42 research outputs found
Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings
This paper aims at providing further studies of the notion of quasi-relative
interior for convex sets introduced by Borwein and Lewis. We obtain new
formulas for representing quasi-relative interiors of convex graphs of
set-valued mappings and for convex epigraphs of extended-real-valued functions
defined on locally convex topological vector spaces. We also show that the
role, which this notion plays in infinite dimensions and the results obtained
in this vein, are similar to those involving relative interior in
finite-dimensional spaces.Comment: This submission replaces our previous version
A DC Programming Approach for Solving Multicast Network Design Problems via the Nesterov Smoothing Technique
This paper continues our effort initiated in [9] to study Multicast
Communication Networks, modeled as bilevel hierarchical clustering problems, by
using mathematical optimization techniques. Given a finite number of nodes, we
consider two different models of multicast networks by identifying a certain
number of nodes as cluster centers, and at the same time, locating a particular
node that serves as a total center so as to minimize the total transportation
cost through the network. The fact that the cluster centers and the total
center have to be among the given nodes makes this problem a discrete
optimization problem. Our approach is to reformulate the discrete problem as a
continuous one and to apply Nesterov smoothing approximation technique on the
Minkowski gauges that are used as distance measures. This approach enables us
to propose two implementable DCA-based algorithms for solving the problems.
Numerical results and practical applications are provided to illustrate our
approach