The Log-Exponential Smoothing Technique and Nesterov’s Accelerated Gradient Method for Generalized Sylvester Problems

Abstract

The Sylvester smallest enclosing circle problem involves finding the smallest circle that encloses a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov's accelerated gradient method, we present an effective numerical algorithm for solving these problems.

A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

“…. , w * n ) and the corresponding index set I the conditions (i)-(vi) are fulfilled, then x is an optimal solution to (P γ G , T ), 1] for all i = 1, . .…”

“…has at least one solution and that dom f 1 . We ran our matlab programs for various step sizes ν and chose always the origin as the starting point and set the initialization parameters to the value 1.…”

“…. , n, and compare the dual method with the numerical algorithm build on the logexponential smoothing technique and Nesterov's accelerated gradient method (log-exp), which was developed in [1] for solving generalized Sylvester problems of the kind of (P T ).…”

“…The motivation to investigate such problems comes from both theoretical and practical reasons, as location type problems arise in various areas of research and real life, such as geometry, physics, economics or health management, applications from these fields being mentioned in our paper as possible interpretations of our results. As suggested, for instance, in [1,21], solving general location problems as considered in this paper could prove to be useful in dealing with some classes of constrained optimization, too, like the ones that appear in machine learning. Actually, given the fact that the algorithm we propose is able to successfully solve location optimization problems with large data sets in high dimensions faster than its counterparts from the literature makes us confident regarding a future usage of this technique on big data problems arising in machine learning, for instance those approached by means of support vector techniques.…”

“…One of these concrete examples was numerically solved in [13,17] by means of a subgradient method and a comparison of the computational results is provided as well, stressing once again the superiority of the algorithm proposed in the present paper. Another comparison is made with the log-exponential smoothing accelerated gradient method proposed in [1] in several examples, one of them including a large data set in high dimensions and our method turns out again to converge faster towards the optimal solution of the considered location problem.…”