Approximation of reachable sets using optimal control algorithms

To appearInternational audienceNumerical experiences with a method for the approximation of reachable sets of nonlinear control systems are reported. The method is based on the formulation of suitable optimal control problems with varying objective functions, whose discretization by Euler's method lead to finite dimensional non-convex nonlinear programs. These are solved by a sequential quadratic programming method. An efficient adjoint method for gradient computation is used to reduce the computational costs. The discretization of the state space is more efficiently than by usual sequential realization of Euler's method and allows adaptive calculations or refinements. The method is illustrated for two test examples. Both examples have non-linear dynamics, the first one has a convex reachable set, whereas the second one has a non-convex reachable set

Nonsmooth Lagrangian mechanics and variational collision integrators

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated

Splitting methods with variable metric for KL functions

We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satisfies the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailled

Dynamical convergence analysis for nonconvex linearized proximal ADMM algorithms

The convergence analysis of optimization algorithms using continuous-time dynamical systems has received much attention in recent years. In this paper, we investigate applications of these systems to analyze the convergence of linearized proximal ADMM algorithms for nonconvex composite optimization, whose objective function is the sum of a continuously differentiable function and a composition of a possibly nonconvex function with a linear operator. We first derive a first-order differential inclusion for the linearized proximal ADMM algorithm, LP-ADMM. Both the global convergence and the convergence rates of the generated trajectory are established with the use of Kurdyka-\L{}ojasiewicz (KL) property. Then, a stochastic variant, LP-SADMM, is delved into an investigation for finite-sum nonconvex composite problems. Under mild conditions, we obtain the stochastic differential equation corresponding to LP-SADMM, and demonstrate the almost sure global convergence of the generated trajectory by leveraging the KL property. Based on the almost sure convergence of trajectory, we construct a stochastic process that converges almost surely to an approximate critical point of objective function, and derive the expected convergence rates associated with this stochastic process. Moreover, we propose an accelerated LP-SADMM that incorporates Nesterov's acceleration technique. The continuous-time dynamical system of this algorithm is modeled as a second-order stochastic differential equation. Within the context of KL property, we explore the related almost sure convergence and expected convergence rates