Luca Schenato

We address the problem of distributed unconstrained convex optimization under separability assumptions, i.e., the framework where a network of agents, each endowed with local private multidimensional convex cost and subject to communication constraints, wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proven, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents distributedly compute and sequentially update an approximated Newton-Raphson direction by means of suitable average consensus ratios. We show with numerical simulations that the speed of convergence of this strategy is comparable with alternative optimization strategies such as the Alternating Direction Method of Multipliers. Finally, we propose some alternative strategies which trade-off communication and computational requirements with convergence speed.

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.

Personal control with occupancy and daylight adaptation is considered in a lighting system with multiple luminaires. Each luminaire is equipped with a co-located occupancy sensor and light sensor that respectively provide local occupancy and illumination information to a central controller. Users may also provide control inputs to indicate a desired illuminance value. Using sensor feedback and user input, the central controller determines dimming values of the luminaires using an optimization framework. The cost function consists of a weighted sum of illumination errors at light sensors and the power consumption of the system. The optimum dimming values are determined with the constraints that the illuminance value at the light sensors are above the reference set-point at the light sensors and the dimming levels are within physical allowable limits. Different approaches to determine the set-points at light sensors associated with multiple user illumination requests are considered. The performance of the proposed constrained optimization problem is compared with a reference stand-alone controller under different simulation scenarios in an open-plan office lighting system.

The knowledge of the size of a network, i.e.\ of the number of nodes composing it, is important for maintenance and organization purposes. In networks where the identity of the nodes or is not unique or cannot be disclosed for privacy reasons, the size-estimation problem is particularly challenging since the exchanged messages cannot be uniquely associated with a specific node. In this work, we propose a totally distributed anonymous strategy based on statistical inference concepts. In our approach, each node starts generating a vector of independent random numbers from a known distribution. Then nodes compute a common function via some distributed consensus algorithms, and finally they compute the Maximum Likelihood (ML) estimate of the network size exploiting opportune statistical inferences. In this work we study the performance that can be obtained following this computational scheme when the consensus strategy is either the maximum or the average. In the max-consensus scenario, when data come from absolutely continuous distributions, we provide a complete characterization of the ML estimator. In particular, we show that the squared estimation error decreases as $1/M$, where $M$ is the amount of random numbers locally generated by each node, independently of the chosen probability distribution. Differently, in the average-consensus scenario, we show that if the locally generated data are independent Bernoulli trials, then the probability for the ML estimator to return a wrong answer decreases exponentially in $M$. Finally, we provide a discussion as how the numerical errors may affect the estimators performance under different scenarios.