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On univariate function identification problems

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Abstract

Applications in the areas of data fitting, forecasting, and estimation naturally lead to a rich class of constrained infinite-dimensional optimization problems over extended real-valued semicontinuous functions. We discuss a framework for dealing with such applications, even in the presence of nearly arbitrary constraints on the functions. We formulate computationally tractable approximating problems relying on piecewise polynomial semicontinuous approximations of the actual functions. The approximations enable the study of evolving problems caused by incrementally arriving data and other information. In contrast to an earlier more general treatment, we focus on optimization over functions defined on a compact interval of the real line, which still addresses a large number of applications. The paper provides an introduction to the subject through simplified derivations and illustrative examples.

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Notes

  1. The set is closed under the epi-topology as described below.

  2. We recall that the outer limit of a sequence of sets \(\{A^\nu \}_{\nu \in {IN}}\) is the collection of points y to which a subsequence of \(\{y^\nu \}_{\nu \in {IN}}\), with \(y^\nu \in A^\nu \), converges. The inner limit is the points to which a sequence of \(\{y^\nu \}_{\nu \in {IN}}\), with \(y^\nu \in A^\nu \), converges. If both limits exist and are identical, we say that the set is the Painlevé-Kuratowski limit of \(\{A^\nu \}_{\nu \in {IN}}\); see [7, 30].

  3. Exponential epi-splines are essentially unrelated to the exponential splines in [27].

  4. A third equation could be introduced, of a similar type, that would model interest rates and would give the option to discount future prices. This would be useful if the system is used to value financial instruments that are copper-related.

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Acknowledgements

In part, this work was carried out while the first author was on a sabbatical leave at the UC Davis, Department of Mathematics. The authors thank R. Sood, M. Casey, I. Rios, D. Woodruff, and N. Sukumar for invaluable support and discussions.

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Authors and Affiliations

  1. Operations Research Department, Naval Postgraduate School, Monterey, CA, USA

    Johannes O. Royset

  2. Department of Mathematics, UC Davis, Davis, CA, USA

    Roger J.-B. Wets

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  1. Johannes O. Royset
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  2. Roger J.-B. Wets
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Correspondence to Johannes O. Royset.

Additional information

This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant Numbers 00101-80683, W911NF-10-1-0246 and W911NF-12-1-0273.

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Royset, J.O., Wets, R.JB. On univariate function identification problems. Math. Program. 168, 449–474 (2018). https://doi.org/10.1007/s10107-017-1130-y

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  • DOI: https://doi.org/10.1007/s10107-017-1130-y

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