Kamy's Website

This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank-Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an ε-suboptimal solution in \mathcalO(ε^-2) iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle (LMO) call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule.

This paper considers online convex optimization (OCO) problems where decisions are constrained by available energy resources. A key scenario is optimal power control for an energy harvesting device with a finite capacity battery. The goal is to minimize a time-average loss function while keeping the used energy less than what is available. In this setup, the distribution of the randomly arriving harvestable energy (which is assumed to be i.i.d.) is unknown, the current loss function is unknown, and the controller is only informed by the history of past observations. A prior algorithm is known to achieve O(sqrtT ) regret by using a battery with an O(sqrtT ) capacity. This paper develops a new algorithm that maintains this asymptotic trade-off with the number of time steps T while improving dependency on the dimension of the decision vector from O(sqrtn ) to O(sqrtłog(n) ). The proposed algorithm introduces a separation of the decision vector into amplitude and direction components. It uses two distinct types of Bregman divergence, together with energy queue information, to make decisions for each component.