tested. It basically consists of a linear programming algorithm, a routine which converts the (relaxed) integer program to a group minimization problem (over the fractional column group or the isomorphic factor group attained via Smith's Normal Form), solving the group problem by dynamic programming or by a shortest path algorithm, and when necessary, uses a branch and bound procedure. Details and computational results are given. Future work regarding other computational strategies available to group theoretic algorithms is also included.

This paper considers heuristic procedures for general mixed integer linear programming with inequality constraints. It focuses on the question of how to most effectively initialize such procedures by constructing an interior path from which to search for good feasible solutions. These paths lead from an optimal solution for the corresponding linear programming problem (i.e., deleting integrality restrictions) into the interior of the feasible region for this problem. Previous methods for constructing linear paths of this kind are analyzed from a statistical viewpoint, which motivates a promising new method. These methods are then extended to piecewise linear paths in order to improve the direction of search in certain cases where constraints that are not binding on the optimal linear programming solution become particularly relevant. Computational experience is reported. (Author)