# Cutting ellipses from area-minimizing rectangles

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## Abstract

A set of ellipses, with given semi-major and semi-minor axes, is to be cut from a rectangular design plate, while minimizing the area of the design rectangle. The design plate is subject to lower and upper bounds of its widths and lengths; the ellipses are free of any orientation restrictions. We present new mathematical programming formulations for this ellipse cutting problem. The key idea in the developed non-convex nonlinear programming models is to use separating hyperlines to ensure the ellipses do not overlap with each other. For small number of ellipses we compute feasible points which are globally optimal subject to the finite arithmetic of the global solvers at hand. However, for more than 14 ellipses none of the local or global NLP solvers available in GAMS can even compute a feasible point. Therefore, we develop polylithic approaches, in which the ellipses are added sequentially in a strip-packing fashion to the rectangle restricted in width, but unrestricted in length. The rectangle’s area is minimized in each step in a greedy fashion. The sequence in which we add the ellipses is random; this adds some GRASP flavor to our approach. The polylithic algorithms allow us to compute good, near optimal solutions for up to 100 ellipses.

## Keywords

Global optimization Non-convex nonlinear programming Mixed integer programming Cutting stock problem Packing problem Shape constraints Non-overlap constraints Design problem Polylithic solution approach Computational geometry## Notes

### Acknowledgments

Thanks are directed to Prof. Dr. Siegfried Jetzke (Ostfalia Hochschule, Salzgitter, Germany) for his interest in this work, comments on the manuscript and discussion about the usefulness of ellipses and ellipsoids in real world problems.

## References

- 1.Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: Global optimization of mixed integer nonlinear problems. AIChE J.
**46**, 1769–1797 (2000)CrossRefGoogle Scholar - 2.Adjiman, C.S., Androulakis, I.P., Maranas, C.D., Floudas, C.A.: A global optimization method aBB for process design. Comput. Chem. Eng.
**20**(Suppl. 1), S419–424 (1996)CrossRefGoogle Scholar - 3.Androulakis, I.P., Maranas, C.D., Floudas, C.A.: aBB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim.
**7**, 337–363 (1995)CrossRefGoogle Scholar - 4.Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res.
**44**, 145–159 (1990)CrossRefGoogle Scholar - 5.Feo, T., Resende, M.: Greedy randomized adaptive search procedures. J. Glob. Optim.
**6**, 109–133 (1995)CrossRefGoogle Scholar - 6.Floudas, CA.: Private communication (2012)Google Scholar
- 7.Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.: Global optimization in the 21st century: advances and challenges for problems with nonlinear dynamics. Comput. Chem. Eng.
**29**, 1185–1202 (2005)CrossRefGoogle Scholar - 8.Gensane, T., Honvault, P.: Optimal Packings of Two Ellipses in a Square. Technical report, L.M.P.A. (2012)Google Scholar
- 9.Hettich, R., Kortanek, K.O.: Semi-infinite programming. SIAM Rev.
**35**, 380–429 (1993)CrossRefGoogle Scholar - 10.Kallrath, J.: Combined strategic design and operative planning in the process industry. Comput. Chem. Eng.
**33**, 1983–1993 (2009)Google Scholar - 11.Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim.
**43**, 299–328 (2009)CrossRefGoogle Scholar - 12.Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett.
**5**, 453–466 (2011). doi: 10.1007/s11590-011-0320-4 CrossRefGoogle Scholar - 13.Liberti, L.: Reformulation and convex relaxation techniques for global optimization. Ph.D. Thesis, Imperial College London, London, UK (2004)Google Scholar
- 14.Liberti, L., Pantelides, C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim.
**36**, 161–189 (2006)CrossRefGoogle Scholar - 15.Lindo Systems: Lindo API: User’s Manual. Lindo Systems, Chicago (2004)Google Scholar
- 16.Lopez, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res.
**180**, 491–518 (2007)CrossRefGoogle Scholar - 17.Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim.
**7**, 143–182 (1995)CrossRefGoogle Scholar - 18.Markót, M.C.: Interval methods for verifying structural optimality of circle packing configurations in the unit square. J. Comput. Appl. Math.
**199**, 353–357 (2007)CrossRefGoogle Scholar - 19.Markót, M.C., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim.
**16**, 193–219 (2005)CrossRefGoogle Scholar - 20.Miller, P.: Globally optimal packing of nonconvex two-dimensional shapes by approximation with ellipses. Princeton University, Princeton, NJ, Senior Thesis (2012)Google Scholar
- 21.Misener, R., Floudas, C.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. pp. 1–48 (2012). doi: 10.1007/s10898-012-9874-7
- 22.Rebennack, S., Kallrath, J.: Continuous piecewise linear \(\delta \)-approximations for MINLP problems. I. Minimal breakpoint systems for univariate functions. Tech. rep., Colorado School of Mines Division of Economics and Business Working Paper Series 2012–12 (2012)Google Scholar
- 23.Rebennack, S., Kallrath, J.: Continuous piecewise linear \(\delta \)-approximations for MINLP problems. II. Bivariate and multivariate functions. Tech. rep., Colorado School of Mines Division of Economics and Business Working Paper Series 2012–13 (2012)Google Scholar
- 24.Rebennack, S., Kallrath, J., Pardalos, P.M.: Column enumeration based decomposition techniques for a class of non-convex MINLP problems. J. Glob. Optim.
**43**, 277–297 (2009)CrossRefGoogle Scholar - 25.Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications. Nonconvex optimization and its applications series. Kluwer, Dordrecht, The Netherlands (2002)CrossRefGoogle Scholar