In this paper we study consensus-based optimization (CBO), which is a metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. Based on an experimentally supported intuition that CBO performs a gradient descent on the convex envelope of a given objective, we derive a novel technique for proving the convergence to the global minimizer in mean-field law for a rich class of objective functions. Our results unveil internal mechanisms of CBO that are responsible for the success of the method. Furthermore, we improve prior analyses by requiring minimal assumptions about the initialization of the method and by covering objectives that are merely locally Lipschitz continuous. As a by-product of our analysis, we establish a quantitative nonasymptotic Laplace principle, which may be of independent interest.

Typically, lane departure warning systems rely on lane lines being present on the road.However, in many scenarios, e.g., secondary roads or some streets in cities, lane lines are eithernot present or not sufficiently well signaled. In this work, we present a vision-based method tolocate a vehicle within the road when no lane lines are present using only RGB images as input.To this end, we propose to fuse together the outputs of a semantic segmentation and a monoculardepth estimation architecture to reconstruct locally a semantic 3D point cloud of the viewed scene.We only retain points belonging to the road and, additionally, to any kind of fences or walls thatmight be present right at the sides of the road. We then compute the width of the road at a certainpoint on the planned trajectory and, additionally, what we denote as the fence-to-fence distance.Our system is suited to any kind of motoring scenario and is especially useful when lane lines arenot present on the road or do not signal the path correctly. The additional fence-to-fence distancecomputation is complementary to the road’s width estimation. We quantitatively test our methodon a set of images featuring streets of the city of Munich that contain a road-fence structure, so asto compare our two proposed variants, namely the road’s width and the fence-to-fence distancecomputation. In addition, we also validate our system qualitatively on the Stuttgart sequence of thepublicly available Cityscapes dataset, where no fences or walls are present at the sides of the road,thus demonstrating that our system can be deployed in a standard city-like environment. For thebenefit of the community, we make our software open source.

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In this paper we provide a rigorous convergence analysis for the renowned Particle Swarm Optimization method using tools from stochastic calculus and the analysis of partial differential equations. Based on a time-continuous formulation of the particle dynamics as a system of stochastic differential equations, we establish the convergence to a global minimizer in two steps. First, we prove the consensus formation of the dynamics by analyzing the time-evolution of the variance of the particle distribution. Consecutively, we show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. Our results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum are satisfied. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.

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