32,184 research outputs found
Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo
This paper presents a detailed theoretical analysis of the Langevin Monte
Carlo sampling algorithm recently introduced in Durmus et al. (Efficient
Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets
Moreau, 2016) when applied to log-concave probability distributions that are
restricted to a convex body . This method relies on a
regularisation procedure involving the Moreau-Yosida envelope of the indicator
function associated with . Explicit convergence bounds in total
variation norm and in Wasserstein distance of order are established. In
particular, we show that the complexity of this algorithm given a first order
oracle is polynomial in the dimension of the state space. Finally, some
numerical experiments are presented to compare our method with competing MCMC
approaches from the literature
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
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