Applications of Linear Programming to Coding Theory

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem. In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such vectors are known as pseudocodewords. In this work, we completely classify the set of linear programming pseudocodewords for the family of cycle codes. For the case of the binary symmetric channel, another approximation of maximum-likelihood decoding was introduced by Omura in 1972. This decoder employs an iterative algorithm whose behavior closely mimics that of the simplex algorithm. We generalize Omura's decoder to operate on any binary-input memoryless channel, thus obtaining a soft-decision decoding algorithm. Further, we prove that the probability of the generalized algorithm returning the maximum-likelihood codeword approaches 1 as the number of iterations goes to infinity.

An integer linear programming approach for approximate string comparison

We introduce a problem called maximum common characters in blocks (MCCB), which arises in applications of approximate string comparison, particularly in the unification of possibly erroneous textual data coming from different sources. We show that this problem is NP-complete, but can nevertheless be solved satisfactorily using integer linear programming for instances of practical interest. Two integer linear formulations are proposed and compared in terms of their linear relaxations. We also compare the results of the approximate matching with other known measures such as the Levenshtein (edit) distance. (C) 2008 Elsevier B.V. All rights reserved.

Heterogeneity correction in the construction of optimized planning in radiotherapy using linear programming

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Application-oriented Mixed Integer Non-Linear Programming

In the most recent years there is a renovate interest for Mixed Integer Non-Linear Programming (MINLP) problems. This can be explained for different reasons: (i) the performance of solvers handling non-linear constraints was largely improved; (ii) the awareness that most of the applications from the real-world can be modeled as an MINLP problem; (iii) the challenging nature of this very general class of problems. It is well-known that MINLP problems are NP-hard because they are the generalization of MILP problems, which are NP-hard themselves. However, MINLPs are, in general, also hard to solve in practice. We address to non-convex MINLPs, i.e. having non-convex continuous relaxations: the presence of non-convexities in the model makes these problems usually even harder to solve. The aim of this Ph.D. thesis is to give a flavor of different possible approaches that one can study to attack MINLP problems with non-convexities, with a special attention to real-world problems. In Part 1 of the thesis we introduce the problem and present three special cases of general MINLPs and the most common methods used to solve them. These techniques play a fundamental role in the resolution of general MINLP problems. Then we describe algorithms addressing general MINLPs. Parts 2 and 3 contain the main contributions of the Ph.D. thesis. In particular, in Part 2 four different methods aimed at solving different classes of MINLP problems are presented. Part 3 of the thesis is devoted to real-world applications: two different problems and approaches to MINLPs are presented, namely Scheduling and Unit Commitment for Hydro-Plants and Water Network Design problems. The results show that each of these different methods has advantages and disadvantages. Thus, typically the method to be adopted to solve a real-world problem should be tailored on the characteristics, structure and size of the problem. Part 4 of the thesis consists of a brief review on tools commonly used for general MINLP problems, constituted an integral part of the development of this Ph.D. thesis (especially the use and development of open-source software). We present the main characteristics of solvers for each special case of MINLP.

Optimizing search engines results using linear programming

When a query is passed to multiple search engines, each search engine returns a ranked list of documents. Researchers have demonstrated that combining results, in the form of a "metasearch engine", produces a significant improvement in coverage and search effectiveness. This paper proposes a linear programming mathematical model for optimizing the ranked list result of a given group of Web search engines for an issued query. An application with a numerical illustration shows the advantages of the proposed method. © 2011 Elsevier Ltd. All rights reserved.

Data Flow Analysis and the Linear Programming Model

* The research is supported partly by INTAS: 04-77-7173 project, http://www.intas.be

Method of Linear Programming as an Idea for High School Curriculum

2010 Mathematics Subject Classification: 97D40, 97M10, 97M40, 97N60, 97N80, 97R80

ECONOMICAL OCCUPATION OF AGRICULTURAL AREA OF PIRACICABA (SAO PAULO STATE, BRAZIL), INCLUDING RISK THROUGH LINEAR PROGRAMMING

The economic occupation of an area of 500 ha for Piracicaba was studied with the irrigated cultures of maize, tomato, sugarcane and beans, having used models of deterministic linear programming and linear programming including risk for the Target-Motad model, where two situations had been analyzed. In the deterministic model the area was the restrictive factor and the water was not restrictive for none of the tested situations. For the first situation the gotten maximum income was of R$ 1,883,372.87 and for the second situation it was of R$ 1,821,772.40. In the model including risk a producer that accepts risk can in the first situation get the maximum income of R$ 1,883,372. 87 with a minimum risk of R$ 350 year(-1), and in the second situation R$ 1,821,772.40 with a minimum risk of R$ 40 year(-1). Already a producer averse to the risk can get in the first situation a maximum income of R$ 1,775,974.81 with null risk and for the second situation R$ 1.707.706, 26 with null risk, both without water restriction. These results stand out the importance of the inclusion of the risk in supplying alternative occupations to the producer, allowing to a producer taking of decision considered the risk aversion and the pretension of income.

Optimal scheduling of a renewable micro-grid in an isolated load area using mixed-integer linear programming

In the energy management of the isolated operation of small power system, the economic scheduling of the generation units is a crucial problem. Applying right timing can maximize the performance of the supply. The optimal operation of a wind turbine, a solar unit, a fuel cell and a storage battery is searched by a mixed-integer linear programming implemented in General Algebraic Modeling Systems (GAMS). A Virtual Power Producer (VPP) can optimal operate the generation units, assured the good functioning of equipment, including the maintenance, operation cost and the generation measurement and control. A central control at system allows a VPP to manage the optimal generation and their load control. The application of methodology to a real case study in Budapest Tech, demonstrates the effectiveness of this method to solve the optimal isolated dispatch of the DC micro-grid renewable energy park. The problem has been converged in 0.09 s and 30 iterations.

Ancillary services dispatch using linear programming and genetic algorithm approaches

Electricity market players operating in a liberalized environment requires access to an adequate decision support tool, allowing them to consider all the business opportunities and take strategic decisions. Ancillary services represent a good negotiation opportunity that must be considered by market players. For this, decision support tools must include ancillary market simulation. This paper proposes two different methods (Linear Programming and Genetic Algorithm approaches) for ancillary services dispatch. The methodologies are implemented in MASCEM, a multi-agent based electricity market simulator. A test case concerning the dispatch of Regulation Down, Regulation Up, Spinning Reserve and Non-Spinning Reserve services is included in this paper.

Mixed integer non linear programming and artificial neural network based approach to ancillary services dispatch in competitive electricity markets

Ancillary services represent a good business opportunity that must be considered by market players. This paper presents a new methodology for ancillary services market dispatch. The method considers the bids submitted to the market and includes a market clearing mechanism based on deterministic optimization. An Artificial Neural Network is used for day-ahead prediction of Regulation Down, regulation-up, Spin Reserve and Non-Spin Reserve requirements. Two test cases based on California Independent System Operator data concerning dispatch of Regulation Down, Regulation Up, Spin Reserve and Non-Spin Reserve services are included in this paper to illustrate the application of the proposed method: (1) dispatch considering simple bids; (2) dispatch considering complex bids.

Non emergency patients transport - a mixed integer linear programming

This work presents a model and a heuristic to solve the non-emergency patients transport (NEPT) service issues given the new rules recently established in Portugal. The model follows the same principle of the Team Orienteering Problem by selecting the patients to be included in the routes attending the maximum reduction in costs when compared with individual transportation. This model establishes the best sets of patients to be transported together. The model was implemented in AMPL and a compact formulation was solved using NEOS Server. A heuristic procedure based on iteratively solving problems with one vehicle was presented, and this heuristic provides good results in terms of accuracy and computation time.

A new compact linear programming formulation for choice network revenue management

The choice network revenue management model incorporates customer purchase behavioras a function of the offered products, and is the appropriate model for airline and hotel networkrevenue management, dynamic sales of bundles, and dynamic assortment optimization.The optimization problem is a stochastic dynamic program and is intractable. A certainty-equivalencerelaxation of the dynamic program, called the choice deterministic linear program(CDLP) is usually used to generate dyamic controls. Recently, a compact linear programmingformulation of this linear program was given for the multi-segment multinomial-logit (MNL)model of customer choice with non-overlapping consideration sets. Our objective is to obtaina tighter bound than this formulation while retaining the appealing properties of a compactlinear programming representation. To this end, it is natural to consider the affine relaxationof the dynamic program. We first show that the affine relaxation is NP-complete even for asingle-segment MNL model. Nevertheless, by analyzing the affine relaxation we derive a newcompact linear program that approximates the dynamic programming value function betterthan CDLP, provably between the CDLP value and the affine relaxation, and often comingclose to the latter in our numerical experiments. When the segment consideration sets overlap,we show that some strong equalities called product cuts developed for the CDLP remain validfor our new formulation. Finally we perform extensive numerical comparisons on the variousbounds to evaluate their performance.

On the throughput-WIP trade-off in queueing systems, diminishing returns and the threshold property: A linear programming approach

We present a new unifying framework for investigating throughput-WIP(Work-in-Process) optimal control problems in queueing systems,based on reformulating them as linear programming (LP) problems withspecial structure: We show that if a throughput-WIP performance pairin a stochastic system satisfies the Threshold Property we introducein this paper, then we can reformulate the problem of optimizing alinear objective of throughput-WIP performance as a (semi-infinite)LP problem over a polygon with special structure (a thresholdpolygon). The strong structural properties of such polygones explainthe optimality of threshold policies for optimizing linearperformance objectives: their vertices correspond to the performancepairs of threshold policies. We analyze in this framework theversatile input-output queueing intensity control model introduced byChen and Yao (1990), obtaining a variety of new results, including (a)an exact reformulation of the control problem as an LP problem over athreshold polygon; (b) an analytical characterization of the Min WIPfunction (giving the minimum WIP level required to attain a targetthroughput level); (c) an LP Value Decomposition Theorem that relatesthe objective value under an arbitrary policy with that of a giventhreshold policy (thus revealing the LP interpretation of Chen andYao's optimality conditions); (d) diminishing returns and invarianceproperties of throughput-WIP performance, which underlie thresholdoptimality; (e) a unified treatment of the time-discounted andtime-average cases.

Implementing TURF analysis through binary linear programming

This paper introduces the approach of using Total Unduplicated Reach and Frequency analysis (TURF) to design a product line through a binary linear programming model. This improves the efficiency of the search for the solution to the problem compared to the algorithms that have been used to date. The results obtained through our exact algorithm are presented, and this method shows to be extremely efficient both in obtaining optimal solutions and in computing time for very large instances of the problem at hand. Furthermore, the proposed technique enables the model to be improved in order to overcome the main drawbacks presented by TURF analysis in practice.