20XX

N. Bastianello, R. Carli, L. Schenato, M. Todescato.

**Asynchronous Distributed Optimization over Lossy Peer-to-Peer Networks via ADMM: Stability and Exponential Convergence.** *IEEE Trans. Automatic Control [submitted]*, 20XX

Abstract:

In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular we are interested in the case in which communications between nodes are lossy and the agents are not synchronized among themselves. We address this problem by proposing a modified version of the relaxed ADMM (R-ADMM), which corresponds to the generalized Douglas-Rachford operator applied to the dual of our problem. By exploiting results from operator theory we are then able to prove the almost sure convergence of the proposed algorithm under i.i.d. random packet losses and asynchronous operation of the agents. By further assuming the cost functions to be strongly convex, we are able to prove that the algorithm converges exponentially fast in mean square in a neighborhood of the optimal solution. Moreover, we provide an upper bound to the convergence rate. Finally, we present numerical simulations of the proposed algorithm over random geometric graphs in the aforementioned lossy and asynchronous scenario.

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N. Bastianello, A. Simonetto, R. Carli.

**Prediction-Correction Splittings for Time-Varying Optimization with Intermittent Observations.** *IEEE Control Systems Letters [submitted]*, 20XX

Abstract:

We study the solution of a time-varying optimization problem which is observed, that is, it is known, only intermittently. We propose three approaches based on the prediction-correction scheme for solving this problem by exploiting splitting methods. We present convergence results in mean to a bounded asymptotical error, and showcase them in a numerical example featuring a regression problem.

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2019

N. Bastianello, A. Simonetto, R. Carli.

**Prediction-Correction for Nonsmooth Time-Varying Optimization via Forward-Backward Envelopes.** *Inernational Conference on Acoustics, Speech, and Signal Processing (ICASSP'19)*, pp. 5581-5585, 2019

Abstract:

We present an algorithm for minimizing the sum of a strongly convex time-varying function with a time-invariant, convex, and nonsmooth function.

The proposed algorithm employs the prediction-correction scheme alongside the forward-backward envelope, and we are able to prove the convergence of the solutions to a neighborhood of the optimizer that depends on the sampling time.

Numerical simulations for a time-varying regression problem with elastic net regularization highlight the effectiveness of the algorithm.

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N. Bastianello, A. Simonetto, R. Carli.

**Prediction-Correction Splittings for Nonsmooth Time-Varying Optimization.** *European Control Conference (ECC'19) [accepted]*, 2019

Abstract:

We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function.

We design an algorithmic framework based on a prediction-correction scheme, which employs splitting methods to solve the sampled instances of the time-varying problem.

We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a novel result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory. Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm.

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2018

N. Bastianello, R. Carli, L. Schenato, M. Todescato.

**A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks.** *IEEE 57th Conference on Decision and Control (CDC'18)*, pp. 3379-3384, 2018

Abstract:

In this paper we propose a distributed implementation

of the relaxed Alternating Direction Method of Multipliers algorithm

(R-ADMM) for optimization of a separable convex cost

function, whose terms are stored by a set of interacting agents,

one for each agent. Specifically the local cost stored by each node is in

general a function of both the state of the node and the states of its

neighbors, a framework that we refer to as `partition-based' optimization.

This framework presents a great flexibility and can be adapted to a large

number of different applications.

By recasting the problem into an operator theoretical framework, the proposed

algorithm is shown to be provably robust against random packet losses that

might occur in the communication between

neighboring nodes. Finally, the effectiveness of the proposed algorithm is

confirmed by a set of compelling numerical simulations run over random

geometric graphs subject to i.i.d. random packet losses.

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N. Bastianello, M. Todescato, R. Carli, L. Schenato.

**Distributed Optimization over Lossy Networks via Relaxed Peaceman-Rachford Splitting: a Robust ADMM Approach.** *European Control Conference (ECC'18)*, pp. 477-482, 2018

Abstract:

In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm, referred to as relaxed ADMM (R-ADMM) via a generalized *Peaceman-Rachford Splitting* operator on the Lagrange dual formulation of the original optimization problem. This algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$ and we show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case. Moreover, first, we reformulate our R-ADMM algorithm into an implementation that presents reduced complexity in terms of memory, communication and computational requirements. Second, we propose a further reformulation which let us provide the first ADMM-like algorithm with guaranteed convergence properties even in the presence of lossy communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over random geometric graphs subject to i.i.d. random packet losses.

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