# Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

A.Y.D. Siem, D. den Hertog, A.L. Hoffmann

Research output: Working paperDiscussion paperOther research output

### Abstract

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.
Original language English Tilburg Operations research 15 2005-132 Published - 2005

### Publication series

Name CentER Discussion Paper 2005-132

### Fingerprint

Smoothing
Norm
Approximation
Evaluation Function
Smoothing Methods
Multivariate Functions
Multivariate Data
Piecewise Linear
Convex function
Upper and Lower Bounds
Polynomial time
Fold
Derivative
Methodology

### Keywords

• approximation theory
• convexity
• data-smoothing

### Cite this

Siem, A. Y. D., den Hertog, D., & Hoffmann, A. L. (2005). Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing. (CentER Discussion Paper; Vol. 2005-132). Tilburg: Operations research.
Siem, A.Y.D. ; den Hertog, D. ; Hoffmann, A.L. / Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing. Tilburg : Operations research, 2005. (CentER Discussion Paper).
title = "Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing",
abstract = "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.",
keywords = "approximation theory, convexity, data-smoothing",
author = "A.Y.D. Siem and {den Hertog}, D. and A.L. Hoffmann",
note = "Subsequently published in Lecture Notes Computer Science (2006) Pagination: 15",
year = "2005",
language = "English",
volume = "2005-132",
series = "CentER Discussion Paper",
publisher = "Operations research",
type = "WorkingPaper",
institution = "Operations research",

}

Siem, AYD, den Hertog, D & Hoffmann, AL 2005 'Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing' CentER Discussion Paper, vol. 2005-132, Operations research, Tilburg.

Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing. / Siem, A.Y.D.; den Hertog, D.; Hoffmann, A.L.

Tilburg : Operations research, 2005. (CentER Discussion Paper; Vol. 2005-132).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

AU - Siem, A.Y.D.

AU - den Hertog, D.

AU - Hoffmann, A.L.

N1 - Subsequently published in Lecture Notes Computer Science (2006) Pagination: 15

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

KW - approximation theory

KW - convexity

KW - data-smoothing

M3 - Discussion paper

VL - 2005-132

T3 - CentER Discussion Paper

BT - Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

PB - Operations research

CY - Tilburg

ER -

Siem AYD, den Hertog D, Hoffmann AL. Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing. Tilburg: Operations research. 2005. (CentER Discussion Paper).