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Leo Liberti

Publications and source records attributed to Leo Liberti.

At least 37 records · Page 2Linked to original sources
arXiv2020

MIP and Set Covering approaches for Sparse Approximation

The Sparse Approximation problem asks to find a solution $x$ such that $||y - Hx|| < α$, for a given norm $||\cdot||$, minimizing the size of the support $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$. We present valid inequalities for Mixed Integer Programming (MIP) formulations for this problem and we show that these families are sufficient to describe the set of feasible supports. This leads to a reformulation of the problem as an Integer Programming (IP) model which in turn represents a Minimum Set Covering formulation, thus yielding many families of valid inequalities which may be used to strengthen the models up. We propose algorithms to solve sparse approximation problems including a branch \& cut for the MIP, a two-stages algorithm to tackle the set covering IP and a heuristic approach based on Local Branching type constraints. These methods are compared in a computational experimentation with the goal of testing their practical potential.

cs.DM↗
arXiv2020

A new algorithm for the $^K$DMDGP subclass of Distance Geometry Problems

The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.

math.CO↗
arXiv2020

Mathematical Programming formulations for the Alternating Current Optimal Power Flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

math.OC↗
arXiv2020

Learning Discontinuous Piecewise Affine Fitting Functions using Mixed Integer Programming for Segmentation and Denoising

Piecewise affine functions are widely used to approximate nonlinear and discontinuous functions. However, most, if not all existing models only deal with fitting continuous functions. In this paper, we investigate the problem of fitting a discontinuous piecewise affine function to given data that lie in an orthogonal grid, where no restriction on the partition is enforced (i.e., its geometric shape can be nonconvex). This is useful for segmentation and denoising when data corresponding to images. We propose a novel Mixed Integer Program (MIP) formulation for the piecewise affine fitting problem, where binary variables determine the location of break-points. To obtain consistent partitions (i.e. image segmentation), we include multi-cut constraints in the formulation. Since the resulting problem is $\mathcal{NP}$-hard, two techniques are introduced to improve the computation. One is to add facet-defining inequalities to the formulation and the other to provide initial integer solutions using a special heuristic algorithm. We conduct extensive experiments by some synthetic images as well as real depth images, and the results demonstrate the feasibility of our model.

math.OC↗
arXiv2019

Distance Geometry and Data Science

Data are often represented as graphs. Many common tasks in data science are based on distances between entities. While some data science methodologies natively take graphs as their input, there are many more that take their input in vectorial form. In this survey we discuss the fundamental problem of mapping graphs to vectors, and its relation with mathematical programming. We discuss applications, solution methods, dimensional reduction techniques and some of their limits. We then present an application of some of these ideas to neural networks, showing that distance geometry techniques can give competitive performance with respect to more traditional graph-to-vector mappings.

cs.LG↗
arXiv2017

Globally Optimal Symbolic Regression

In this study we introduce a new technique for symbolic regression that guarantees global optimality. This is achieved by formulating a mixed integer non-linear program (MINLP) whose solution is a symbolic mathematical expression of minimum complexity that explains the observations. We demonstrate our approach by rediscovering Kepler's law on planetary motion using exoplanet data and Galileo's pendulum periodicity equation using experimental data.

stat.ML↗
arXiv2017

Random projections for trust region subproblems

The trust region method is an algorithm traditionally used in the field of derivative free optimization. The method works by iteratively constructing surrogate models (often linear or quadratic functions) to approximate the true objective function inside some neighborhood of a current iterate. The neighborhood is called "trust region in the sense that the model is trusted to be good enough inside the neighborhood. Updated points are found by solving the corresponding trust region subproblems. In this paper, we describe an application of random projections to solving trust region subproblems approximately.

math.OC↗
arXiv2017

Random projections for linear programming

Random projections are random linear maps, sampled from appropriate distributions, that approx- imately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are random ma- trices with surprisingly few rows that approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to solve linear programs approximately. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.

math.OC↗
arXiv2016

Open research areas in distance geometry

Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open and promising research areas.

math.MG↗
arXiv2016

New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver$+$formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates.

cs.CG↗
arXiv2015

Gaussian random projections for Euclidean membership problems

We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and infeasibility of this problem are preserved with high probability when the problem data is projected to a lower dimensional space. Our results are applicable to any algorithmic setting which needs to solve Euclidean membership problems in a high-dimensional space.

math.OC↗
arXiv2015

Using the Johnson-Lindenstrauss lemma in linear and integer programming

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we introduce a first attempt at using this lemma in the context of feasibility problems in linear and integer programming, which cannot be expressed only in function of Euclidean distances.

math.OC↗
arXiv2015

Six mathematical gems from the history of Distance Geometry

This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semi-metric spaces, a result of Goedel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.

math.HO↗
arXiv2012

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.

q-bio.QM↗
arXiv2011

Polynomial cases of the Discretizable Molecular Distance Geometry Problem

An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between corresponding point pairs. When the graph represents the backbone of a protein, one can exploit the natural vertex order to show that the search space for feasible embeddings is discrete. The corresponding decision problem can be solved using a binary tree based search procedure which is exponential in the worst case. We discuss assumptions that bound the search tree width to a polynomial size.

cs.CG↗
arXiv2010

On the number of solutions of the discretizable molecular distance geometry problem

The Generalized Discretizable Molecular Distance Geometry Problem is a distance geometry problems that can be solved by a combinatorial algorithm called ``Branch-and-Prune''. It was observed empirically that the number of solutions of YES instances is always a power of two. We give a proof that this event happens with probability one.

cs.DM↗
arXiv2008

A local branching heuristic for MINLPs

Local branching is an improvement heuristic, developed within the context of branch-and-bound algorithms for MILPs, which has proved to be very effective in practice. For the binary case, it is based on defining a neighbourhood of the current incumbent solution by allowing only a few binary variables to flip their value, through the addition of a local branching constraint. The neighbourhood is then explored with a branch-and-bound solver. We propose a local branching scheme for (nonconvex) MINLPs which is based on iteratively solving MILPs and NLPs. Preliminary computational experiments show that this approach is able to improve the incumbent solution on the majority of the test instances, requiring only a short CPU time. Moreover, we provide algorithmic ideas for a primal heuristic whose purpose is to find a first feasible solution, based on the same scheme.

math.CO↗