Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a(0) + a(1)x(1) + ... + a(n)x(n) subject to certain constraints to solve the problem of minimizing a rational function of the form (a(0) + a(1)x(1) + ... + a(n)x(n))/(b(0) + b(1)x(1) + ... + b(n)x(n)) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo`s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo`s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an alpha-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an alpha-approximation (1 1/alpha-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.

Newton`s iterates can converge to non-stationary points

In this note we discuss the convergence of Newton`s method for minimization. We present examples in which the Newton iterates satisfy the Wolfe conditions and the Hessian is positive definite at each step and yet the iterates converge to a non-stationary point. These examples answer a question posed by Fletcher in his 1987 book Practical methods of optimization.

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

Algoritmos para o problema de localização de uma facilidade com distâncias limitadas e restrições de atendimento

The objective in the facility location problem with limited distances is to minimize the sum of distance functions from the facility to the customers, but with a limit on each distance, after which the corresponding function becomes constant. The problem has applications in situations where the service provided by the facility is insensitive after a given threshold distance (eg. fire station location). In this work, we propose a global optimization algorithm for the case in which there are lower and upper limits on the numbers of customers that can be served

KT-invexity in optimal control problems

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Relação entre modelos de programação não-linear com incerteza no conjunto de restrições

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Programação linear por partes: revisão teórica e aplicações

Procura-se resgatar a importância de uma subárea da Programação Matemática conhecida como Programação Linear Por Partes - PLP. de fato a PLP tem inúmeras aplicações tanto na área teórica como em situações reais. Este trabalho apresenta os resultados de uma pesquisa bibliográfica, efetuada nas principais revistas técnicas e livros disponíveis relacionados com Pesquisa Operacional, que visou situar o estado da'arte da Programação Linear por Partes, bem como a abrangência de sua aplicabilidade. Particularmente, no contexto da PLP, este texto deslaca a Programação em Redes Lineares por Partes devido a sua relevância em muitas situações práticas.

Aggregation-based decomposition for multi-divisional models

The aggregation theory of mathematical programming is used to study decentralization in convex programming models. A two-level organization is considered and a aggregation-disaggregation scheme is applied to such a divisionally organized enterprise. In contrast to the known aggregation techniques, where the decision variables/production planes are aggregated, it is proposed to aggregate resources allocated by the central planning department among the divisions. This approach results in a decomposition procedure, in which the central unit has no optimization problem to solve and should only average local information provided by the divisions.

Algorithms for network piecewise-linear programs: A comparative study

Piecewise-Linear Programming (PLP) is an important area of Mathematical Programming and concerns the minimisation of a convex separable piecewise-linear objective function, subject to linear constraints. In this paper a subarea of PLP called Network Piecewise-Linear Programming (NPLP) is explored. The paper presents four specialised algorithms for NPLP: (Strongly Feasible) Primal Simplex, Dual Method, Out-of-Kilter and (Strongly Polynomial) Cost-Scaling and their relative efficiency is studied. A statistically designed experiment is used to perform a computational comparison of the algorithms. The response variable observed in the experiment is the CPU time to solve randomly generated network piecewise-linear problems classified according to problem class (Transportation, Transshipment and Circulation), problem size, extent of capacitation, and number of breakpoints per arc. Results and conclusions on performance of the algorithms are reported.

Modelo de otimização para seleção de árvores matrizes de Eucalyptus grandis

Individual data of basic density and volume of wood, pulp kappa number, soluble lignin, cost of pulping process as well as gravimetric gross yield of pulping process were used from 64 trees of Eucalyptus grandis W. Hill ex Maiden from a commercial population at Lençóis Paulista, SP. The Eucalyptus grandis's seeds were originally from a Seed Production Area (SPA) of Duratex S/A at Botucatu, SP. Obtained data were quantified considering objective of maximization no-bleaching pulp and volume and mass wood restriction, mass of residual and soluble lignin, planted area and pulping process cost. it has also been aimed a selection method for matrix trees through mathematical programming techniques. Obtained strategy maximized the economical result, selected matrix trees and followed all limits of technological and organizing productivities imposed by the company. It also aimed the production of no-bleaching pulp within the planned time.

Bound constrained smooth optimization for solving variational inequalities and related problems

Variational inequalities and related problems may be solved via smooth bound constrained optimization. A comprehensive discussion of the important features involved with this strategy is presented. Complementarity problems and mathematical programming problems with equilibrium constraints are included in this report. Numerical experiments are commented. Conclusions and directions of future research are indicated.

Branch and bound algorithm for transmission network expansion planning using DC model

This paper presents an algorithm to solve the network transmission system expansion planning problem using the DC model which is a mixed non-linear integer programming problem. The major feature of this work is the use of a Branch-and-Bound (B&B) algorithm to directly solve mixed non-linear integer problems. An efficient interior point method is used to solve the non-linear programming problem at each node of the B&B tree. Tests with several known systems are presented to illustrate the performance of the proposed method. ©2007 IEEE.

Wireless mesh networks planning based on parameters of quality of service

The use of QoS parameters to evaluate the quality of service in a mesh network is essential mainly when providing multimedia services. This paper proposes an algorithm for planning wireless mesh networks in order to satisfy some QoS parameters, given a set of test points (TPs) and potential access points (APs). Examples of QoS parameters include: probability of packet loss and mean delay in responding to a request. The proposed algorithm uses a Mathematical Programming model to determine an adequate topology for the network and Monte Carlo simulation to verify whether the QoS parameters are being satisfied. The results obtained show that the proposed algorithm is able to find satisfactory solutions.

A hybrid multi-population genetic algorithm applied to solve the multi-level capacitated lot sizing problem with backlogging

The present paper proposes a new hybrid multi-population genetic algorithm (HMPGA) as an approach to solve the multi-level capacitated lot sizing problem with backlogging. This method combines a multi-population based metaheuristic using fix-and-optimize heuristic and mathematical programming techniques. A total of four test sets from the MULTILSB (Multi-Item Lot-Sizing with Backlogging) library are solved and the results are compared with those reached by two other methods recently published. The results have shown that HMPGA had a better performance for most of the test sets solved, specially when longer computing time is given. © 2012 Elsevier Ltd.

Uma abordagem para problemas e controle ótimo via métodos de Runge-Kutta e análise de erro

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