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Application of general semi-infinite programming to lapidary cutting problems

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  • Winterfeld, Anton

Abstract

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interior-point method developed by Stein [O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers, Boston, 2003]. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on real-world data are also presented.

Suggested Citation

  • Winterfeld, Anton, 2008. "Application of general semi-infinite programming to lapidary cutting problems," European Journal of Operational Research, Elsevier, vol. 191(3), pages 838-854, December.
    • Handle: RePEc:eee:ejores:v:191:y:2008:i:3:p:838-854
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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    2. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    3. David F. Rogers & Robert D. Plante & Richard T. Wong & James R. Evans, 1991. "Aggregation and Disaggregation Techniques and Methodology in Optimization," Operations Research, INFORMS, vol. 39(4), pages 553-582, August.
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    Cited by:

    1. Stuart M. Harwood & Paul I. Barton, 2017. "How to solve a design centering problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 215-254, August.
    2. Jan Schwientek & Tobias Seidel & Karl-Heinz Küfer, 2021. "A transformation-based discretization method for solving general semi-infinite optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 83-114, February.
    3. Daniel Jungen & Hatim Djelassi & Alexander Mitsos, 2022. "Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 83-112, August.
    4. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    5. Volker Maag, 2015. "A collision detection approach for maximizing the material utilization," Computational Optimization and Applications, Springer, vol. 61(3), pages 761-781, July.

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