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This paper proposes a set of novel optimization algorithms for solving a class of convex optimization problems with time-varying streaming cost functions. We develop an approach to track the optimal solution with a bounded error. Unlike prior work, our algorithm is executed only by using the first-order derivatives of the cost function, which makes it computationally efficient for optimization with time-varying cost function. We compare our algorithms to the gradient descent algorithm and show why gradient descent is not an effective solution for optimization problems with time-varying cost. Several examples, including solving a model predictive control problem cast as a convex optimization problem with a streaming time-varying cost function, demonstrate our results. 

We propose a continuous-time second-order optimization algorithm for solving unconstrained convex optimization problems with bounded Hessian. We show that this alternative algorithm has a comparable convergence rate to that of the continuous-time Newton–Raphson method, however structurally, it is amenable to a more efficient distributed implementation. We present a distributed implementation of our proposed optimization algorithm and prove its convergence via Lyapunov analysis. A numerical example demonstrates our results. 

This paper is a comprehensive study of a long-observed phenomenon of an increase in the stability margin and so the rate of convergence of a class of linear systems due to time delay. We use Lambert W function to determine (a) in what systems the delay can lead to increase in the rate of convergence, (b) the exact range of time delay for which the rate of convergence is greater than that of the delay-free system, and (c) an estimate on the value of the delay that leads to the maximum rate of convergence. For the special case when the system matrix eigenvalues are all negative real numbers, we expand our results to show that the rate of convergence in the presence of delay depends only on the eigenvalues with minimum and maximum real parts. Moreover, we determine the exact value of the maximum rate of convergence and the corresponding maximizing time delay. We demonstrate our results through a numerical example on the practical application in accelerating an agreement algorithm for networked~systems by use of delayed feedback. 

This paper examines accelerating the well-known Laplacian average consensus algorithm by breaking its conventional delay-free input into two weighted parts and replacing one of these parts by an outdated feedback. We determine for what weighted sum there exists a range of time delay that leads to an increase in the rate of convergence of the algorithm. For such weights, using the Lambert W function, we obtain the rate increasing range of the time delay and also the maximum reachable rate and its corresponding maximizer delay. We also specify what combinations of the current and an outdated feedback increase the rate of convergence without increasing the control effort of the agents. Lastly, we determine the optimum combination of the current and the outdated feedback weights to achieve maximum increase in the rate of convergence without increasing the control effort. We demonstrate our results through a numerical example.