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Ragnar Frisch and interior-point methods

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Abstract

The distinguished econometrician Ragnar Frisch (1895–1973) also played an important role in optimization theory. In fact, he was a pioneer of interior-point methods. This note reconsiders his contribution, relating it to history and modern developments.

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Notes

  1. Later [3] introduced the inverse barrier function. He referred to Frisch’s [21] multiplex method, also evoked by [48].

  2. For an exception, see [1] who already knew and appreciated the logarithmic potential and multiplex methods of [17, 21]. See also [48].

  3. To wit, there are time gaps between [16], the 1968 classic of Fiacco and McCormick, and the [35, 43] papers on pathways in linear programming.

  4. Many optimizers cite Frisch with an extra initial letter M—probably transmitted from the many papers or presentations in France of Monsieur Frisch [11] somewhat mysteriously cited Frisch, a misnomer which has propagated through citation trees.

  5. Other studies of interior-point methods, from that period, include [7] and [25].

  6. See also [19, 20].

  7. After Frisch’s time, optimization has grown impressively, in theory and applications. But alas, it is now less emphasized in economists’ training.

  8. See also [6].

  9. Frisch, also a trained goldsmith and passionate bee-keeper (1952), was editor of Econometrica where the 1949 paper of Dantzig appeared.

  10. http://www.sv.uio.no/econ/om/tall-og-fakta/nobelprivinnere/ragnar-frisch/memoranda-and-others/.

  11. The discrete nature of the simplex method has taken some hold. However, [5], Sect. 7.2, pages 156-160] also stressed its continuous nature, seen as a (reduced) gradient procedure.

  12. For nice presentations of the Khachiyan and Karmarkar methods, as well as for historical remarks, see [41].

  13. The logarithm also enters stochastic programming. There, constraints often come as reliability requirements-say, in the form \(\Pr \left\{ c(x)\ge \underline{c}\right\} \ge p>0\) where \(\underline{c}\) is a random vector. Then, it’s very convenient if that vector has log-concave distribution, to the effect that \(\ln \Pr \left\{ c(x)\ge \underline{c}\right\} \) becomes concave; see [39].

  14. Henceforth, to assist economic intuition, the symbol \(*\) signals a price transform or variable.

  15. Sonnevend [43] first proposed these specially for linear programs.

  16. Conf. [32, 33, 36]. For implementation see Gondzio and Terlaky (1996).

  17. The editor of a conservative Oslo magazine felt compelled to inform Frisch, who had socialist inclinations, that there were competing concepts of freedom.

  18. In essence, this is the pivot step—or Jordan exchange—that drives the simplex algorithm; see [45] or [9].

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  1. Economics Department, University of Oslo, Oslo, Norway

    Olav Bjerkholt

  2. Economics Department, University of Bergen, box 7800 , Foss gt 14, 5020, Bergen, Norway

    Sjur Didrik Flåm

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Bjerkholt, O., Flåm, S.D. Ragnar Frisch and interior-point methods. Optim Lett 9, 1053–1061 (2015). https://doi.org/10.1007/s11590-014-0807-x

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