A Fast Active Set Block Coordinate Descent Algorithm for ℓ1\ell_1-regularized least squares

The problem of finding sparse solutions to underdetermined systems of linear equations arises in several applications (e.g. signal and image processing, compressive sensing, statistical inference). A standard tool for dealing with sparse recovery is the $\ell_1$-regularized least-squares approach that has been recently attracting the attention of many researchers. In this paper, we describe an active set estimate (i.e. an estimate of the indices of the zero variables in the optimal solution) for the considered problem that tries to quickly identify as many active variables as possible at a given point, while guaranteeing that some approximate optimality conditions are satisfied. A relevant feature of the estimate is that it gives a significant reduction of the objective function when setting to zero all those variables estimated active. This enables to easily embed it into a given globally converging algorithmic framework. In particular, we include our estimate into a block coordinate descent algorithm for $\ell_1$-regularized least squares, analyze the convergence properties of this new active set method, and prove that its basic version converges with linear rate. Finally, we report some numerical results showing the effectiveness of the approach.Comment: 28 pages, 5 figure

Valorizzazione degli scarti della produzione industriale di legumi processati per la preparazione di biocompositi

Negli ultimi anni la ricerca indirizzata verso lo sviluppo di materiali a ridotto impatto ambientale e la valorizzazione di materiali di scarti è cresciuta per via di una più forte sensibilità ambientale e per i vantaggi economici che derivano da una chimica più ‘green’. Nel presente lavoro di tesi è osservato il comportamento e le proprietà del biocomposito all’aumentare della quantità di una fibra vegetale di scarto per determinare se questo porta ad una diminuzione o significativa variazione delle proprietà del composito. Questo perchè l’utilizzo di una percentuale maggiore di fibra è vantaggiosa in quanto permette di risparmiare sulla matrice polimerica adoperando una quantità maggiore di materiale di scarto il cui smaltimento è oneroso per l’industria. La fibra vegetale utilizzata nella presente tesi è stata la fibra di piselli (Pisum Sativum). Come matrice polimerica è stato utilizzato l’acido polilattico (PLA), polimero biodegradabile e interessante anche dal punto di vista commerciale. Allo scopo di migliorare l’adesione tra la matrice polimerica e la fibra di scarto sono stati sperimentati due compatibilizzanti: l’anidride maleica aggraffata al PLA e il polietilenglicole 1500. In questo lavoro sono studiati gli effetti sulle proprietà termiche, chimiche e meccaniche in funzione dell’uso di un compatibilizzante nella preparazione dei biocompositi, della differente percentuale di fibra di scarto utilizzata e del metodo di miscelazione (diretta o previa preparazione di un masterbatch). Sono in particolare studiati gli eventuali effetti degradativi prodotti da un maggiore utilizzo della fibra di scarto e del compatibilizzante sulla matrice polimerica e la dispersione e miscelazione della fibra nei vari compositi studiati

An Active-Set Algorithmic Framework for Non-Convex Optimization Problems over the Simplex

In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new "nonorthogonality" type of condition. We prove global convergence to stationary points when using an Armijo line search in the given framework. We further describe three different examples of active-set gradient related directions that guarantee linear convergence rate (under suitable assumptions). Finally, we report numerical experiments showing the effectiveness of the approach.Comment: 29 pages, 3 figure

Total variation based community detection using a nonlinear optimization approach

Maximizing the modularity of a network is a successful tool to identify an important community of nodes. However, this combinatorial optimization problem is known to be NP-complete. Inspired by recent nonlinear modularity eigenvector approaches, we introduce the modularity total variation $TV_Q$ and show that its box-constrained global maximum coincides with the maximum of the original discrete modularity function. Thus we describe a new nonlinear optimization approach to solve the equivalent problem leading to a community detection strategy based on $TV_Q$. The proposed approach relies on the use of a fast first-order method that embeds a tailored active-set strategy. We report extensive numerical comparisons with standard matrix-based approaches and the Generalized RatioDCA approach for nonlinear modularity eigenvectors, showing that our new method compares favourably with state-of-the-art alternatives

A derivative-free approach for a simulation-based optimization problem in healthcare

Hospitals have been challenged in recent years to deliver high quality care with limited resources. Given the pressure to contain costs,developing procedures for optimal resource allocation becomes more and more critical in this context. Indeed, under/overutilization of emergency room and ward resources can either compromise a hospital's ability to provide the best possible care, or result in precious funding going toward underutilized resources. Simulation--based optimization tools then help facilitating the planning and management of hospital services, by maximizing/minimizing some specific indices (e.g. net profit) subject to given clinical and economical constraints. In this work, we develop a simulation--based optimization approach for the resource planning of a specific hospital ward. At each step, we first consider a suitably chosen resource setting and evaluate both efficiency and satisfaction of the restrictions by means of a discrete--event simulation model. Then, taking into account the information obtained by the simulation process, we use a derivative--free optimization algorithm to modify the given setting. We report results for a real--world problem coming from the obstetrics ward of an Italian hospital showing both the effectiveness and the efficiency of the proposed approach