Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

  • Alex Y. D. Siem
  • Dick den Hertog
  • Aswin L. Hoffmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3982)


The main contents of this paper is two-fold. First, we present a method to approximate multivariate convex functions by piecewise linear upper and lower bounds. We consider a method that is based on function evaluations only. However, to use this method, the data have to be convex. Unfortunately, even if the underlying function is convex, this is not always the case due to (numerical) errors. Therefore, secondly, we present a multivariate data-smoothing method that smooths nonconvex data. We consider both the case that we have only function evaluations and the case that we also have derivative information. Furthermore, we show that our methods are polynomial time methods. We illustrate this methodology by applying it to some examples.


Multiobjective Optimization Derivative Information Naval Research Logistics Isotonic Regression Pareto Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alex Y. D. Siem
    • 1
  • Dick den Hertog
    • 1
  • Aswin L. Hoffmann
    • 2
  1. 1.Department of Econometrics and Operations Research/Center for Economic Research (CentER)Tilburg UniversityTilburgThe Netherlands
  2. 2.Department of Radiation OncologyRadboud University Nijmegen Medical CentreNijmegenThe Netherlands

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