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Mathematically Rigorous Global Optimization and Fuzzy Optimization

A Brief Comparison of Paradigms, Methods, Similarities, and Differences

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Part of the Springer Optimization and Its Applications book series (SOIA,volume 170)

Abstract

Mathematically rigorous global optimization and fuzzy optimization have different philosophical underpinnings, goals, and applications. However, some of the tools used in implementations are similar or identical. We review, compare and contrast basic ideas and applications behind these areas, referring to some of the work in the very large literature base.

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Fig. 1

Notes

  1. 1.

    Our interval details are more comprehensive, since fuzzy technology is such a large field, and since our primary expertise lies in interval analysis.

  2. 2.

    Typically, through control of the rounding mode, as specified in the IEEE 754 standard for floating point arithmetic [19].

  3. 3.

    Most desktops and laptops with appropriate programming language support, and some supercomputers, adhere to this standard.

  4. 4.

    First observed in [39, pp. 90–93] and treated by many since then.

  5. 5.

    The clustering effect is actually in common to most basic branch and bound algorithms, whether or not interval technology is used. However, its cause is closely related to the interval dependency problem.

  6. 6.

    But with complicated interpretation of results in general settings.

  7. 7.

    This is not to say that mathematical implications of particular procedures are not highly analyzed.

  8. 8.

    An interval \(\mathbb {X}\) corresponds to a fuzzy set as in Definition 5 with \(\mathcal {A} = \mathbb {X}\) and μ A(x) = 1 if \(x\in \mathbb {X}\), μ A(x) = 0 otherwise.

  9. 9.

    However, data, and even statistics, can sometimes be used in the design of membership functions.

  10. 10.

    The terms “degree of truth” and “degree of belief” are common in the literature concerning handling uncertain knowledge. Also, “belief functions,” like membership functions, are defined. See [53], for example. We do not guarantee that our definition of “degree of truth” is the same as that of all others.

  11. 11.

    The results of (1) and (2) are sharp to within measurement and rounding errors but are sometimes increasingly un-sharp when operations are combined.

  12. 12.

    See [42], etc.

  13. 13.

    For details and generalizations, see a text on interval analysis, such as [4, 43], or our work [24].

  14. 14.

    Under mild conditions on the interval extension of F′ and for most ways of computing v.

  15. 15.

    Various researchers refer to similar lists as code lists (Rall), directed acyclic graphs (DAGs, Neumaier), etc. Although, for a given problem, such lists are not unique, they may be generated automatically with computer language compilers or by other means.

  16. 16.

    To obtain a possibly better upper bound on the global optimum, we replace φ in Problem 10 by − φ and then form and solve a relaxation of the resulting problem. However, for mathematical rigor, computing an upper bound on φ is more complicated than computing a lower bound over \(\mathcal {D}\), since the region \(\mathcal {D}\) may not contain a feasible point.

  17. 17.

    Two-variable relaxations correspond to multiplication; the literature describes various relaxations to these.

  18. 18.

    Another group previously published similar ideas, with a slightly different perspective; see [15].

  19. 19.

    Notably, quadratics, since quadratic programs have been extensively studied.

  20. 20.

    Derived from μ X and f or designed some other way, to take account of perceived goodness of values of φ.

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Kearfott, R.B. (2021). Mathematically Rigorous Global Optimization and Fuzzy Optimization. In: Pardalos, P.M., Rasskazova, V., Vrahatis, M.N. (eds) Black Box Optimization, Machine Learning, and No-Free Lunch Theorems. Springer Optimization and Its Applications, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-030-66515-9_7

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