612 research outputs found
2-irreducible and strongly 2-irreducible ideals of commutative rings
An ideal I of a commutative ring R is said to be irreducible if it cannot be
written as the intersection of two larger ideals. A proper ideal I of a ring R
is said to be strongly irreducible if for each ideals J, K of R, J\cap
K\subseteq I implies that J\subset I or K\subset I. In this paper, we introduce
the concepts of 2-irreducible and strongly 2-irreducible ideals which are
generalizations of irreducible and strongly irreducible ideals, respectively.
We say that a proper ideal I of a ring R is 2-irreducible if for each ideals J,
K and L of R, I= J\cap K\cap L implies that either I=J\cap K or I=J\cap L or
I=K\cap L. A proper ideal I of a ring R is called strongly 2-irreducible if for
each ideals J, K and L of R, J\cap K\cap L\subseteq I implies that either J\cap
K\subseteq I or J\cap L\subseteq I or K\cap L\subseteq I.Comment: 15 page
A distributed adaptive steplength stochastic approximation method for monotone stochastic Nash Games
We consider a distributed stochastic approximation (SA) scheme for computing
an equilibrium of a stochastic Nash game. Standard SA schemes employ
diminishing steplength sequences that are square summable but not summable.
Such requirements provide a little or no guidance for how to leverage
Lipschitzian and monotonicity properties of the problem and naive choices
generally do not preform uniformly well on a breadth of problems. While a
centralized adaptive stepsize SA scheme is proposed in [1] for the optimization
framework, such a scheme provides no freedom for the agents in choosing their
own stepsizes. Thus, a direct application of centralized stepsize schemes is
impractical in solving Nash games. Furthermore, extensions to game-theoretic
regimes where players may independently choose steplength sequences are limited
to recent work by Koshal et al. [2]. Motivated by these shortcomings, we
present a distributed algorithm in which each player updates his steplength
based on the previous steplength and some problem parameters. The steplength
rules are derived from minimizing an upper bound of the errors associated with
players' decisions. It is shown that these rules generate sequences that
converge almost surely to an equilibrium of the stochastic Nash game.
Importantly, variants of this rule are suggested where players independently
select steplength sequences while abiding by an overall coordination
requirement. Preliminary numerical results are seen to be promising.Comment: 8 pages, Proceedings of the American Control Conference, Washington,
201
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