1,557 research outputs found
Stampacchia's property, self-duality and orthogonality relations
We show that if the conclusion of the well known Stampacchia Theorem, on
variational inequalities, holds on a Banach space X, then X is isomorphic to a
Hilbert space. Motivated by this we obtain a relevant result concerning
self-dual Banach spaces and investigate some connections between existing
notions of orthogonality and self-duality. Moreover, we revisit the notion of
the cosine of a linear operator and show that it can be used to characterize
Hilbert space structure. Finally, we present some consequences of our results
to quadratic forms and to evolution triples
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
Generalizations of the Lax-Milgram theorem
We prove a linear and a nonlinear generalization of the Lax-Milgram theorem.
In particular we give sufficient conditions for a real-valued function defined
on the product of a reflexive Banach space and a normed space to represent all
bounded linear functionals of the latter. We also give two applications to
singular differential equations
Subspaces with a common complement in a Banach space
We study the problem of the existence of a common algebraic complement for a
pair of closed subspaces of a Banach space. We prove the following two
characterizations: (1) The pairs of subspaces of a Banach space with a common
complement coincide with those pairs which are isomorphic to a pair of graphs
of bounded linear operators between two other Banach spaces. (2) The pairs of
subspaces of a Banach space X with a common complement coincide with those
pairs for which there exists an involution S on X exchanging the two subspaces,
such that I+S is bounded from below on their union. Moreover we show that, in a
separable Hilbert space, the only pairs of subspaces with a common complement
are those which are either equivalently positioned or not completely asymptotic
to one another. We also obtain characterizations for the existence of a common
complement for subspaces with closed sum
The Complexity of Non-Monotone Markets
We introduce the notion of non-monotone utilities, which covers a wide
variety of utility functions in economic theory. We then prove that it is
PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets
with linear and non-monotone utilities. Building on this result, we settle the
long-standing open problem regarding the computation of an approximate
Arrow-Debreu market equilibrium in markets with CES utility functions, by
proving that it is PPAD-complete when the Constant Elasticity of Substitution
parameter \rho is any constant less than -1
Polynomial Time Algorithms for Multi-Type Branching Processes and Stochastic Context-Free Grammars
We show that one can approximate the least fixed point solution for a
multivariate system of monotone probabilistic polynomial equations in time
polynomial in both the encoding size of the system of equations and in
log(1/\epsilon), where \epsilon > 0 is the desired additive error bound of the
solution. (The model of computation is the standard Turing machine model.)
We use this result to resolve several open problems regarding the
computational complexity of computing key quantities associated with some
classic and heavily studied stochastic processes, including multi-type
branching processes and stochastic context-free grammars
Recursive Concurrent Stochastic Games
We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent
analysis of recursive simple stochastic games to a concurrent setting where the
two players choose moves simultaneously and independently at each state. For
multi-exit games, our earlier work already showed undecidability for basic
questions like termination, thus we focus on the important case of single-exit
RCSGs (1-RCSGs).
We first characterize the value of a 1-RCSG termination game as the least
fixed point solution of a system of nonlinear minimax functional equations, and
use it to show PSPACE decidability for the quantitative termination problem. We
then give a strategy improvement technique, which we use to show that player 1
(maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM)
strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM
strategies. Thus, such games are r-SM-determined. These results mirror and
generalize in a strong sense the randomized memoryless determinacy results for
finite stochastic games, and extend the classic Hoffman-Karp strategy
improvement approach from the finite to an infinite state setting. The proofs
in our infinite-state setting are very different however, relying on subtle
analytic properties of certain power series that arise from studying 1-RCSGs.
We show that our upper bounds, even for qualitative (probability 1)
termination, can not be improved, even to NP, without a major breakthrough, by
giving two reductions: first a P-time reduction from the long-standing
square-root sum problem to the quantitative termination decision problem for
finite concurrent stochastic games, and then a P-time reduction from the latter
problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure
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