8,491 research outputs found
Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control
This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York
Implicit Langevin Algorithms for Sampling From Log-concave Densities
For sampling from a log-concave density, we study implicit integrators
resulting from -method discretization of the overdamped Langevin
diffusion stochastic differential equation. Theoretical and algorithmic
properties of the resulting sampling methods for and a
range of step sizes are established. Our results generalize and extend prior
works in several directions. In particular, for , we prove
geometric ergodicity and stability of the resulting methods for all step sizes.
We show that obtaining subsequent samples amounts to solving a strongly-convex
optimization problem, which is readily achievable using one of numerous
existing methods. Numerical examples supporting our theoretical analysis are
also presented
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