This work proposes an accelerated first-order algorithm we call the Robust Momentum Method for optimizing smooth strongly convex functions. The algorithm has a single scalar parameter that can be tuned to trade off robustness to gradient noise versus worst-case convergence rate. At one extreme, the algorithm is faster than Nesterov's Fast Gradient Method by a constant factor but more fragile to noise. At the other extreme, the algorithm reduces to the Gradient Method and is very robust to noise. The algorithm design technique is inspired by methods from classical control theory and the resulting algorithm has a simple analytical form. Algorithm performance is verified on a series of numerical simulations in both noise-free and relative gradient noise cases.Notation. The set of functions that are m-strongly convex and L-smooth is denoted F(m, L). In particular, f ∈ F(m, L) if for all x, y ∈ R n ,The condition ratio is defined as κ := L/m.2 A numerical study in [3] revealed that the standard rate bound for FGM derived in [2] is conservative. Nevertheless, the bound has a simple algebraic form and is asymptotically tight.

Classical conditions for ensuring the robust stability of a linear system in feedback with a sector-bounded nonlinearity include small gain, circle, passivity, and conicity theorems. In this work, we present a similar stability condition, but expressed in terms of relations defined on a general semi-inner product space. This increased generality leads to a clean result that can be specialized in a variety of ways. First, we show how to recover both sufficient and necessary-and-sufficient versions of the aforementioned classical results. Second, we show that suitably choosing the semi-inner product space leads to a new necessary and sufficient condition for weighted stability, which is in turn sufficient for exponential stability. Finally, in the spirit of classical robust stability analysis, we provide linear matrix inequalities that allow for the efficient verification of the conditions of our theorem.

This paper introduces locational prices associated with power injection variability. The sensitivity of uncertainty in injected power on system cost is calculated using a probabilistic DC optimal power flow (DC-OPF). This sensitivity may be used as a price for regulation to equitably allocate costs associated with tracking power injections. In this context the price also provides a locational value to energy storage and hence can be used as a tool to investigate best locations for energy storage. As a case study, a 24-bus test system with uncertain variable wind power and uncertain variable loads is analyzed.