A nonlinear programming model with implicit variables for packing ellipsoids

E. G. Birgin, R. D. Lobato, J. M. Martínez
2016 Journal of Global Optimization  
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
more » ... psoids 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. In the last few decades, many works [5, 7, 14, 15, 16, 19, 29, 37, 38, 39, 40] addressed the problem of packing non-overlapping spheres in the n-dimensional space within a variety of fixed-or variable-dimensions containers of different shapes by means of nonlinear programming (NLP) models and methods. However, to the best of our knowledge, only six very recent works [8, 25, 30, 31, 35, 36] have considered the problem of packing non-overlapping ellipses, spheroids, or ellipsoids using nonlinear programming models and optimization techniques. In 2013, the problem of packing as many identical ellipses as possible within a rectangular container was considered in [25] . By restricting the ellipses to have their semi-axes parallel to the Cartesian axes and allowing the centers of the ellipses to belong to a finite set of points in the plane (grid), the problem was modelled as a mixed-integer linear programming problem (MILP). As expected, only small instances of the MILP model could be solved to optimality within an affordable time. In 2014, the problem of packing a given set of (non-necessarily identical) freely-rotated ellipses within a rectangle of minimum area was considered [31] . The non-overlapping between the ellipses was modelled using the idea of separating lines. State-of-the-art global optimization solvers were able to find solutions for instances with up to 14 ellipses. For instances with up to 100 ellipses, the authors presented solutions obtained by a constructive heuristic method. The same problem was addressed in [36] . The problem was modelled using "quasi-phi-functions" that is an extension of the phi-functions [17] extensively and successfully used to model a large variety of complicated packing problems. As well as in [31] , the non-overlapping between ellipses was modelled based on the idea of separating lines. Models were tackled by a local optimization solver combined with ad hoc initial points and a multi-start strategy. Most of the solutions presented in [31] were improved in [36] , where numerical experiments with additional instances with up to 120 ellipses were also shown. In 2015, the work [30] extended the ideas presented for the two-dimensional case in [31] to deal with the three-dimensional problem of packing a given set of (non-necessarily identical) freely-rotated ellipsoids within a rectangular container of minimum volume. The idea of separating lines to model the non-overlapping between ellipses was naturally extended to separating planes to model the non-overlapping between ellipsoids. Resulting NLP models are non-convex and highly complex. Numerical experiments in [30] , that considered instances with up to 100 ellipsoids, showed that state-of-the-art global optimization solvers were only able to deliver feasible solutions within an affordable prescribed CPU time limit. Heuristic methods were also proposed in [30] . Also in 2015, [35] extended the methods and methodology proposed in [36] from the two-to the three-dimensional case; but only spheroids, instead of arbitrary ellipsoids, were considered in the three-dimensional case. In that work, quasi-phi-functions were defined, NLP models proposed (based on separating planes), and solutions were delivered by applying a multi-start strategy associated with a local NLP solver. Illustrative numerical experiments in [35] describe solutions obtained for instances with up to 12 spheroids. Still in 2015, continuous and differentiable NLP models for n-dimensional ellipsoids' packing problems were proposed in [8] . The non-overlapping between ellipsoids was formulated in two different ways: (i) based on separating hyperplanes (and, in this sense, similarly to the already mentioned approaches) and (ii) based on linear transformations. In the latter case, the nonoverlapping between a pair of ellipsoids reduces to the non-overlapping between a sphere and an ellipsoid. The solution to this simpler problem was inspired in the models proposed in [5]
doi:10.1007/s10898-016-0483-8 fatcat:azehg47drfgrvlzd25n4o6k4ja