In this paper we solve the problem of finding a trajectory that shows that a given hybrid dynamical system with deterministic evolution leaves a given set of states considered to be safe. The algorithm combines local with global search for achieving both efficiency and global convergence. In local search, it exploits derivatives for efficient computation. Unlike other methods for falsification of hybrid systems with deterministic evolution, we do not restrict our search to trajectories of a certain bounded length but search for error trajectories of arbitrary length.
Our algorithm can be viewed as an adaptation of the Best Start two-phase method for global optimization @cite_0 to our context.
@cite_0
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We present a search-based planning approach for controlling a quadrupedal robot over rough terrain. Given a start and goal position, we consider the problem of generating a complete joint trajectory that will result in the legged robot successfully moving from the start to the goal. We decompose the problem into two main phases: an initial global planning phase, which results in a footstep trajectory; and an execution phase, which dynamically generates a joint trajectory to best execute the footstep trajectory. We show how R* search can be employed to generate high-quality global plans in the high-dimensional space of footstep trajectories. Results show that the global plans coupled with the joint controller result in a system robust enough to deal with a variety of terrains. This paper presents dynamic feedback, a technique that enables computations to adapt dynamically to different execution environments. A compiler that uses dynamic feedback produces several different versions of the same source code; each version uses a different optimization policy. The generated code alternately performs sampling phases and production phases. Each sampling phase measures the overhead of each version in the current environment. Each production phase uses the version with the least overhead in the previous sampling phase. The computation periodically resamples to adjust dynamically to changes in the environment.We have implemented dynamic feedback in the context of a parallelizing compiler for object-based programs. The generated code uses dynamic feedback to automatically choose the best synchronization optimization policy. Our experimental results show that the synchronization optimization policy has a significant impact on the overall performance of the computation, that the best policy varies from program to program, that the compiler is unable to statically choose the best policy, and that dynamic feedback enables the generated code to exhibit performance that is comparable to that of code that has been manually tuned to use the best policy. We have also performed a theoretical analysis which provides, under certain assumptions, a guaranteed optimality bound for dynamic feedback relative to a hypothetical (and unrealizable) optimal algorithm that uses the best policy at every point during the execution. Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the "high noise" regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees. The fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras, and the eight-point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches. It has the advantage of simplicity of implementation. The prevailing view is, however, that it is extremely susceptible to noise and hence virtually useless for most purposes. This paper challenges that view, by showing that by preceding the algorithm with a very simple normalization (translation and scaling) of the coordinates of the matched points, results are obtained comparable with the best iterative algorithms. This improved performance is justified by theory and verified by extensive experiments on real images.