18,006 research outputs found
Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?
In the literature, necessary and sufficient conditions in terms of
variational inequalities are introduced to characterize minimizers of convex
set valued functions with values in a conlinear space. Similar results are
proved for a weaker concept of minimizers and weaker variational inequalities.
The implications are proved using scalarization techniques that eventually
provide original problems, not fully equivalent to the set-valued counterparts.
Therefore, we try, in the course of this note, to close the network among the
various notions proposed. More specifically, we prove that a minimizer is
always a weak minimizer, and a solution to the stronger variational inequality
always also a solution to the weak variational inequality of the same type. As
a special case we obtain a complete characterization of efficiency and weak
efficiency in vector optimization by set-valued variational inequalities and
their scalarizations. Indeed this might eventually prove the usefulness of the
set-optimization approach to renew the study of vector optimization
Variational inequalities characterizing weak minimality in set optimization
We introduce the notion of weak minimizer in set optimization. Necessary and
sufficient conditions in terms of scalarized variational inequalities of
Stampacchia and Minty type, respectively, are proved. As an application, we
obtain necessary and sufficient optimality conditions for weak efficiency of
vector optimization in infinite dimensional spaces. A Minty variational
principle in this framework is proved as a corollary of our main result.Comment: Includes an appendix summarizing results which are submitted but not
published at this poin
Algorithms for Stochastic Games on Interference Channels
We consider a wireless channel shared by multiple transmitter-receiver pairs.
Their transmissions interfere with each other. Each transmitter-receiver pair
aims to maximize its long-term average transmission rate subject to an average
power constraint. This scenario is modeled as a stochastic game. We provide
sufficient conditions for existence and uniqueness of a Nash equilibrium (NE).
We then formulate the problem of finding NE as a variational inequality (VI)
problem and present an algorithm to solve the VI using regularization. We also
provide distributed algorithms to compute Pareto optimal solutions for the
proposed game
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