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Finding an Envelope is an Optimization Problem

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Abstract

The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding optimization problem is established. Extensions of the envelope problem to multiple parameters and families of surfaces are considered, as well as envelope problems that cannot be solved by the standard approach, for example, problems involving multiple parameters subject to constraints on the parameters.

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References

  1. Courant, R.: Differential and Integral Calculus, vol. II, pp. 169–183. Interscience Publishers, New York (1936)

    Google Scholar 

  2. Taylor, A.E.: Advanced Calculus, pp. 393–404. Ginn and Company, New York (1955)

    MATH  Google Scholar 

  3. Greenberg, M.P.: Foundations of Applied Mathematics, pp. 399–400. Prentice-Hall, Upper Saddle River (1978)

    MATH  Google Scholar 

  4. Bronshtein, I.N., et al.: Handbook of Mathematics, vol. 83, pp. 237–238. Springer, Upper Saddle River (2004)

    Book  Google Scholar 

  5. Google Search.: “envelope of a family of curves”. Found are many websites from sources such as Wikipedia, Math24, Math Curve, Mathematics Stack Exchange, and more, containing examples and figures of envelopes (2020)

  6. Lin, J.G.: Multiple-objective problems: pareto-optimal solutions by method of proper equality constraints. IEEE Trans. Automat. Contr. 21, 641–650 (1976)

    Article  MathSciNet  Google Scholar 

  7. Li, D., Haimes, Y.Y.: The envelope approach for multiobjective optimization problems. IEEE Trans. Syst. Man Cybern. 17, 1026–1038 (1987)

    Article  MathSciNet  Google Scholar 

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  1. University of Texas at Austin, Austin, TX, USA

    David G. Hull

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  1. David G. Hull
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Correspondence to David G. Hull.

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Hull, D.G. Finding an Envelope is an Optimization Problem. J Optim Theory Appl 186, 453–458 (2020). https://doi.org/10.1007/s10957-020-01707-9

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  • DOI: https://doi.org/10.1007/s10957-020-01707-9

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