Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows

This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differential equations. This includes Riemannian gradient flow systems which occur naturally in optimization problems. The Itoh--Abe discrete gradient is formulated and applied to gradient systems, yielding a derivative-free optimization algorithm. The algorithm is tested on two eigenvalue problems and two problems from manifold valued imaging: InSAR denoising and DTI denoising.Comment: Post-revision version. To appear in SIAM Journal on Scientific Computin

Randomized Solutions to Convex Programs with Multiple Chance Constraints

The scenario-based optimization approach (`scenario approach') provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach is that it neither assumes knowledge of the uncertainty set, as it is common in robust optimization, nor of its probability distribution, as it is usually required in stochastic optimization. Moreover, the scenario approach is computationally efficient as its solution is based on a deterministic optimization program that is canonically convex, even when the original chance-constrained problem is not. Recently, researchers have obtained theoretical foundations for the scenario approach, providing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint. However, this paper shows that these bounds can be improved in situations where the constraints have a limited `support rank', a new concept that is introduced for the first time. This property is typically found in a large number of practical applications---most importantly, if the problem originally contains multiple chance constraints (e.g. multi-stage uncertain decision problems), or if a chance constraint belongs to a special class of constraints (e.g. linear or quadratic constraints). In these cases the quality of the scenario solution is improved while the same bound on the constraint violation probability is maintained, and also the computational complexity is reduced.Comment: This manuscript is the preprint of a paper submitted to the SIAM Journal on Optimization and it is subject to SIAM copyright. SIAM maintains the sole rights of distribution or publication of the work in all forms and media. If accepted, the copy of record will be available at http://www.siam.or

Weak Dynamic Programming for Generalized State Constraints

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.Comment: 36 pages;forthcoming in 'SIAM Journal on Control and Optimization