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We present exact mixed-integer programming linear formulations for verifying the performance of first-order methods for parametric quadratic optimization. We formulate the verification problem as a mixed-integer linear program where the objective is to maximize the infinity norm of the fixed-point residual after a given number of iterations. Our approach captures a wide range of gradient, projection, proximal iterations through affine or piecewise affine constraints. We derive tight polyhedral convex hull formulations of the constraints representing the algorithm iterations. To improve the scalability, we develop a custom bound tightening technique combining interval propagation, operator theory, and optimization-based bound tightening. Numerical examples, including linear and quadratic programs from network optimization and sparse coding using Lasso, show that our method provides several orders of magnitude reductions in the worst-case fixed-point residuals, closely matching the true worst-case performance.

In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (DCUT) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the DCUT formulation on metric instances. A key contribution of our work is extending of the Gap problem, previously explored in the context of the Traveling Salesman problems, to the metric Steiner Tree problem. To tackle the Gap problem for Steiner Tree instances, we first establish several structural properties of the CM formulation. We then classify the isomorphism classes of the vertices within the CM polytope, revealing a correspondence between the vertices of the DCUT and CM polytopes. Computationally, we exploit these structural properties to design two complementary heuristics for finding nontrivial small metric Steiner instances with a large integrality gap. We present several vertices for graphs with a number of nodes $\leq 10$, which realize the best-known lower bounds on the integrality gap for the CM and the DCUT formulations. We conclude the paper by presenting three new conjectures on the integrality gap of the DCUT and CM formulations for small graphs., Comment: 27 pages, 4 figures

Training neural networks (NNs) using combinatorial optimization solvers has gained attention in recent years. In low-data settings, state-of-the-art mixed integer linear programming solvers can train exactly a NN, avoiding intensive GPU-based training and hyper-parameter tuning and simultaneously training and sparsifying the network. We study the case of few-bit discrete-valued neural networks, both Binarized Neural Networks (BNNs), whose values are restricted to +-1, and Integer Neural Networks (INNs), whose values lie in a range {-P, ..., P}. Few-bit NNs receive increasing recognition due to their lightweight architecture and ability to run on low-power devices. This paper proposes new methods to improve the training of BNNs and INNs. Our contribution is a multi-objective ensemble approach based on training a single NN for each possible pair of classes and applying a majority voting scheme to predict the final output. Our approach results in training robust sparsified networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible. We compare this BeMi approach to the current state-of-the-art in solver-based NN training and gradient-based training, focusing on BNN learning in few-shot contexts. We compare the benefits and drawbacks of INNs versus BNNs, bringing new light to the distribution of weights over the {-P, ..., P} interval. Finally, we compare multi-objective versus single-objective training of INNs, showing that robustness and network simplicity can be acquired simultaneously, thus obtaining better test performances. While the previous state-of-the-art approaches achieve an average accuracy of 51.1% on the MNIST dataset, the BeMi ensemble approach achieves an average accuracy of 68.4% when trained with 10 images per class and 81.8% when trained with 40 images per class, having up to 75.3% NN links removed., Comment: 22 pages, 6 figure, 5 tables

In this paper, by resorting to classical methods of statistical mechanics, we build a kinetic model able to reproduce the observed statistical weight distribution of many diverse species. The kinetic description of the time variations of the weight distribution is based on elementary interactions that describe in a qualitative and quantitative way successive evolutionary updates, and determine explicit equilibrium distributions. Numerical fittings on mammalian eutherians of the order Chiroptera population illustrates the effectiveness of the approach.

In Mathematical Music theory, the Aperiodic Tiling Complements Problem consists in finding all the possible aperiodic complements of a given rhythm $A$. The complexity of this problem depends on the size of the period $n$ of the canon and on the cardinality of the given rhythm $A$. The current state-of-the-art algorithms can solve instances with $n$ smaller than $180$. In this paper we propose an ILP formulation and a SAT Encoding to solve this mathemusical problem, and we use the Maplesat solver to enumerate all the aperiodic complements. We validate our SAT Encoding using several different periods and rhythms and we compute for the first time the complete list of aperiodic tiling complements of standard Vuza rhythms for canons of period $n=\{180,420,900\}$.

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

A total coloring of a graph $G = (V, E)$ is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of $G$ into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements. This covering formulation can be solved by a Column Generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce three families of nontrivial valid inequalities: vertex-clique inequalities based on standard clique inequalities of the Stable Set Polytope, congruent-$2k3$ cycle inequalities based on the parity of the vertex set induced by the cycle, and even-clique inequalities induced by complete subgraphs of even order. We prove that congruent-$2k3$ cycle inequalities are facet-defining only when $k = 4$, while the vertex-clique and even-cliques are always facet-defining. Finally, we present preliminary computational results of a Column Generation algorithm for the Total Coloring Problem and a Cutting Plane algorithm for the Total Matching Problem., Comment: 29 pages, 5 figures

This paper introduces the Gene Mover's Distance, a measure of similarity between a pair of cells based on their gene expression profiles obtained via single-cell RNA sequencing. The underlying idea of the proposed distance is to interpret the gene expression array of a single cell as a discrete probability measure. The distance between two cells is hence computed by solving an Optimal Transport problem between the two corresponding discrete measures. In the Optimal Transport model, we use two types of cost function for measuring the distance between a pair of genes. The first cost function exploits a gene embedding, called gene2vec, which is used to map each gene to a high dimensional vector: the cost of moving a unit of mass of gene expression from a gene to another is set to the Euclidean distance between the corresponding embedded vectors. The second cost function is based on a Pearson distance among pairs of genes. In both cost functions, the more two genes are correlated, the lower is their distance. We exploit the Gene Mover's Distance to solve two classification problems: the classification of cells according to their condition and according to their type. To assess the impact of our new metric, we compare the performances of a $k$-Nearest Neighbor classifier using different distances. The computational results show that the Gene Mover's Distance is competitive with the state-of-the-art distances used in the literature., Comment: 16 pages, 8 figures. The data used in this paper is available online at: https://zenodo.org/record/4604569

We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equivalent either to the Euclidean-Wasserstein distance $W_2$, or to the Kantorovich-Wasserstein distance $W_1$, with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones., Comment: 18 pages, 2 figures, 1 table

In recent years it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences. In social sciences, the appearance of lognormal distribution has been noticed, among others, when looking at body weight, and at women's age at first marriage. Likewise, in economics, lognormal distribution appears when looking at consumption in a western society, at call-center service times, and others. The common feature of these situations, which describe the distribution of a certain people's hallmark, is the presence of a desired target to be reached by repeated choices. In this paper we discuss a possible explanation of lognormal distribution forming in human activities by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic variation of the hallmark around its target value, leading to a linear Fokker--Planck type equation with lognormal equilibrium density, is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker--Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf's law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland.

This paper presents a novel method to compute the exact Kantorovich-Wasserstein distance between a pair of $d$-dimensional histograms having $n$ bins each. We prove that this problem is equivalent to an uncapacitated minimum cost flow problem on a $(d+1)$-partite graph with $(d+1)n$ nodes and $dn^{\frac{d+1}{d}}$ arcs, whenever the cost is separable along the principal $d$-dimensional directions. We show numerically the benefits of our approach by computing the Kantorovich-Wasserstein distance of order 2 among two sets of instances: gray scale images and $d$-dimensional biomedical histograms. On these types of instances, our approach is competitive with state-of-the-art optimal transport algorithms., Comment: 12 pages, 4 figures, 3 tables

In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring Wasserstein distances of order one between histograms with $n$ bins, by solving a classical transportation problem on very large complete bipartite graphs with $n$ nodes and $n^2$ edges. The main contribution of our work is to approximate the original transportation problem by an uncapacitated min cost flow problem on a reduced flow network of size $O(n)$ that exploits the geometric structure of the cost function. More precisely, when the distance among the bin centers is measured with the 1-norm or the $\infty$-norm, our approach provides an optimal solution. When the distance among bins is measured with the 2-norm: (i) we derive a quantitative estimate on the error between optimal and approximate solution; (ii) given the error, we construct a reduced flow network of size $O(n)$. We numerically show the benefits of our approach by computing Wasserstein distances of order one on a set of grey scale images used as benchmark in the literature. We show how our approach scales with the size of the images with 1-norm, 2-norm and $\infty$-norm ground distances, and we compare it with other two methods which are largely used in the literature., Comment: 27 pages, 35 figures, 5 tables

Call centers are service networks in which agents provide telephone-based services. An important part of call center operations is represented by service durations. In recent statistical analysis of real data, it has been noticed that the distribution of service times reveals a remarkable fit to the lognormal distribution. In this paper we discuss a possible source of this behavior by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic service time variation leading to a linear kinetic equation with lognormal equilibrium density is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.