Nonsmooth Approaches Research Articles

Exponentially Convergent Algorithm Design for Constrained Distributed Optimization via Nonsmooth Approach

We develop an exponentially convergent distributed algorithm to minimize a sum of nonsmooth cost functions with a set constraint. The set constraint generally leads to the nonlinearity in distributed algorithms, and results in difficulties to derive an exponential rate. In this article, we remove the consensus constraints by an exact penalty method, and then propose a distributed projected subgradient algorithm by virtue of a differential inclusion and a differentiated projection operator. Resorting to nonsmooth approaches, we prove the convergence for this algorithm, and moreover, provide both the sublinear and exponential rates under some mild assumptions.

Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning

Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic differentiation oracles, as for instance the famous backpropagation algorithm in deep learning. Our differential model is applied to establish the convergence in values of nonsmooth stochastic gradient methods as they are implemented in practice.

A reformulation of the nonsmooth approaches of M. Frémond and J.J. Moreau for rigid body dynamics

This paper deals with the nonsmooth dynamics of a rigid bodies system. The proposed theory is inspired by the formalism of J.J. Moreau and that of M. Frémond and relies on the notion of percussion which is the integral of the contact force during the duration of the collision. Contrary to classical discrete element models, it is here assumed that percussions can be expressed as a function of only the velocity before the impact. This assumption is checked for the usual mechanical constitutive laws for collisions derived from a pseudopotential of dissipation or the Coulomb friction law. Motion equations are then reformulated taking into account simultaneous collisions of solids. A mathematical study of the new model is presented: the existence and uniqueness of the solution are discussed according to the regularity of both the forces (Lebesgue‐density occurring during the regular evolution of the system) and the percussions (Dirac‐density describing the collision). In the light of the principles of thermodynamics, a condition on the internal percussion assuring that the collision is thermodynamically admissible, is established. Finally, an application of this new model to the motion of a system of rigid disks, including simultaneous collisions is presented.

Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time

While non-smooth approaches (including sliding mode control) provide explicit feedback laws that ensure finite-time stabilization but in terminal time that depends on the initial condition, fixed-time optimal control with a terminal constraint ensures regulation in prescribed time but lacks the explicit character in the presence of nonlinearities and uncertainties. In this paper we present an alternative to these approaches, which, while lacking optimality, provides explicit time-varying feedback laws that achieve regulation in prescribed finite time, even in the presence of non-vanishing (though matched) uncertain nonlinearities. Our approach employs a scaling of the state by a function of time that grows unbounded towards the terminal time and is followed by a design of a controller that stabilizes the system in the scaled state representation, yielding regulation in prescribed finite time for the original state. The achieved robustness to right-hand-side disturbances is not accompanied by robustness to measurement noise, which is also absent from all controllers that are nonsmooth or discontinuous at the origin.

Examining the smooth and nonsmooth discrete element approaches to granular matter

SUMMARYThe smooth and nonsmooth approaches to the discrete element method (DEM) are examined from a computational perspective. The main difference can be understood as using explicit versus implicit time integration. A formula is obtained for estimating the computational effort depending on error tolerance, system geometric shape and size, and on the dynamic state. For the nonsmooth DEM (NDEM), a regularized version mapping to the Hertz contact law is presented. This method has the conventional nonsmooth and smooth DEM as special cases depending on size of time step and value of regularization. The use of the projected Gauss‐Seidel solver for NDEM simulation is studied on a range of test systems. The following characteristics are found. First, the smooth DEM is computationally more efficient for soft materials, wide and tall systems, and with increasing flow rate. Secondly, the NDEM is more beneficial for stiff materials, shallow systems, static or slow flow, and with increasing error tolerance. Furthermore, it is found that just as pressure saturates with depth in a granular column, due to force arching, also the required number of iterations saturates and become independent of system size. This effect make the projected Gauss‐Seidel solver scale much better than previously thought. Copyright © 2014 John Wiley & Sons, Ltd.

Smooth/non-smooth contact modeling of human crowds movement: numerical aspects and application to emergency evacuations

The aim of this paper is to develop on discrete models that reproduce the behavior of a crowd of people in several emergency evacuation situations. The first step in this study is to determine how to treat contacts between pedestrians. For that, three already existing discrete approaches, one smooth and two non-smooth, originally proposed to simulate the collisions of granular assemblies, are first analyzed both from the theoretical and the numerical point of view. The solving algorithms are presented and the numerical formulation of the two non-smooth approaches is compared to standard plasticity in order to point out the common theoretical framework. The next step is to adapt these discrete approaches to represent pedestrians. The key point is to introduce a “willingness” for each particle through a specific desired velocity. These adapted discrete approaches are able to handle local interactions, like pedestrian-pedestrian or pedestrian-obstacle contacts, in order to reproduce the global dynamic of pedestrian traffic. Finally, results of several simulations in emergency configurations are presented as well as compared to real exercise ones.

The smooth Colonel meets the Reverend

Kernel smoothing techniques have attracted much attention and some notoriety in recent years. The attention is well deserved as kernel methods free researchers from having to impose rigid parametric structure on their data. The notoriety arises from the fact that the amount of smoothing (i.e., local averaging) that is appropriate for the problem at hand is under the control of the researcher. In this study we provide a deeper understanding of kernel smoothing methods for discrete data by leveraging the unexplored links between hierarchical Bayes models and kernel methods for discrete processes. Several potentially useful results are thereby obtained, including bounds on when kernel smoothing can be expected to dominate non-smooth (e.g., parametric) approaches in mean squared error and suggestions for thinking about the appropriate amount of smoothing.

Nonsmooth optimization approaches to VDA of models with on/off parameterizations: Theoretical issues

Some variational data assimilation problems of time-and space-discrete models with on/ off parameterizations can be regarded as nonsmooth optimization problems. Some theoretical issues related to those problems is systematically addressed. One of the basic concept in nonsmooth optimization is subgradient, a generalized notation of a gradient of the cost function. First it is shown that the concept of subgradient leads to a clear definition of the adjoint variables in the conventional adjoint model at singular points caused by on/ off switches. Using an illustrated example of a multi-layer diffusion model with the convective adjustment, it is proved that the solution of the conventional adjoint model can not be inter-preted as Gateaux derivatives or directional derivatives, at singular points, but can be interpreted as a subgradient of the cost function.