TAILIEUCHUNG - Continuous Piecewise Linear -Approximations for MINLP Problems. II. Bivariate and Multivariate Functions

As is well known, a large part of the remittances is channelled via the informal financial system through money transfer companies. So far, the access of migrants and their families to the formal financial system has been very limited due, in part, to the fact that a large proportion of the remittance senders are in an irregular situation in the developed countries. Cost and insecurity are usually pointed out as the main disadvantages to transferring remittances via the informal system. As far as cost is concerned, there are many different estimates: according to data collected. | 0COLORADOSCHOOLOFMINES _ . EARTH ENERGY ENVIRONMENT Division OF Economics and Business Working Paper Series Continuous Piecewise Linear -Approximations for MINLP Problems. II. Bivariate and Multivariate Functions Steffen Rebennack Josef Kallrath Working Paper 2012-13 http working-papers Colorado School of Mines Division of Economics and Business 1500 Illinois Street Golden CO 80401 October 2012 2012 by the listed authors. All rights reserved. Colorado School of Mines Division of Economics and Business Working Paper No. 2012-13 October 2012 Title Continuous Piecewise Linear -Approximations for MINLP Problems. II. Bivariate and Multivariate Functions Author s Steffen Rebennack Division of Economics and Business Colorado School of Mines Golden CO 80401-1887 srebenna@ Josef Kallrath Department of Astronomy University of Florida kallrath@ ABSTRACT Following up on Rebennack Kallrath 2012 in this paper for functions depending on two variables using refinement heuristics we automatically construct triangulations subject to the condition that the continuous piecewise linear approximation under- or overestimation never deviates more than a given J-tolerance from the original function over a given domain. This tolerance is proven by solving subproblems over each triangle to global optimality. The continuous piecewise linear approximators under- and overestimators involve shift variables at the vertices of the triangles leading to a small number of triangles while still ensuring continuity over the full domain. On a set of test functions we demonstrate the numerical behavior of our approach. For functions depending on more than two variables we provide appropriate transformations and substitutions which allow to use one- or two-dimensional J-approximators. We address the problem of error propagation when using these dimensionality reduction routines. The automatic refinement triangulation provides an alternative to .

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