203 research outputs found
Idempotent structures in optimization
Consider the set A = R āŖ {+ā} with the binary operations o1 = max
and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries
in A. Let the generalised sum u o1 v of two vectors denote the vector with
entries uj o1 vj , and the product a o2 v of an element a ā A and a vector
v ā An denote the vector with the entries a o2 vj . With these operations,
the set An provides the simplest example of an idempotent semimodule.
The study of idempotent semimodules and their morphisms is the subject
of idempotent linear algebra, which has been developing for about
40 years already as a useful tool in a number of problems of discrete optimisation.
Idempotent analysis studies infinite dimensional idempotent
semimodules and is aimed at the applications to the optimisations problems
with general (not necessarily finite) state spaces. We review here
the main facts of idempotent analysis and its major areas of applications
in optimisation theory, namely in multicriteria optimisation, in turnpike
theory and mathematical economics, in the theory of generalised solutions
of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games
and controlled Marcov processes, in financial mathematics
Corruption and botnet defense : a mean field game approach
Recently developed toy models for the mean-field games of corruption and botnet defence in cyber-security with three or four states of agents are extended to a more general mean-field-game model with 2d states, dāN . In order to tackle new technical difficulties arising from a larger state-space we introduce new asymptotic regimes, namely small discount and small interaction asymptotics. Moreover, the link between stationary and time-dependent solutions is established rigorously leading to a performance of the turnpike theory in a mean-field-game setting
On the relationship between stochastic turnpike and dissipativity notions
In this paper, we introduce and study different dissipativity notions and
different turnpike properties for discrete-time stochastic nonlinear optimal
control problems. The proposed stochastic dissipativity notions extend the
classic notion of Jan C. Willems to random variables and to probability
measures. Our stochastic turnpike properties range from a formulation for
random variables via turnpike phenomena in probability and in probability
measures to the turnpike property for the moments. Moreover, we investigate how
different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in
the analysis. Our results are built upon stationarity concepts in distribution
and in random variables and on the formulation of the stochastic optimal
control problem as a finite-horizon Markov decision process. We investigate how
the proposed dissipativity notions connect to the various stochastic turnpike
properties and we work out the link between these two different forms of
dissipativity
Pathwise turnpike and dissipativity results for discrete-time stochastic linear-quadratic optimal control problems
We investigate pathwise turnpike behavior of discrete-time stochastic
linear-quadratic optimal control problems. Our analysis is based on a novel
strict dissipativity notion for such problems, in which a stationary stochastic
process replaces the optimal steady state of the deterministic setting. The
analytical findings are illustrated by a numerical example
Euler-Lagrange equations of stochastic differential games: application to a game of a productive asset
This paper analyzes a noncooperative and symmetric dynamic game where players have free access to a productive asset whose evolution is a diffusion process with Brownian uncertainty. A Euler-Lagrange equation is found and used to provide necessary and sufficient conditions for the existence and uniqueness of a smooth Markov Perfect Nash Equilibrium. The Euler-Lagrange equation also provides a stochastic Keynes-Ramsey rule, which has the form of a forward-backward stochastic differential equation. It is used to study the properties of the equilibrium and to make some comparative statics exercises.The authors were also supported by the Spanish Ministerio de Ciencia e InnovaciĆ³n under Projects ECO2008-02358 and ECO2011-24200, and first author from Junta de Castilla y LeĆ³n under Project VA056A09
Turnpike and dissipativity in generalized discrete-time stochastic linear-quadratic optimal control
We investigate different turnpike phenomena of generalized discrete-time
stochastic linear-quadratic optimal control problems. Our analysis is based on
a novel strict dissipativity notion for such problems, in which a stationary
stochastic process replaces the optimal steady state of the deterministic
setting. We show that from this time-varying dissipativity notion, we can
conclude turnpike behaviors concerning different objects like distributions,
moments, or sample paths of the stochastic system and that the distributions of
the stationary pair can be characterized by a stationary optimization problem.
The analytical findings are illustrated by numerical simulations
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