In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions—a wide class of functions which includes the additive and convex composite classes. At a high-level, the method is an inexact proximal point iteration in which the strongly convex proximal subproblems are quickly solved with a specialized stochast...

In this paper we address the problem of multi-agent optimization for convex functions expressible as sums of convex functions. Each agent has access to only one function in the sum and can use only local information to update its current estimate of the optimal solution. We consider two consensus-based iterative algorithms, based on a combination between a consensus step and a subgradient decen...

We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradientbased learning. Metaphorically, the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features. Our paradigm stems from recent advances in stochastic optimization and on...

In this paper we propose a purely distributed dynamic network routing algorithm that simultaneously regulates queue sizes across the network. The algorithm is distributed since each node decides on its outgoing link flows based only on its own and its immediate neighbors’ information. Therefore, this routing method will be adaptive and robust to changes in the network topology, such as the node...

We introduce a novel subgradient optimization-based framework for iterative peak-to-average power ratio (PAR) reduction for multicarrier systems, such as wireless orthogonal frequency division multiplexing (OFDM) and wireline discrete multitone (DMT) very high-speed digital subscriber line (DMT-VDSL) systems. The proposed approach uses reserved or unused tones to minimize the peak magnitude of ...

We study primal solutions obtained as a by-product of subgradient methods when solving the Lagrangian dual of a primal convex constrained optimization problem (possibly nonsmooth). The existing literature on the use of subgradient methods for generating primal optimal solutions is limited to the methods producing such solutions only asymptotically (i.e., in the limit as the number of subgradien...

Time series averaging in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems. This article proposes and analyzes subgradient methods for the problem of finding a sample mean in DTW spaces. The class of subgradient methods generalizes existing sample mean algorithms such as DTW Barycenter Averaging (DBA). We show that DBA is a majorize-minimize ...

In this lecture, we discuss first order methods for the minimization of convex functions. We focus almost exclusively on subgradient-based methods, which are essentially universally applicable for convex optimization problems, because they rely very little on the structure of the problem being solved. This leads to effective but slow algorithms in classical optimization problems, however, in la...

Bundle methods have been used frequently to solve nonsmooth optimization problems. In these methods, subgradient directions from past iterations are accumulated in a bundle, and a trial direction is obtained by performing quadratic programming based on the information contained in the bundle. A line search is then performed along the trial direction, generating a serious step if the function va...