Publications

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Amaral, Paula. "O algoritmo dos k-Caminhos mais curtos na relaxação do espaço de estados." MsC Disertation, Faculty of Science, UL, Lisbon, Portugal (in Portuguese) (1993). Abstract
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Amaral, P. "On Fractional Quadratic Problems." XII global optimization workshop MAGO 2014, 1-4 September 2014. 2014. 113-116. Abstract
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Amaral, P., and P. Barahona. "After infeasibility in linear programming." Proceedings of CP-AI-OR99 workshop on integration of AI and OR techniques in Constraint Programming for Combinatorial Optimization problems. Vol. 1. Universit, 1999. Abstract

This work is focused on the correction of Infeasible Linear problems. Its motivation is not difficult to understand if one thinks on the complexity of model building in large real problems. The inconsistency can arise in the definition of the model due, for instance, to structural or data type errors. The identification of conflict sets of constraints is very useful but might not be enough to overcome the problem, since the implementation of a solution may require the definition of a new feasible model. We present a short review on known procedures for the diagnosis of these problems. The approach we propose is based on the correction of (potentially) all the parameters of the model restrictions. We present a pure algebraic methodology based on the Singular Value Decomposition of a matrix. This method is quite rigid in the changes of the matrix coefficients changes, so we give insights on a heuristic based approach in order to attain more flexibility.

Amaral, Paula, Joaquim Júdice, and Hanif D. Sherali. "A reformulation–linearization–convexification algorithm for optimal correction of an inconsistent system of linear constraints." Computers & Operations Research. 35 (2008): 1494-1509. Abstract
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Amaral, P., L. M. Fernandes, J. Júdice, and H. D. Sherali. "On optimal zero-preserving corrections for inconsistent linear systems." Journal of Global Optimization. 45 (2009): 645-666. Abstract

This paper addresses the problem of finding an optimal correction of an inconsistent linear system, where only the nonzero coefficients of the constraint matrix are allowed to be perturbed for reconstructing a consistent system. Using the Frobenius norm as a measure of the distance to feasibility, a nonconvex minimization problem is formulated, whose objective function is a sum of fractional functions. A branch-and-bound algorithm for solving this nonconvex program is proposed, based on suitably overestimating the denominator function for computing lower bounds. Computational experience is presented to demonstrate the efficacy of this approach.

Amaral, P., M. W. Trosset, and P. Barahona Correcting an Inconsistent System of Linear Inequalities by Nonlinear Programming. TX 77005, Houston: Department of Computational & Applied Mathematics, Rice University, 2000. Abstract
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Amaral, Paula, and Pedro Barahona. "Connections between the total least squares and the correction of an infeasible system of linear inequalities." Linear algebra and its applications. 395 (2005): 191-210. Abstract
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Amaral, Paula A., and Immanuel M. Bomze. "Copositivity-based approximations for mixed-integer fractional quadratic optimization." Pacific Journal of Optimization. 11 (2015): 225-238. Abstract
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Amaral, P. "Matemática Industrial em Rede." Gazeta de Matemática. 181 (2017). Abstract
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Amaral, P. "O admirável Mundo Novo do Big Data." Gazeta de Matemática. 182 (2017). Abstract
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Amaral, P., M. W. Trosset, and P. Barahona Correcting an inconsistent system of linear inequalities by nonlinear programming. Houston, TX 77005: Department of Computational & Applied Mathematics, Rice University, 2000. Abstract
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Amaral, Paula, and Tiago Cardal Pais. "Compromise ratio with weighting functions in a Tabu Search multi-criteria approach to examination timetabling." Computers & Operations Research. 72 (2016): 160-174. Abstract
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Amaral, P., and P. Barahona. "A Framework for Optimal Correction of Inconsistent Linear Constraints." Constraints. 10:1 (2005): 67-86. Abstract

The problem of inconsistency between constraints often arises in practice as the result, among others, of the complexity of real models or due to unrealistic requirements and preferences. To overcome such inconsistency two major actions may be taken: removal of constraints or changes in the coefficients of the model. This last approach, that can be generically described as ``model corre\-ction" is the problem we address in this paper in the context of linear constraints over the reals. The correction of the right hand side alone, which is very close to a fuzzy constraints approach, was one of the first proposals to deal with inconsistency, as it may be mapped into a linear problem. The correction of both the matrix of coefficients and the right hand side introduces non linearity in the constraints. The degree of difficulty in solving the problem of the optimal correction depends on the objective function, whose purpose is to measure the closeness between the original and corrected model. Contrary to other norms, that provide corrections with quite rigid patterns, the optimization of the important Frobenius norm was still an open problem. We have analyzed the problem using the KKT conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima, in terms of the solution of the Total Least Squares and the set of active constraints. These conditions justify a set of pruning rules, which proved, in preliminary experimental results, quite successful in a tree search procedure for determining the global minimizer.

Amaral, Paula, Immanuel M. Bomze, and Joaquim Júdice. "Copositivity and constrained fractional quadratic problems." Mathematical Programming. 146 (2014): 325-350. Abstract
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Amaral, Paula, and Pedro Barahona. "K-Best Feasible Clusters- Ranking optimal solutions from an infeasible LP." under revision (2017). Abstract
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Amaral, Paula. "Contribuições para o estudo de sistemas lineares inconsistentes." PhD Disertation, Faculty of Science and Technology, UNL, Lisbon, Portugal (in Portuguese) (2001). Abstract
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Amaral, P., and P. Barahona. "On Optimal Correction of Inconsistent Linear Constraints." Principles and Practice of Constraint Programming, CP'2002. Ed. Pascal Van Hentenryck. Vol. 2470. Lecture Notes in Computer Science, 2470. Springer, 2002. 33-46. Abstract

In practice one has often to deal with the problem of inconsistency between constraints, as the result, among others, of the comple\-xi\-ty of real models. To overcome these conflicts we can outline two major \mbox{actions}: removal of constraints or changes in the coefficients of the model. This last approach, that can be generically described as ``model corre\-ction" is the problem we address in this paper. The correction of the right hand side alone was one of the first approaches. The correction of both the matrix of coefficients and the right hand side introduces non linearity in the constraints. The degree of difficulty in solving the problem of the optimal correction depends on the objective function, whose purpose is to measure the closeness between the original and corrected model. Contrary to other norms, the optimization of the important Frobenius was still an open problem. We have analyzed the problem using the KKT conditions and derived necessary and sufficient conditions which enabled us to unequivocally characterize local optima, in terms of the solution of the Total Least Squares and the set of active constraints. These conditions justify a set of pruning rules, which proved, in preliminary experimental results, quite successful in a tree search procedure for determining the global minimizer.

Amaral, Paula, and Pedro Barahona. "About infeasibility in the constraints of a linear model." Ricerca Operativa. 92 (1999): 49-67. Abstract
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Gomes, Isabel, Jorge Santos, Luís Cavique, Nelson C. Martins, Manuel Vieira, Paula Amaral, Raquel Barreira, and Vitor H. Fernandes AMT (airline maintenance technicians) timetabling optimization- ESGI101 - Relatório Final- TAP-AMT. ESGI 101, FCT, UNL, 2014, 2014. Abstract
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Gomes, Margarida M., Rita A. Ribeiro, and Paula Amaral. "Reducing the Number of Membership Functions in Linguistic Variables." Livro de actas do 14º Congresso da Associação Portuguesa de Investigação Operacional, IO 2009, 7-9 September 2009. 2009. 75-82. Abstract
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Hendrix, Eligius M. T., Leocadio G. Casado, and Paula Amaral. "Global Optimization Simplex Bisection Revisited Based on Considerations by Reiner Horst." Lecture Notes in Computer Science - ICSSA2012 . 7335 (2012): 159-173. AbstractWebsite

In this paper, the use of non-optimality spheres in a simplicial branch and bound (B&B) algorithm is investigated. In this context, some considerations regarding the use of bisection on the longest edge in relation with ideas of Reiner Horst are reminded. Three arguments highlight the merits of bisection of simplicial subsets in B&B schemes.

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Pais, Tiago Cardal, and Paula Amaral. "Managing the tabu list length using a fuzzy inference system: an application to exams timetabling." The 7th International Conference for the Practice and Theory of Automated Timetabling. 2008. 1-6. Abstract
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