arXiv subjects

Francesco Rinaldi

Publications and source records attributed to Francesco Rinaldi.

At least 37 records · Page 2Linked to original sources

arXiv2023

Collaboration and topic switches in science

Collaboration is a key driver of science and innovation. Mainly motivated by the need to leverage different capacities and expertise to solve a scientific problem, collaboration is also an excellent source of information about the future behavior of scholars. In particular, it allows us to infer the likelihood that scientists choose future research directions via the intertwined mechanisms of selection and social influence. Here we thoroughly investigate the interplay between collaboration and topic switches. We find that the probability for a scholar to start working on a new topic increases with the number of previous collaborators, with a pattern showing that the effects of individual collaborators are not independent. The higher the productivity and the impact of authors, the more likely their coworkers will start working on new topics. The average number of coauthors per paper is also inversely related to the topic switch probability, suggesting a dilution of this effect as the number of collaborators increases.

physics.soc-ph↗

arXiv2022

Avoiding bad steps in Frank Wolfe variants

The analysis of Frank Wolfe (FW) variants is often complicated by the presence of different kinds of "good" and "bad" steps. In this article we aim to simplify the convergence analysis of some of these variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a sizable decrease of the objective at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Lojasiewicz (KL) property. While this setting has been widely studied for proximal gradient type methods, to our knowledge, it has not been analyzed before for the Frank Wolfe variants under study. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that this condition is satisfied on polytopes by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).

math.OC↗

arXiv2022

A Variance-aware Multiobjective Louvain-like Method for Community Detection in Multiplex Networks

In this paper, we focus on the community detection problem in multiplex networks, i.e., networks with multiple layers having same node sets and no inter-layer connections. In particular, we look for groups of nodes that can be recognized as communities consistently across the layers. To this end, we propose a new approach that generalizes the Louvain method by (a) simultaneously updating average and variance of the modularity scores across the layers, and (b) reformulating the greedy search procedure in terms of a filter-based multiobjective optimization scheme. Unlike many previous modularity maximization strategies, which rely on some form of aggregation of the various layers, our multiobjective approach aims at maximizing the individual modularities on each layer simultaneously. We report experiments on synthetic and real-world networks, showing the effectiveness and the robustness of the proposed strategies both in the informative case, where all layers show the same community structure, and in the noisy case, where some layers represent only noise.

cs.SI↗

arXiv2022

Minimization over the l1-ball using an active-set non-monotone projected gradient

The l1-ball is a nicely structured feasible set that is widely used in many fields (e.g., machine learning, statistics and signal analysis) to enforce some sparsity in the model solutions. In this paper, we devise an active-set strategy for efficiently dealing with minimization problems over the l1-ball and embed it into a tailored algorithmic scheme that makes use of a non-monotone first-order approach to explore the given subspace at each iteration. We prove global convergence to stationary points. Finally, we report numerical experiments, on two different classes of instances, showing the effectiveness of the algorithm.

math.OC↗

arXiv2022

Retraction based Direct Search Methods for Derivative Free Riemannian Optimization

Direct search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, we explore the application of those strategies to Riemannian optimization, wherein minimization is to be performed with respect to variables restricted to lie on a manifold. More specifically, we consider classic and line search extrapolated variants of direct search, and, by making use of retractions, we devise tailored strategies for the minimization of both smooth and nonsmooth functions. As such we analyze, for the first time in the literature, a class of retraction based algorithms for minimizing nonsmooth objectives on a Riemannian manifold without having access to (sub)derivatives. Along with convergence guarantees we provide a set of numerical performance illustrations on a standard set of problems.

math.OC↗

arXiv2022

An oracle-based framework for robust combinatorial optimization

We propose a general solution approach for min-max-robust counterparts of combinatorial optimization problems with uncertain linear objectives. We focus on the discrete scenario case, but our approach can be extended to other types of uncertainty sets such as polytopes or ellipsoids. Concerning the underlying certain problem, the algorithm is entirely oracle-based, i.e., our approach only requires a (primal) algorithm for solving the certain problem. It is thus particularly useful in case the underlying problem is hard to solve, or only defined implicitly by a given software addressing the certain case. The idea of our algorithm is to solve the convex relaxation of the robust problem by a simplicial decomposition approach, the main challenge being the non-differentiability of the objective function in the case of discrete or polytopal uncertainty. The resulting dual bounds are then used within a tailored branch-and-bound framework for solving the robust problem to optimality. By a computational evaluation, we show that our method outperforms straightforward linearization approaches on the robust minimum spanning tree problem. Moreover, using the Concorde solver for the certain oracle, our approach computes much better dual bounds for the robust traveling salesman problem in the same amount of time.

math.OC↗

arXiv2021

An Improved Penalty Algorithm using Model Order Reduction for MIPDECO problems with partial observations

This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improved penalty algorithm proposed by the authors, to the time-dependent setting. The main contribution is a novel combination of an interior point method, preconditioning, and model order reduction yielding a tailored local optimization solver at the heart of the overall solution procedure. A thorough numerical investigation is carried out both for a Poisson problem as well as a convection-diffusion problem demonstrating the versatility of the approach.

math.OC↗

arXiv2021

Improved penalty algorithm for Mixed Integer PDE Constrained Optimization Problems

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.

math.NA↗

arXiv2021

Derivative-free methods for mixed-integer nonsmooth constrained optimization

In this paper, we consider mixed-integer nonsmooth constrained optimization problems whose objective/constraint functions are available only as the output of a black-box zeroth-order oracle (i.e., an oracle that does not provide derivative information) and we propose a new derivative-free linesearch-based algorithmic framework to suitably handle those problems. We first describe a scheme for bound constrained problems that combines a dense sequence of directions (to handle the nonsmoothness of the objective function) with primitive directions (to handle discrete variables). Then, we embed an exact penalty approach in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of the proposed algorithms toward stationary points and we report the results of an extensive numerical experience on a set of mixed-integer test problems.

math.OC↗

arXiv2021

Frank-Wolfe and friends: a journey into projection-free first-order optimization methods

Invented some 65 years ago in a seminal paper by Marguerite Straus-Frank and Philip Wolfe, the Frank-Wolfe method recently enjoys a remarkable revival, fuelled by the need of fast and reliable first-order optimization methods in Data Science and other relevant application areas. This review tries to explain the success of this approach by illustrating versatility and applicability in a wide range of contexts, combined with an account on recent progress in variants, both improving on the speed and efficiency of this surprisingly simple principle of first-order optimization.

math.OC↗

arXiv2021

Fast cluster detection in networks by first-order optimization

Cluster detection plays a fundamental role in the analysis of data. In this paper, we focus on the use of s-defective clique models for network-based cluster detection and propose a nonlinear optimization approach that efficiently handles those models in practice. In particular, we introduce an equivalent continuous formulation for the problem under analysis, and we analyze some tailored variants of the Frank-Wolfe algorithm that enable us to quickly find maximal s-defective cliques. The good practical behavior of those algorithmic tools, which is closely connected to their support identification properties, makes them very appealing in practical applications. The reported numerical results clearly show the effectiveness of the proposed approach.

math.OC↗

arXiv2021

A derivative-free method for structured optimization problems

Structured optimization problems are ubiquitous in fields like data science and engineering. The goal in structured optimization is using a prescribed set of points, called atoms, to build up a solution that minimizes or maximizes a given function. In the present paper, we want to minimize a black-box function over the convex hull of a given set of atoms, a problem that can be used to model a number of real-world applications. We focus on problems whose solutions are sparse, i.e., solutions that can be obtained as a proper convex combination of just a few atoms in the set, and propose a suitable derivative-free inner approximation approach that nicely exploits the structure of the given problem. This enables us to properly handle the dimensionality issues usually connected with derivative-free algorithms, thus getting a method that scales well in terms of both the dimension of the problem and the number of atoms. We analyze global convergence to stationary points. Moreover, we show that, under suitable assumptions, the proposed algorithm identifies a specific subset of atoms with zero weight in the final solution after finitely many iterations. Finally, we report numerical results showing the effectiveness of the proposed method.

math.OC↗

arXiv2020

A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition

The analysis of projection-free first order methods is often complicated by the presence of different kinds of "good" and "bad" steps. In this article, we propose a unifying framework for projection-free methods, aiming to simplify the converge analysis by getting rid of such a distinction between steps. The main tool employed in our framework is the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as convergence rates under a Kurdyka-Lojasiewicz (KL) property, a setting that, to our knowledge, has not been analyzed before for the projection-free methods under study. In this context, we prove local convergence rates comparable to those of projected gradient methods under the same conditions. Our analysis relies on a sufficient slope condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant among feasible directions. This condition is satisfied, among others, by several Frank-Wolfe (FW) variants on polytopes, and by some projection-free methods on convex sets with smooth boundary.

math.OC↗

arXiv2020

Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-cuts) that defines the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program min{cx : x \in S \cap Z^n }, where S \subset R^n is a compact set and c \in Z^n . We analyze the number of iterations of our algorithm.

math.OC↗

arXiv2020

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.

cs.SI↗

arXiv2020

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.

math.OC↗

arXiv2020

Solving non-monotone equilibrium problems via a DIRECT-type approach

A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of the \texttt{DIRECT} algorithm, which exploits local bounds of the Lipschitz constant of the objective function, with local minimizations. Unlike most existing solution methods for EPs, no monotonicity-type condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.

math.OC↗

arXiv2019

Active set complexity of the Away-step Frank-Wolfe Algorithm

In this paper, we study active set identification results for the away-step Frank-Wolfe algorithm in different settings. We first prove a local identification property that we apply, in combination with a convergence hypothesis, to get an active set identification result. We then prove, in the nonconvex case, a novel $O(1/\sqrt{k})$ convergence rate result and active set identification for different stepsizes (under suitable assumptions on the set of stationary points). By exploiting those results, we also give explicit active set complexity bounds for both strongly convex and nonconvex objectives. While we initially consider the probability simplex as feasible set, in the appendix we show how to adapt some of our results to generic polytopes.

math.OC↗