5,328 research outputs found
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
Solving Factored MDPs with Hybrid State and Action Variables
Efficient representations and solutions for large decision problems with
continuous and discrete variables are among the most important challenges faced
by the designers of automated decision support systems. In this paper, we
describe a novel hybrid factored Markov decision process (MDP) model that
allows for a compact representation of these problems, and a new hybrid
approximate linear programming (HALP) framework that permits their efficient
solutions. The central idea of HALP is to approximate the optimal value
function by a linear combination of basis functions and optimize its weights by
linear programming. We analyze both theoretical and computational aspects of
this approach, and demonstrate its scale-up potential on several hybrid
optimization problems
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
Perron vector optimization applied to search engines
In the last years, Google's PageRank optimization problems have been
extensively studied. In that case, the ranking is given by the invariant
measure of a stochastic matrix. In this paper, we consider the more general
situation in which the ranking is determined by the Perron eigenvector of a
nonnegative, but not necessarily stochastic, matrix, in order to cover
Kleinberg's HITS algorithm. We also give some results for Tomlin's HOTS
algorithm. The problem consists then in finding an optimal outlink strategy
subject to design constraints and for a given search engine.
We study the relaxed versions of these problems, which means that we should
accept weighted hyperlinks. We provide an efficient algorithm for the
computation of the matrix of partial derivatives of the criterion, that uses
the low rank property of this matrix. We give a scalable algorithm that couples
gradient and power iterations and gives a local minimum of the Perron vector
optimization problem. We prove convergence by considering it as an approximate
gradient method.
We then show that optimal linkage stategies of HITS and HOTS optimization
problems verify a threshold property. We report numerical results on fragments
of the real web graph for these search engine optimization problems.Comment: 28 pages, 5 figure
Ergodic Control and Polyhedral approaches to PageRank Optimization
We study a general class of PageRank optimization problems which consist in
finding an optimal outlink strategy for a web site subject to design
constraints. We consider both a continuous problem, in which one can choose the
intensity of a link, and a discrete one, in which in each page, there are
obligatory links, facultative links and forbidden links. We show that the
continuous problem, as well as its discrete variant when there are no
constraints coupling different pages, can both be modeled by constrained Markov
decision processes with ergodic reward, in which the webmaster determines the
transition probabilities of websurfers. Although the number of actions turns
out to be exponential, we show that an associated polytope of transition
measures has a concise representation, from which we deduce that the continuous
problem is solvable in polynomial time, and that the same is true for the
discrete problem when there are no coupling constraints. We also provide
efficient algorithms, adapted to very large networks. Then, we investigate the
qualitative features of optimal outlink strategies, and identify in particular
assumptions under which there exists a "master" page to which all controlled
pages should point. We report numerical results on fragments of the real web
graph.Comment: 39 page
From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming
We consider linear programming (LP) problems in infinite dimensional spaces
that are in general computationally intractable. Under suitable assumptions, we
develop an approximation bridge from the infinite-dimensional LP to tractable
finite convex programs in which the performance of the approximation is
quantified explicitly. To this end, we adopt the recent developments in two
areas of randomized optimization and first order methods, leading to a priori
as well as a posterior performance guarantees. We illustrate the generality and
implications of our theoretical results in the special case of the long-run
average cost and discounted cost optimal control problems for Markov decision
processes on Borel spaces. The applicability of the theoretical results is
demonstrated through a constrained linear quadratic optimal control problem and
a fisheries management problem.Comment: 30 pages, 5 figure
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