Here you can find the abstracts of my publications as well as links to the respective full texts.


E. G. Birgin, R. D. Lobato and J. M. Martínez. A nonlinear programming model with implicit variables for packing ellipsoids. Journal of Global Optimization (2017) 68, 467–499.

Abstract: The problem of packing ellipsoids is considered in the present work. Usually, the computational effort associated with numerical optimization methods devoted to packing ellipsoids grows quadratically with respect to the number of ellipsoids being packed. The reason is that the number of variables and constraints of ellipsoids' packing models is associated with the requirement that every pair of ellipsoids must not overlap. As a consequence, it is hard to solve the problem when the number of ellipsoids is large. In this paper, we present a nonlinear programming model for packing ellipsoids that contains a linear number of variables and constraints. The proposed model finds its basis in a transformation-based non-overlapping model recently introduced by Birgin, Lobato, and Martínez [Journal of Global Optimization (2015), DOI: 10.1007/s10898-015-0395-z]. Numerical experiments show the efficiency and effectiveness of the proposed model.

Keywords: Cutting and packing ellipsoids, optimization, nonlinear programming, models, numerical experiments.

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DOI: 10.1007/s10898-016-0483-8

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E. G. Birgin, R. D. Lobato and J. M. Martínez. Constrained optimization with integer and continuous variables using inexact restoration and projected gradients. Bulletin of Computational Applied Mathematics (2016) 4, 55–70.

Abstract: Inexact restoration (IR) is a well established technique for continuous minimization problems with constraints that can be applied to constrained optimization problems with specific structures. When some variables are restricted to be integer, an IR strategy seems to be appropriate. The IR strategy employs a restoration procedure in which one solves a standard nonlinear programming problem and an optimization procedure in which the constraints are linearized and techniques for mixed-integer (linear or quadratic) programming can be employed.

Keywords: Inexact restoration, mixed-integer nonlinear programming (MINLP), projected gradients.

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E. G. Birgin, R. D. Lobato and J. M. Martínez. Packing ellipsoids by nonlinear optimization. Journal of Global Optimization (2016) 65, 709–743.

Abstract: In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the $n$-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

Keywords: Cutting and packing ellipsoids, nonlinear programming, models, numerical experiments.

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DOI: 10.1007/s10898-015-0395-z

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E. G. Birgin, R. D. Lobato and R. Morabito. Generating unconstrained two-dimensional non-guillotine cutting patterns by a recursive partitioning algorithm. Journal of the Operational Research Society (2012) 63, 183–200.

Abstract: In this study, a dynamic programming approach to deal with the unconstrained twodimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. Since a counterexample for which the approach fails to find known optimal solutions was not found, it is conjectured that it always finds an optimal unconstrained non-guillotine cutting. The method is able to find the optimal cutting pattern of a large number of problem instances of moderate sizes known in the literature. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method.

Keywords: Cutting and packing, two-dimensional non-guillotine cutting pattern, dynamic programming, recursive approach, distributor's pallet loading problem.

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DOI: 10.1057/jors.2011.6

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E. G. Birgin and R. D. Lobato. Orthogonal packing of identical rectangles within isotropic convex regions. Computers & Industrial Engineering (2010) 59, 595–602.

Abstract: A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and active-set strategies for bound-constrained minimization of smooth functions. Numerical results show the reliability of the presented approach.

Keywords: Packing and cutting of rectangles, orthogonal packing, isotropic convex regions, feasibility problems, nonlinear programming, models.

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DOI: 10.1016/j.cie.2010.07.004

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E. G. Birgin, R. D. Lobato and R. Morabito. An effective recursive partitioning approach for the packing of identical rectangles in a rectangle. Journal of the Operational Research Society (2010) 61, 306–320.

Abstract: In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an $L$-approach for packing rectangles into larger rectangles and $L$-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50000 instances). It is also effective for solving the instances of problem set Cover III (almost 100000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.

Keywords: Cutting and packing, manufacturer's pallet loading problem, woodpulp stowage problem, non-guillotine cutting pattern, dynamic programming, raster points.

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DOI: 10.1057/jors.2008.141

Web site and source code:

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