Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for several first-order algorithms on Hadamard manifolds. Specifically, we prove ...

This paper introduces the paradigm of optimization under uncertainty for modelling and solving matrix nearness problems. In particular, it considers the concrete problem of recovering correlation matrices from uncertain observations by introducing two different approaches to tackling uncertainty. The first approach invokes the framework of robust optimization to construct low error solutions th...

We develop a uniied framework for convergence analysis of subgradient and subgradient projection methods for minimization of nonsmooth convex functionals in Banach spaces. The important novel features of our analysis are that we neither assume that the functional is uniformly or strongly convex, nor use regularization techniques. Moreover, no boundedness assumptions are made on the level sets o...

Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimiza...

We consider the nonsmooth convex composition optimization problem where the objective isa composition of two finite-sum functions and analyze stochastic compositional variance reducedgradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much atten-tion given their edge over stochastic compositional gradient descent (SCGD); but the theoreticalanalysis ...

Stochastic coordinate descent, due to its practicality and efficiency, is increasingly popular in machine learning and signal processing communities as it has proven successful in several large-scale optimization problems , such as l1 regularized regression, Support Vector Machine, to name a few. In this paper, we consider a composite problem where the nonsmoothness has a general structure that...

An algorithm is presented for topology selection in graphical models of autoregressive Gaussian time series. The graph topology of the model represents the sparsity pattern of the inverse spectrum of the time series and characterizes conditional independence relations between the variables. The method proposed in the paper is based on an l1-type nonsmooth regularization of the conditional maxim...