4,972 research outputs found
on the efficiency of convex polyhedra
Abstract The domain of convex polyhedra plays a special role in the collection of numerical domains considered for program analysis and verification. As far as precision is concerned, it would be the most natural choice in many contexts but, due to its worst case exponential complexity, it is sometimes considered an unaffordable option. This has led to a systematic quest for simpler domains that are capable of reasonable precision using less computational resources. There are anyway cases where the use of the domain of convex polyhedra turns out to be feasible, also due to recent progress in their implementation. After reviewing a few known approaches to decrease the amount of resources needed when computing on this domain, we will introduce a couple of novel techniques that can be used to further improve its efficiency, without incurring precision losses
Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra in convex pieces
We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small amount of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r 2) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron
The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems
Since its inception as a student project in 2001, initially just for the
handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library
has been continuously improved and extended by joining scrupulous research on
the theoretical foundations of (possibly non-convex) numerical abstractions to
a total adherence to the best available practices in software development. Even
though it is still not fully mature and functionally complete, the Parma
Polyhedra Library already offers a combination of functionality, reliability,
usability and performance that is not matched by similar, freely available
libraries. In this paper, we present the main features of the current version
of the library, emphasizing those that distinguish it from other similar
libraries and those that are important for applications in the field of
analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table
LNCS
Template polyhedra generalize intervals and octagons to polyhedra whose facets are orthogonal to a given set of arbitrary directions. They have been employed in the abstract interpretation of programs and, with particular success, in the reachability analysis of hybrid automata. While previously, the choice of directions has been left to the user or a heuristic, we present a method for the automatic discovery of directions that generalize and eliminate spurious counterexamples. We show that for the class of convex hybrid automata, i.e., hybrid automata with (possibly nonlinear) convex constraints on derivatives, such directions always exist and can be found using convex optimization. We embed our method inside a CEGAR loop, thus enabling the time-unbounded reachability analysis of an important and richer class of hybrid automata than was previously possible. We evaluate our method on several benchmarks, demonstrating also its superior efficiency for the special case of linear hybrid automata
Experiments with a Convex Polyhedral Analysis Tool for Logic Programs
Convex polyhedral abstractions of logic programs have been found very useful
in deriving numeric relationships between program arguments in order to prove
program properties and in other areas such as termination and complexity
analysis. We present a tool for constructing polyhedral analyses of
(constraint) logic programs. The aim of the tool is to make available, with a
convenient interface, state-of-the-art techniques for polyhedral analysis such
as delayed widening, narrowing, "widening up-to", and enhanced automatic
selection of widening points. The tool is accessible on the web, permits user
programs to be uploaded and analysed, and is integrated with related program
transformations such as size abstractions and query-answer transformation. We
then report some experiments using the tool, showing how it can be conveniently
used to analyse transition systems arising from models of embedded systems, and
an emulator for a PIC microcontroller which is used for example in wearable
computing systems. We discuss issues including scalability, tradeoffs of
precision and computation time, and other program transformations that can
enhance the results of analysis.Comment: Paper presented at the 17th Workshop on Logic-based Methods in
Programming Environments (WLPE2007
Convex Hull of Planar H-Polyhedra
Suppose are planar (convex) H-polyhedra, that is, $A_i \in
\mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i =
\{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 +
n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron
with the smallest such that
Polyhedral Predictive Regions For Power System Applications
Despite substantial improvement in the development of forecasting approaches,
conditional and dynamic uncertainty estimates ought to be accommodated in
decision-making in power system operation and market, in order to yield either
cost-optimal decisions in expectation, or decision with probabilistic
guarantees. The representation of uncertainty serves as an interface between
forecasting and decision-making problems, with different approaches handling
various objects and their parameterization as input. Following substantial
developments based on scenario-based stochastic methods, robust and
chance-constrained optimization approaches have gained increasing attention.
These often rely on polyhedra as a representation of the convex envelope of
uncertainty. In the work, we aim to bridge the gap between the probabilistic
forecasting literature and such optimization approaches by generating forecasts
in the form of polyhedra with probabilistic guarantees. For that, we see
polyhedra as parameterized objects under alternative definitions (under
and norms), the parameters of which may be modelled and predicted.
We additionally discuss assessing the predictive skill of such multivariate
probabilistic forecasts. An application and related empirical investigation
results allow us to verify probabilistic calibration and predictive skills of
our polyhedra.Comment: 8 page
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