# Computing tight bounds via piecewise linear functions through the example of circle cutting problems

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## Abstract

This paper discusses approximations of continuous and mixed-integer non-linear optimization problems via piecewise linear functions. Various variants of circle cutting problems are considered, where the non-overlap of circles impose a non-convex feasible region. While the paper is written in an “easy-to-understand” and “hands-on” style which should be accessible to graduate students, also new ideas are presented. Specifically, piecewise linear functions are employed to yield mixed-integer linear programming problems which provide lower and upper bounds on the original problem, the circle cutting problem. The piecewise linear functions are modeled by five different formulations, containing the incremental and logarithmic formulations. Another variant of the cutting problem involves the assignment of circles to pre-defined rectangles. We introduce a new global optimization algorithm, based on piecewise linear function approximations, which converges in finitely many iterations to a globally optimal solution. The discussed formulations are implemented in GAMS. All GAMS-files are available for download in the Electronic supplementary material. Extensive computational results are presented with various illustrations.

## Keywords

Piecewise linear functions Circle cutting Non-convex optimization Global optimization Non-linear programming (NLP) Quadratically constrained programming (QCP) Mixed-integer linear programming (MILP) Outer approximation Inner approximation Incremental formulation Logarithmic formulation## Notes

### Acknowledgments

The author thanks Ed Klotz (IBM CPLEX) and two anonymous reviewers for their valuable suggestions.

## Supplementary material

## References

- Beale ELM (1963) Two transportation problems. In: Proceedings of the third international conference on operational research 1963, Paris and English Universities Press, London, Dunod, pp 780–788Google Scholar
- Beale ELM, Tomlin JA (1970) Special facilities in a general mathematical programming system for nonconvex problem using ordered sets of variables. In: Lawrence J (ed) Proceedings of the fifth international conference on operational research 1969, Tavistock Publishing, London, pp 447–454Google Scholar
- Beale EML, Forrest JJH (1976) Global optimization using special ordered sets. Math Program 10:52–69MathSciNetCrossRefzbMATHGoogle Scholar
- Cerisola S, Latorre JM, Ramos A (2012) Stochastic dual dynamic programming applied to nonconvex hydrothermal models. Eur J Oper Res 218(3):687–697MathSciNetCrossRefzbMATHGoogle Scholar
- Correa-Posada CM, Sanchez-Martin P (2014) Gas network optimization: a comparison of piecewise linear models. Optimization. http://www.optimization-online.org/DB_HTML/2014/10/4580.html
- Frank S, Rebennack S (2012) A primer on optimal power flow: theory, formulation, and practical examples. CSM working paperGoogle Scholar
- Frank SM, Rebennack S (2015) Optimal design of mixed AC-DC distribution systems for commercial buildings: a nonconvex generalized Benders decomposition approach. Eur J Oper Res 242(3):710–729CrossRefzbMATHGoogle Scholar
- Geißler B (2011) Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. Dissertation, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen-Nürnberg, GermanyGoogle Scholar
- Geißler B, Kolb O, Lang J, Leugering G, Martin A, Morsi A (2011) Mixed integer linear models for the optimization of dynamical transport networks. Math Methods Oper Res 73:339–362MathSciNetCrossRefzbMATHGoogle Scholar
- Hettich R, Kortanek KO (1993) Semi-infinite programming. SIAM Rev 35:380–429MathSciNetCrossRefzbMATHGoogle Scholar
- Horst R, Pardalos PM, Thoai NV (2000) Introduction to global optimization, 2nd edn. Kluwer, LondonCrossRefzbMATHGoogle Scholar
- Kallrath J (2009) Cutting circles and polygons from area-minimizing rectangles. J Global Optim 43:299–328MathSciNetCrossRefzbMATHGoogle Scholar
- Kallrath J, Rebennack S (2014a) Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate functions. In: Butenko S, Floudas CA, Rassias TM (eds) Optimization in science and engineering. Springer, Berlin, pp 273–292CrossRefGoogle Scholar
- Kallrath J, Rebennack S (2014b) Cutting ellipses from area-minimizing rectangles. J Global Optim 59(2–3):405–437MathSciNetCrossRefzbMATHGoogle Scholar
- Keha AB, de Farias IR, Nemhauser GL Jr (2004) Models for representing piecewise linear cost functions. Oper Res Lett 32:44–48MathSciNetCrossRefzbMATHGoogle Scholar
- Keha AB, de Farias IR, Nemhauser GL Jr (2006) A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper Res 54(5):847–858MathSciNetCrossRefzbMATHGoogle Scholar
- Koch T, Achterberg T, Andersen E, Bastert O, Berthold T, Bixby RE, Danna E, Gamrath G, Gleixner AM, Heinz S, Lodi A, Mittelmann H, Ralphs T, Salvagnin D, Steffy DE, Wolter K (2011) MIPLIB 2010. Math Program Comput 3(2):103–163MathSciNetCrossRefGoogle Scholar
- Li X, Tomasgard A, Barton PI (2011) Nonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programs. J Optim Theory Appl 151(3):425–454MathSciNetCrossRefzbMATHGoogle Scholar
- Li X, Chen Y, Barton PI (2012a) Nonconvex generalized Benders decomposition with piecewise convex relaxations for global optimization of integrated process design and operation problems. Ind Eng Chem Res 51(21):7287–7299CrossRefGoogle Scholar
- Li X, Tomasgard A, Barton PI (2012b) Decomposition strategy for the stochastic pooling problem. J Global Optim 4(4):765–790MathSciNetCrossRefzbMATHGoogle Scholar
- Lodi A, Tramontani A (2013) Performance variability in mixed-integer programming, chapter 2, pp 1–12. INFORMSGoogle Scholar
- Misener R, Floudas CA (2014) Antigone: Algorithms for continuous/integer global optimization of nonlinear equations. J Global Optim 59(2–3):503–526MathSciNetCrossRefzbMATHGoogle Scholar
- Padberg M (2000) Approximating separable nonlinear functions via mixed zero–one programs. Oper Res Lett 27:1–5MathSciNetCrossRefzbMATHGoogle Scholar
- Pardalos PM, Rosen JB (1987) Constrained global optimization: algorithms and applications. Lecture notes in computer science. Springer, BerlinGoogle Scholar
- Pereira MVF, Pinto LMVG (1991) Multi-stage stochastic optimization applied to energy planning. Math Program 52:359–375MathSciNetCrossRefzbMATHGoogle Scholar
- Rebennack S (2016) Combining sampling-based and scenario-based nested benders decomposition methods: application to stochastic dual dynamic programming. Math Program 156(1):343–389MathSciNetCrossRefzbMATHGoogle Scholar
- Rebennack S, Kallrath J (2015a) Continuous piecewise linear delta-approximations for bivariate and multivariate functions. J Optim Theory Appl 167(1):102–117MathSciNetCrossRefzbMATHGoogle Scholar
- Rebennack S, Kallrath J (2015b) Continuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systems. J Optim Theory Appl 167(2):617–643MathSciNetCrossRefzbMATHGoogle Scholar
- Rebennack S, Kallrath J, Pardalos PM (2009) Column enumeration based decomposition techniques for a class of non-convex MINLP problems. J Global Optim 43(2–3):277–297MathSciNetCrossRefzbMATHGoogle Scholar
- Rosen JB, Pardalos PM (1986) Global minimization of large-scale constrained concave quadratic problems by separable programming. Math Program 34:163–174MathSciNetCrossRefzbMATHGoogle Scholar
- Sherali HD (2001) On mixed-integer zero–one representations for separable lower-semicontinuous piecewise-linear functions. Oper Res Lett 28:155–160MathSciNetCrossRefzbMATHGoogle Scholar
- Tomlin JA (1988) Special ordered sets and an application to gas supply operating planning. Math Program 45:69–84MathSciNetCrossRefGoogle Scholar
- Vielma JP, Nemhauser G (2011) Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math Program 128:49–72MathSciNetCrossRefzbMATHGoogle Scholar
- Vielma JP, Ahmed S, Nemhauser G (2010) Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper Res 58(2):303–315MathSciNetCrossRefzbMATHGoogle Scholar