Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing
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.
KeywordsMultiobjective Optimization Derivative Information Naval Research Logistics Isotonic Regression Pareto Surface
Unable to display preview. Download preview PDF.
- 1.Kuijt, F.: Convexity preserving interpolation – Stationary nonlinear subdivision and splines. PhD thesis, University of Twente, Enschede, The Netherlands (1998)Google Scholar
- 2.Siem, A.Y.D., de Klerk, E., den Hertog, D.: Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions. CentER Discussion Paper 2005-73, Tilburg University, Tilburg (2005)Google Scholar
- 7.Siem, A.Y.D., den Hertog, D., Hoffmann, A.L.: A method for approximating univariate convex functions using only function evaluations. Working paper, Tilburg University, Tilburg (2005)Google Scholar
- 16.Hoffmann, A.L., Siem, A.Y.D., den Hertog, D., Kaanders, J.H.A.M., Huizenga, H.: Dynamic generation and interpolation of pareto optimal IMRT treatment plans for convex objective functions. Working paper, Radboud University Nijmegen Medical Centre, Nijmegen (2005)Google Scholar
- 17.den Boef, E., den Hertog, D.: Efficient line searching for convex functions. CentER Discussion Paper 2004-52. Tilburg University, Tilburg (2004)Google Scholar