We propose a unified framework for time-varying convex optimization based on the prediction-correction paradigm, both in the primal and dual spaces. In this framework, a continuously varying optimization problem is sampled at fixed intervals, and each problem is approximately solved with a primal or dual correction step. The solution method is warm-started with the output of a prediction step, which solves an approximation of a future problem using past information. Prediction approaches are studied and compared under different sets of assumptions. Examples of algorithms covered by this framework are time-varying versions of the gradient method, splitting methods, and the celebrated alternating direction method of multipliers (ADMM).

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 prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean square in a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.

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.

We propose a non-linear model predictive scheme for planning fuel efficient maneuvers of small spacecrafts that shall rendezvous space debris. The paper addresses the specific issues of potential limited on-board computational capabilities and low-thrust actuators in the chasing spacecraft, and solves them by using a novel MatLab-based toolbox for real-time non-linear model predictive control (MPC) called MATMPC. This tool computes the MPC rendezvous maneuvering solution in a numerically efficient way, and this allows to greatly extend the prediction horizon length. This implies that the overall MPC scheme can compute solutions that account for the long time-scales that usually characterize the low-thrust rendezvous maneuvers. The so-developed controller is then tested in a realistic scenario that includes all the near-Earth environmental disturbances. We thus show, through numerical simulations, that this MPC method can successfully be used to perform a fuel-efficient rendezvous maneuver with an uncontrolled object, plus evaluate performance indexes such as mission duration, fuel consumption, and robustness against sensor and process noises.

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.

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.

In this paper we present a novel algorithm to solve the robot kinematic structure identification problem. Given a time series of data, typically obtained processing a set of visual observations, the proposed approach identifies the ordered sequence of links associated to the kinematic chain, the joint type interconnecting each couple of consecutive links, and the input signal influencing the relative motion. Compared to the state of the art, the proposed algorithm has reduced computational costs, and is able to identify also the joints' type sequence. 

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.

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.

In this paper, we study the problem of real-time optimal distributed partitioning for perimeter patrolling in the context of multicamera networks for surveillance, where each camera has limited mobility range and speed, and the communication is unreliable. The objective is to coordinate the cameras in order to minimize the time elapsed between two different visits of each point of the perimeter. We address this problem by casting it into a convex problem in which the perimeter is partitioned into nonoverlapping segments, each patrolled by a camera that sweeps back and forth at the maximum speed. We then propose an asynchronous distributed algorithm that guarantees that these segments cover the whole patrolling perimeter at any time and asymptotically converge to the optimal centralized solution under reliable communication. We finally modify the proposed algorithm in order to attain the same convergence and covering properties even in the more challenging scenario, where communication is lossy and there is no channel feedback, i.e., the transmitting camera is not aware whether a packet has been received or not by its neighbors.

In this paper we analyze the performances of some popular algorithms used to solve distributed optimization problems involving quadratic cost functions in a multi agent system. Namely, we study the performances of standard consensus, accelerated consensus and ADMM.  We analyze the scalar quadratic function case, under different scenarios and with structured graphs. We find that accelerated consensus is the algorithm with the best performance in all the cases analyzed. On the other hand, ADMM has performance comparable to the accelerated consensus when the graph is scarcely connected, while for dense graphs its performance deteriorates and becomes worse than the one of standard consensus. The results therefore suggest that the choice of the algorithm to solve the problem we analyze strongly depends on the graph, and that accelerated consensus should always be preferred.

We consider the problem of building a transitional model of an initially uncalibrated camera network. More specifically, we discuss an Hidden Markov Model (HMM) based strategy in which the model’s statespace is defined in terms of a partition of the physical network coverage. Transitions between any two such states are modeled by the distribution of the underlying Markov Process. Extending previous work in (Cenedese et al., 2010), we show how it is possible to infer the model structure and parameters from coordinate free observations and introduce a novel performance index that is used for model validation. We moreover show the predictive power of this HMM approach in simulated and real settings that comprise Pan-Tilt- Zoom (PTZ) cameras.

In this paper we study the problem of real-time optimal distributed partitioning for perimeter patrolling in the context of multi-camera networks for surveillance. The objective is to partition a given segment into non-overlapping sub-segments, each assigned to a different camera to patrol. Each camera has both physical mobility range and limited speed, and it must patrol its assigned sub-segment by sweeping it back and forth at maximum speed. Here we first review the solution for the centralized optimal partitioning. Then we propose two different distributed control strategies to determine the extremes of the optimal patrolling areas of each camera. Both these strategies require only local communication with the neighboring cameras but adopt different communication schemes, respectively, symmetric gossip and asynchronous asymmetric broadcast. The first scheme is shown to be provably convergent to the optimal solution. Some theoretical insights are provided also for the second scheme whose effectiveness is validated through numerical simulations.

This paper proposes a real-time distributed algorithm for a team of smart camerasto self-organize and perform video surveillance of an open boundary. In particular, our algorithmsimultaneously partitions the boundary among the cameras, and synchronizes the motion of thecameras to optimize the surveillance performance. We focus on the detection of smart intruders,who are aware of the cameras configuration at each time instant, and who schedule their motionto avoid detection for as long as possible. We consider both the worst-case and the averagedetection times of smart intruders. Our algorithm achieves minimum worst-case detection time,and, under some reasonable assumptions, constant-factor optimal average detection time.

In this paper a distributed clock synchronization algorithm is proposed. The algorithm requires gossip asynchronous communication between the nodes of the network, and because of its proportional-integral (PI) structure it is able to compensate both initial offsets and different clock speeds. Convergence of the algorithm is proved and analysed with respect to the controller parameter, while scalability issues are addressed by simulations.