Total variation on a tree

We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the non-convex case and derive worst case complexities that are equal or better than existing methods. We show applications to total variation based 2D image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very efficient, offer a high degree of parallelism and come along with memory requirements which are only in the order of the number of image pixels.Comment: accepted to SIAM Journal on Imaging Sciences (SIIMS

Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization

In this paper, we present a new stochastic algorithm, namely the stochastic block mirror descent (SBMD) method for solving large-scale nonsmooth and stochastic optimization problems. The basic idea of this algorithm is to incorporate the block-coordinate decomposition and an incremental block averaging scheme into the classic (stochastic) mirror-descent method, in order to significantly reduce the cost per iteration of the latter algorithm. We establish the rate of convergence of the SBMD method along with its associated large-deviation results for solving general nonsmooth and stochastic optimization problems. We also introduce different variants of this method and establish their rate of convergence for solving strongly convex, smooth, and composite optimization problems, as well as certain nonconvex optimization problems. To the best of our knowledge, all these developments related to the SBMD methods are new in the stochastic optimization literature. Moreover, some of our results also seem to be new for block coordinate descent methods for deterministic optimization

Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, 15(1):15-27, March 2014. This is an extended version with Appendices VIII and IX that provide some mathematical preliminaries and proofs of the main result