56,716 research outputs found
Branch-and-Prune Search Strategies for Numerical Constraint Solving
When solving numerical constraints such as nonlinear equations and
inequalities, solvers often exploit pruning techniques, which remove redundant
value combinations from the domains of variables, at pruning steps. To find the
complete solution set, most of these solvers alternate the pruning steps with
branching steps, which split each problem into subproblems. This forms the
so-called branch-and-prune framework, well known among the approaches for
solving numerical constraints. The basic branch-and-prune search strategy that
uses domain bisections in place of the branching steps is called the bisection
search. In general, the bisection search works well in case (i) the solutions
are isolated, but it can be improved further in case (ii) there are continuums
of solutions (this often occurs when inequalities are involved). In this paper,
we propose a new branch-and-prune search strategy along with several variants,
which not only allow yielding better branching decisions in the latter case,
but also work as well as the bisection search does in the former case. These
new search algorithms enable us to employ various pruning techniques in the
construction of inner and outer approximations of the solution set. Our
experiments show that these algorithms speed up the solving process often by
one order of magnitude or more when solving problems with continuums of
solutions, while keeping the same performance as the bisection search when the
solutions are isolated.Comment: 43 pages, 11 figure
Performance guarantees for model-based Approximate Dynamic Programming in continuous spaces
We study both the value function and Q-function formulation of the Linear
Programming approach to Approximate Dynamic Programming. The approach is
model-based and optimizes over a restricted function space to approximate the
value function or Q-function. Working in the discrete time, continuous space
setting, we provide guarantees for the fitting error and online performance of
the policy. In particular, the online performance guarantee is obtained by
analyzing an iterated version of the greedy policy, and the fitting error
guarantee by analyzing an iterated version of the Bellman inequality. These
guarantees complement the existing bounds that appear in the literature. The
Q-function formulation offers benefits, for example, in decentralized
controller design, however it can lead to computationally demanding
optimization problems. To alleviate this drawback, we provide a condition that
simplifies the formulation, resulting in improved computational times.Comment: 18 pages, 5 figures, journal pape
Optimal control in Markov decision processes via distributed optimization
Optimal control synthesis in stochastic systems with respect to quantitative
temporal logic constraints can be formulated as linear programming problems.
However, centralized synthesis algorithms do not scale to many practical
systems. To tackle this issue, we propose a decomposition-based distributed
synthesis algorithm. By decomposing a large-scale stochastic system modeled as
a Markov decision process into a collection of interacting sub-systems, the
original control problem is formulated as a linear programming problem with a
sparse constraint matrix, which can be solved through distributed optimization
methods. Additionally, we propose a decomposition algorithm which automatically
exploits, if exists, the modular structure in a given large-scale system. We
illustrate the proposed methods through robotic motion planning examples.Comment: 8 pages, 5 figures, submitted to CDC 2015 conferenc
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
Open quantum systems are harder to track than open classical systems
For a Markovian open quantum system it is possible, by continuously
monitoring the environment, to know the stochastically evolving pure state of
the system without altering the master equation. In general, even for a system
with a finite Hilbert space dimension , the pure state trajectory will
explore an infinite number of points in Hilbert space, meaning that the
dimension of the classical memory required for the tracking is infinite.
However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed
that tracking of a qubit () is always possible with a bit (), and
gave a heuristic argument implying that a finite should be sufficient for
any , although beyond it would be necessary to have . Our paper
is concerned with rigorously investigating the relationship between and
, the smallest feasible . We confirm the long-standing
conjecture of Karasik and Wiseman that, for generic systems with , , by a computational proof (via Hilbert Nullstellensatz certificates of
infeasibility). That is, beyond , -dimensional open quantum systems are
provably harder to track than -dimensional open classical systems. Moreover,
we develop, and better justify, a new heuristic to guide our expectation of
as a function of , taking into account the number of
Lindblad operators as well as symmetries in the problem. The use of invariant
subspace and Wigner symmetries makes it tractable to conduct a numerical
search, using the method of polynomial homotopy continuation, to find finite
physically realizable ensembles (as they are known) in . The results of
this search support our heuristic. We thus have confidence in the most
interesting feature of our heuristic: in the absence of symmetries, , implying a quadratic gap between the classical and quantum
tracking problems.Comment: 35 pages, 3 figures, Accepted in Quantum Journal, minor change
A Combined Approach for Constraints over Finite Domains and Arrays
Arrays are ubiquitous in the context of software verification. However,
effective reasoning over arrays is still rare in CP, as local reasoning is
dramatically ill-conditioned for constraints over arrays. In this paper, we
propose an approach combining both global symbolic reasoning and local
consistency filtering in order to solve constraint systems involving arrays
(with accesses, updates and size constraints) and finite-domain constraints
over their elements and indexes. Our approach, named FDCC, is based on a
combination of a congruence closure algorithm for the standard theory of arrays
and a CP solver over finite domains. The tricky part of the work lies in the
bi-directional communication mechanism between both solvers. We identify the
significant information to share, and design ways to master the communication
overhead. Experiments on random instances show that FDCC solves more formulas
than any portfolio combination of the two solvers taken in isolation, while
overhead is kept reasonable
A New Distributed DC-Programming Method and its Applications
We propose a novel decomposition framework for the distributed optimization
of Difference Convex (DC)-type nonseparable sum-utility functions subject to
coupling convex constraints. A major contribution of the paper is to develop
for the first time a class of (inexact) best-response-like algorithms with
provable convergence, where a suitably convexified version of the original DC
program is iteratively solved. The main feature of the proposed successive
convex approximation method is its decomposability structure across the users,
which leads naturally to distributed algorithms in the primal and/or dual
domain. The proposed framework is applicable to a variety of multiuser DC
problems in different areas, ranging from signal processing, to communications
and networking. As a case study, in the second part of the paper we focus on
two examples, namely: i) a novel resource allocation problem in the emerging
area of cooperative physical layer security; ii) and the renowned sum-rate
maximization of MIMO Cognitive Radio networks. Our contribution in this context
is to devise a class of easy-to-implement distributed algorithms with provable
convergence to stationary solution of such problems. Numerical results show
that the proposed distributed schemes reach performance close to (and sometimes
better than) that of centralized methods.Comment: submitted to IEEE Transactions on Signal Processin
Monte-Carlo optimizations for resource allocation problems in stochastic network systems
Real-world distributed systems and networks are often unreliable and subject
to random failures of its components. Such a stochastic behavior affects
adversely the complexity of optimization tasks performed routinely upon such
systems, in particular, various resource allocation tasks. In this work we
investigate and develop Monte Carlo solutions for a class of two-stage
optimization problems in stochastic networks in which the expected value of
resource allocations before and after stochastic failures needs to be
optimized. The limitation of these problems is that their exact solutions are
exponential in the number of unreliable network components: thus, exact methods
do not scale-up well to large networks often seen in practice. We first prove
that Monte Carlo optimization methods can overcome the exponential bottleneck
of exact methods. Next we support our theoretical findings on resource
allocation experiments and show a very good scale-up potential of the new
methods to large stochastic networks.Comment: Appears in Proceedings of the Nineteenth Conference on Uncertainty in
Artificial Intelligence (UAI2003
A Comparison of Logic Programming Approaches for Representation and Solving of Constraint Satisfaction Problems
Many logic programming based approaches can be used to describe and solve
combinatorial search problems. On the one hand there are definite programs and
constraint logic programs that compute a solution as an answer substitution to
a query containing the variables of the constraint satisfaction problem. On the
other hand there are approaches based on stable model semantics, abduction, and
first-order logic model generation that compute solutions as models of some
theory. This paper compares these different approaches from point of view of
knowledge representation (how declarative are the programs) and from point of
view of performance (how good are they at solving typical problems).Comment: 9 pages, 3 figures submitted to NMR 2000, April 9-11, Breckenridge,
Colorad
Propagation by Selective Initialization and Its Application to Numerical Constraint Satisfaction Problems
Numerical analysis has no satisfactory method for the more realistic
optimization models. However, with constraint programming one can compute a
cover for the solution set to arbitrarily close approximation. Because the use
of constraint propagation for composite arithmetic expressions is
computationally expensive, consistency is computed with interval arithmetic. In
this paper we present theorems that support, selective initialization, a simple
modification of constraint propagation that allows composite arithmetic
expressions to be handled efficiently
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