Ivano Notarnicola

Ivano Notarnicola, Giuseppe Notarstefano

In this paper we consider a general, challenging distributed optimization set-up arising in several important network control applications. Agents of a network want to minimize the sum of local cost functions, each one depending on a local variable, subject to local and coupling constraints, with the latter involving all the decision variables. We propose a novel fully distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation, based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node finds a primal-dual optimal solution pair of a local, relaxed version of the original problem, and then updates suitable auxiliary local variables. We prove that agents asymptotically compute their portion of an optimal (feasible) solution of the original problem. This primal recovery property is obtained without any averaging mechanism typically used in dual decomposition methods. To corroborate the theoretical results, we show how the methodology applies to an instance of a Distributed Model Predictive Control scheme in a microgrid control scenario.

Ivano Notarnicola, Ying Sun, Gesualdo Scutari et al.

In this paper, we study distributed big-data nonconvex optimization in multi-agent networks. We consider the (constrained) minimization of the sum of a smooth (possibly) nonconvex function, i.e., the agents' sum-utility, plus a convex (possibly) nonsmooth regularizer. Our interest is in big-data problems wherein there is a large number of variables to optimize. If treated by means of standard distributed optimization algorithms, these large-scale problems may be intractable, due to the prohibitive local computation and communication burden at each node. We propose a novel distributed solution method whereby at each iteration agents optimize and then communicate (in an uncoordinated fashion) only a subset of their decision variables. To deal with non-convexity of the cost function, the novel scheme hinges on Successive Convex Approximation (SCA) techniques coupled with i) a tracking mechanism instrumental to locally estimate gradient averages; and ii) a novel block-wise consensus-based protocol to perform local block-averaging operations and gradient tacking. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Finally, numerical results show the effectiveness of the proposed algorithm and highlight how the block dimension impacts on the communication overhead and practical convergence speed.

Ivano Notarnicola, Ying Sun, Gesualdo Scutari et al.

We study distributed multi-agent large-scale optimization problems, wherein the cost function is composed of a smooth possibly nonconvex sum-utility plus a DC (Difference-of-Convex) regularizer. We consider the scenario where the dimension of the optimization variables is so large that optimizing and/or transmitting the entire set of variables could cause unaffordable computation and communication overhead. To address this issue, we propose the first distributed algorithm whereby agents optimize and communicate only a portion of their local variables. The scheme hinges on successive convex approximation (SCA) to handle the nonconvexity of the objective function, coupled with a novel block-signal tracking scheme, aiming at locally estimating the average of the agents' gradients. Asymptotic convergence to stationary solutions of the nonconvex problem is established. Numerical results on a sparse regression problem show the effectiveness of the proposed algorithm and the impact of the block size on its practical convergence speed and communication cost.

Ivano Notarnicola, Ruggero Carli, Giuseppe Notarstefano

In this paper we consider a novel partitioned framework for distributed optimization in peer-to-peer networks. In several important applications the agents of a network have to solve an optimization problem with two key features: (i) the dimension of the decision variable depends on the network size, and (ii) cost function and constraints have a sparsity structure related to the communication graph. For this class of problems a straightforward application of existing consensus methods would show two inefficiencies: poor scalability and redundancy of shared information. We propose an asynchronous distributed algorithm, based on dual decomposition and coordinate methods, to solve partitioned optimization problems. We show that, by exploiting the problem structure, the solution can be partitioned among the nodes, so that each node just stores a local copy of a portion of the decision variable (rather than a copy of the entire decision vector) and solves a small-scale local problem.

Ivano Notarnicola, Giuseppe Notarstefano

In this paper we consider distributed optimization problems in which the cost function is separable, i.e., a sum of possibly non-smooth functions all sharing a common variable, and can be split into a strongly convex term and a convex one. The second term is typically used to encode constraints or to regularize the solution. We propose a class of distributed optimization algorithms based on proximal gradient methods applied to the dual problem. We show that, by choosing suitable primal variable copies, the dual problem is itself separable when written in terms of conjugate functions, and the dual variables can be stacked into non-overlapping blocks associated to the computing nodes. We first show that a weighted proximal gradient on the dual function leads to a synchronous distributed algorithm with local dual proximal gradient updates at each node. Then, as main paper contribution, we develop asynchronous versions of the algorithm in which the node updates are triggered by local timers without any global iteration counter. The algorithms are shown to be proper randomized block-coordinate proximal gradient updates on the dual function.

Ivano Notarnicola, Florian A. Bayer, Giuseppe Notarstefano et al.

In this paper we develop a numerical method to solve nonlinear optimal control problems with final-state constraints. Specifically, we extend the PRojection Operator based Netwon's method for Trajectory Optimization (PRONTO), which was proposed by Hauser for unconstrained optimal control problems. While in the standard method final-state constraints can be only approximately handled by means of a terminal penalty, in this work we propose a methodology to meet the constraints exactly. Moreover, our method guarantees recursive feasibility of the final-state constraint. This is an appealing property especially in realtime applications in which one would like to be able to stop the computation even if the desired tolerance has not been reached, but still satisfy the constraints. Following the same conceptual idea of PRONTO, the proposed strategy is based on two main steps which (differently from the standard scheme) preserve the feasibility of the final-state constraints: (i) solve a quadratic approximation of the nonlinear problem to find a descent direction, and (ii) get a (feasible) trajectory by means of a feedback law (which turns out to be a nonlinear projection operator). To find the (feasible) descent direction we take advantage of final-state constrained Linear Quadratic optimal control methods, while the second step is performed by suitably designing a constrained version of the trajectory tracking projection operator. The effectiveness of the proposed strategy is tested on the optimal state transfer of an inverted pendulum.

Andrea Camisa, Ivano Notarnicola, Giuseppe Notarstefano

In this paper we deal with a network of agents seeking to solve in a distributed way Mixed-Integer Linear Programs (MILPs) with a coupling constraint (modeling a limited shared resource) and local constraints. MILPs are NP-hard problems and several challenges arise in a distributed framework, so that looking for suboptimal solutions is of interest. To achieve this goal, the presence of a linear coupling calls for tailored decomposition approaches. We propose a fully distributed algorithm based on a primal decomposition approach and a suitable tightening of the coupling constraints. Agents repeatedly update local allocation vectors, which converge to an optimal resource allocation of an approximate version of the original problem. Based on such allocation vectors, agents are able to (locally) compute a mixed-integer solution, which is guaranteed to be feasible after a sufficiently large time. Asymptotic and finite-time suboptimality bounds are established for the computed solution. Numerical simulations highlight the efficacy of the proposed methodology.

Ivano Notarnicola, Mauro Franceschelli, Giuseppe Notarstefano

In this paper we consider a distributed optimization scenario in which a set of processors aims at minimizing the maximum of a collection of "separable convex functions" subject to local constraints. This set-up is motivated by peak-demand minimization problems in smart grids. Here, the goal is to minimize the peak value over a finite horizon with: (i) the demand at each time instant being the sum of contributions from different devices, and (ii) the local states at different time instants being coupled through local dynamics. The min-max structure and the double coupling (through the devices and over the time horizon) makes this problem challenging in a distributed set-up (e.g., well-known distributed dual decomposition approaches cannot be applied). We propose a distributed algorithm based on the combination of duality methods and properties from min-max optimization. Specifically, we derive a series of equivalent problems by introducing ad-hoc slack variables and by going back and forth from primal and dual formulations. On the resulting problem we apply a dual subgradient method, which turns out to be a distributed algorithm. We prove the correctness of the proposed algorithm and show its effectiveness via numerical computations.

Ivano Notarnicola, Giuseppe Notarstefano

In this paper we consider a distributed optimization scenario in which a set of agents has to solve a convex optimization problem with separable cost function, local constraint sets and a coupling inequality constraint. We propose a novel distributed algorithm based on a relaxation of the primal problem and an elegant exploration of duality theory. Despite its complex derivation based on several duality steps, the distributed algorithm has a very simple and intuitive structure. That is, each node solves a local version of the original problem relaxation, and updates suitable dual variables. We prove the algorithm correctness and show its effectiveness via numerical computations.

Ivano Notarnicola, Giuseppe Notarstefano

In this paper we consider a distributed optimization scenario in which the aggregate objective function to minimize is partitioned, big-data and possibly non-convex. Specifically, we focus on a set-up in which the dimension of the decision variable depends on the network size as well as the number of local functions, but each local function handled by a node depends only on a (small) portion of the entire optimization variable. This problem set-up has been shown to appear in many interesting network application scenarios. As main paper contribution, we develop a simple, primal distributed algorithm to solve the optimization problem, based on a randomized descent approach, which works under asynchronous gossip communication. We prove that the proposed asynchronous algorithm is a proper, ad-hoc version of a coordinate descent method and thus converges to a stationary point. To show the effectiveness of the proposed algorithm, we also present numerical simulations on a non-convex quadratic program, which confirm the theoretical results.

Andrea Testa, Ivano Notarnicola, Giuseppe Notarstefano

Submodular optimization is a special class of combinatorial optimization arising in several machine learning problems, but also in cooperative control of complex systems. In this paper, we consider agents in an asynchronous, unreliable and time-varying directed network that aim at cooperatively solving submodular minimization problems in a fully distributed way. The challenge is that the (submodular) objective set-function is only partially known by agents, that is, each one is able to evaluate the function only for subsets including itself. We propose a distributed algorithm based on a proper linear programming reformulation of the combinatorial problem. Our algorithm builds on a column generation approach in which each agent maintains a local candidate basis and locally generates columns with a suitable greedy inner routine. A key interesting feature of the proposed algorithm is that the pricing problem, which involves an exponential number of constraints, is solved by the agents through a polynomial time greedy algorithm. We prove that the proposed distributed algorithm converges in finite time to an optimal solution of the submodular minimization problem and we corroborate the theoretical results by performing numerical computations on instances of the $s$--$t$ minimum graph cut problem.

Andrea Simonetto, Ivano Notarnicola

Equipping current decision-making tools with notions of fairness, equitability, or other ethically motivated outcomes, is one of the top priorities in recent research efforts in machine learning, AI, and optimization. In this paper, we investigate how to allocate limited resources to {locally interacting} communities in a way to maximize a pertinent notion of equitability. In particular, we look at the dynamic setting where the allocation is repeated across multiple periods (e.g., yearly), the local communities evolve in the meantime (driven by the provided allocation), and the allocations are modulated by feedback coming from the communities themselves. We employ recent mathematical tools stemming from data-driven feedback online optimization, by which communities can learn their (possibly unknown) evolution, satisfaction, as well as they can share information with the deciding bodies. We design dynamic policies that converge to an allocation that maximize equitability in the long term. We further demonstrate our model and methodology with realistic examples of healthcare and education subsidies design in Sub-Saharian countries. One of the key empirical takeaways from our setting is that long-term equitability is fragile, in the sense that it can be easily lost when deciding bodies weigh in other factors (e.g., equality in allocation) in the allocation strategy. Moreover, a naive compromise, while not providing significant advantage to the communities, can promote inequality in social outcomes.

Xu Du, Boyu Han, Ivano Notarnicola et al.

This paper investigates distributed resource allocation optimization over directed graphs with limited communication bandwidth. We develop a novel distributed algorithm that integrates the centralized Proximal Jacobian Alternating Direction Method of Multipliers (PJ-ADMM) with a finite-level quantized consensus scheme, enabling nodes to cooperatively solve the optimization in a distributed fashion. Under the assumption of convex objective functions, we establish that the proposed algorithm achieves sublinear convergence to a neighborhood of the optimal solution, with the convergence accuracy explicitly bounded by the quantization level. Numerical experiments validate that the algorithm achieves competitive performance compared to existing approaches while exhibiting communication efficiency.

Lorenzo Sforni, Guido Carnevale, Ivano Notarnicola et al.

In this paper, we investigate a data-driven framework to solve Linear Quadratic Regulator (LQR) problems when the dynamics is unknown, with the additional challenge of providing stability certificates for the overall learning and control scheme. Specifically, in the proposed on-policy learning framework, the control input is applied to the actual (unknown) linear system while iteratively optimized. We propose a learning and control procedure, termed Relearn LQR, that combines a recursive least squares method with a direct policy search based on the gradient method. The resulting scheme is analyzed by modeling it as a feedback-interconnected nonlinear dynamical system. A Lyapunov-based approach, exploiting averaging and timescale separation theories for nonlinear systems, allows us to provide formal stability guarantees for the whole interconnected scheme. The effectiveness of the proposed strategy is corroborated by numerical simulations, where Relearn LQR is deployed on an aircraft control problem, with both static and drifting parameters.

Guido Carnevale, Francesco Farina, Ivano Notarnicola et al.

This paper deals with a network of computing agents aiming to solve an online optimization problem in a distributed fashion, i.e., by means of local computation and communication, without any central coordinator. We propose the gradient tracking with adaptive momentum estimation (GTAdam) distributed algorithm, which combines a gradient tracking mechanism with first and second order momentum estimates of the gradient. The algorithm is analyzed in the online setting for strongly convex cost functions with Lipschitz continuous gradients. We provide an upper bound for the dynamic regret given by a term related to the initial conditions and another term related to the temporal variations of the objective functions. Moreover, a linear convergence rate is guaranteed in the static setup. The algorithm is tested on a time-varying classification problem, on a (moving) target localization problem, and in a stochastic optimization setup from image classification. In these numerical experiments from multi-agent learning, GTAdam outperforms state-of-the-art distributed optimization methods.