arXiv subjects

Andres Gomez

Publications and source records attributed to Andres Gomez.

30 records · Page 2Linked to original sources

arXiv2020

Supermodularity and valid inequalities for quadratic optimization with indicators

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtaining from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

math.OC↗

arXiv2020

Fractional 0-1 programming and submodularity

In this note we study multiple-ratio fractional 0--1 programs, a broad class of NP-hard combinatorial optimization problems. In particular, under some relatively mild assumptions we provide a complete characterization of the conditions, which ensure that a single-ratio function is submodular. Then we illustrate our theoretical results with the assortment optimization and facility location problems, and discuss practical situations that guarantee submodularity in the considered application settings. In such cases, near-optimal solutions for multiple-ratio fractional 0--1 programs can be found via simple greedy algorithms.

math.OC↗

arXiv2020

Sparse and Smooth Signal Estimation: Convexification of L0 Formulations

Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper, we propose a new iterative (convex) conic quadratic relaxations that exploit not only the $\ell_0$-"norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the $\ell_0$-"norm" and its $\ell_1$ surrogate. These stronger relaxations lead to significantly better estimators than $\ell_1$-norm approaches and also allow one to utilize affine sparsity priors. In addition, the parameters of the model and the resulting estimators are easily interpretable. Experiments with a tailored Lagrangian decomposition method indicate that the proposed iterative convex relaxations \rev{yield solutions within 1\% of the exact $\ell_0$ approach, and can tackle instances with up to 100,000 variables under one minute.

stat.ML↗

arXiv2020

Rank-one Convexification for Sparse Regression

Sparse regression models are increasingly prevalent due to their ease of interpretability and superior out-of-sample performance. However, the exact model of sparse regression with an $\ell_0$ constraint restricting the support of the estimators is a challenging (\NP-hard) non-convex optimization problem. In this paper, we derive new strong convex relaxations for sparse regression. These relaxations are based on the ideal (convex-hull) formulations for rank-one quadratic terms with indicator variables. The new relaxations can be formulated as semidefinite optimization problems in an extended space and are stronger and more general than the state-of-the-art formulations, including the perspective reformulation and formulations with the reverse Huber penalty and the minimax concave penalty functions. Furthermore, the proposed rank-one strengthening can be interpreted as a \textit{non-separable, non-convex, unbiased} sparsity-inducing regularizer, which dynamically adjusts its penalty according to the shape of the error function without inducing bias for the sparse solutions. In our computational experiments with benchmark datasets, the proposed conic formulations are solved within seconds and result in near-optimal solutions (with 0.4\% optimality gap) for non-convex $\ell_0$-problems. Moreover, the resulting estimators also outperform alternative convex approaches from a statistical perspective, achieving high prediction accuracy and good interpretability.

stat.ML↗

arXiv2020

Soft chemistry assisted On-chip Integration of Nanostructured quartz-based Piezoelectric Microelectromechanical System

The development of advanced piezoelectric quartz MEMS for sensing and precise frequency control applications requires the nanostructuration and on chip integration on silicon of this material. However, the current quartz manufacturing methods are based on bonding bulk micromachined crystals on silicon, which limits the size, the performance, the integration cost and the scalability of quartz micro devices. Here, we combine chemical solution deposition, soft nanoimprint lithography and top down microfabrication processes to develop the first nanostructured epitaxial 100 quartz 100 Si piezoelectric cantilevers. The coherent Si quartz interface and film thinness combined with a controlled nanostructuration on silicon insulator silicon technology substrates provides high force and mass sensitivity while preserving the mechanical quality factor of the microelectromechanical systems. This work proves that biocompatible nanostructured epitaxial piezoelectric quartz based MEMS on silicon can be engineered at low cost by combining soft chemistry and top down lithographic techniques.

physics.app-ph↗

arXiv2020

Learning Optimal Classification Trees: Strong Max-Flow Formulations

We consider the problem of learning optimal binary classification trees. Literature on the topic has burgeoned in recent years, motivated both by the empirical suboptimality of heuristic approaches and the tremendous improvements in mixed-integer programming (MIP) technology. Yet, existing approaches from the literature do not leverage the power of MIP to its full extent. Indeed, they rely on weak formulations, resulting in slow convergence and large optimality gaps. To fill this gap in the literature, we propose a flow-based MIP formulation for optimal binary classification trees that has a stronger linear programming relaxation. Our formulation presents an attractive decomposable structure. We exploit this structure and max-flow/min-cut duality to derive a Benders' decomposition method, which scales to larger instances. We conduct extensive computational experiments on standard benchmark datasets on which we show that our proposed approaches are 50 times faster than state-of-the art MIP-based techniques and improve out of sample performance up to 13.8%.

stat.ML↗

arXiv2018

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm with extended formulations.

math.OC↗

arXiv2018

Submodularity in conic quadratic mixed 0-1 optimization

We describe strong convex valid inequalities for conic quadratic mixed 0-1 optimization. These inequalities can be utilized for solving numerous practical nonlinear discrete optimization problems from value-at-risk minimization to queueing system design, from robust interdiction to assortment optimization through appropriate conic quadratic mixed 0-1 relaxations. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint over binary variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints of the optimization problem. Computational experiments on mean-risk optimization with correlations, assortment optimization, and robust conic quadratic optimization indicate that the new inequalities strengthen the convex relaxations substantially and lead to significant performance improvements.

math.OC↗

arXiv2018

Strong Formulations for Quadratic Optimization with M-matrices and Indicator Variables

We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.

math.OC↗

arXiv2017

Intrusion Prevention and Detection in Grid Computing - The ALICE Case

Grids allow users flexible on-demand usage of computing resources through remote communication networks. A remarkable example of a Grid in High Energy Physics (HEP) research is used in the ALICE experiment at European Organization for Nuclear Research CERN. Physicists can submit jobs used to process the huge amount of particle collision data produced by the Large Hadron Collider (LHC). Grids face complex security challenges. They are interesting targets for attackers seeking for huge computational resources. Since users can execute arbitrary code in the worker nodes on the Grid sites, special care should be put in this environment. Automatic tools to harden and monitor this scenario are required. Currently, there is no integrated solution for such requirement. This paper describes a new security framework to allow execution of job payloads in a sandboxed context. It also allows process behavior monitoring to detect intrusions, even when new attack methods or zero day vulnerabilities are exploited, by a Machine Learning approach. We plan to implement the proposed framework as a software prototype that will be tested as a component of the ALICE Grid middleware.

cs.DC↗

arXiv2016

Diminish Electrostatic in Piezoresponse Force Microscopy through longer ultra-stiff tips

Piezoresponse Force Microscopy is a powerful but delicate nanoscale technique that measures the mechanical response resulting from the application of a highly localized electric field. Though mechanical response is normally due to piezoelectricity, other physical phenomena, especially electrostatic interaction, can contribute to the signal read. We address this problematic through the use of longer ultra-stiff probes providing state of the art sensitivity, with the lowest electrostatic interaction and avoiding working in high frequency regime. In order to find this solution we develop a theoretical description addressing the effects of electrostatic contributions in the total cantilever vibration and its quantification for different setups. The theory is subsequently tested in a Periodically Poled Lithium Niobate (PPLN) crystal, a sample with well-defined 0deg and 180deg domains, using different commercial available conductive tips. We employ the theoretical description to compare the electrostatic contribution effects into the total phase recorded. Through experimental data our description is corroborated for each of the tested commercially available probes. We propose that a larger probe length can be a solution to avoid electrostatic forces, so the cantilever-sample electrostatic interaction is reduced. For hard oxide samples we propose an ultra-stiff cantilever which avoids the use of high frequency voltage but still diminishing electrostatic forces. Our proposed commercially available solution have great implications into avoiding artifacts while studying soft biological samples, multiferroic oxides, and thin film ferroelectric materials and it opens a new window into tip engineering.

cond-mat.mtrl-sci↗

arXiv2016

Piezo-generated charge mapping revealed through Direct Piezoelectric Force Microscopy

While piezoelectrics and ferroelectrics are playing a key role in many everyday applications, there are still a number of open questions related to the physics of those materials. In order to foster the understanding of piezoelectrics and ferroelectric and pave the way to future applications, the nanoscale characterization of these materials is essential. In this light, we have developed a novel AFM based mode that obtains a direct quantitative analysis of the piezoelectric coefficient d33. This nanoscale tool is capable of detecting and reveal piezo-charge generation through the direct piezoelectric effect at the surface of the piezoelectric and ferroelectric materials. We report the first nanoscale images of the charge generated in a thick single crystal of Periodically Poled Lithium Niobate (PPLN) and a Bismuth Ferrite (BiFO3) thin film by applying a force and recording the current produced by the materials. The quantification of both d33 coefficients for PPLN and BFO are 13 +- 2 pC/N and 46 +- 7 pC/N respectively, in agreement with the values reported in the literature. This new mode can operate simultaneously with PFM mode providing a powerful tool for the electromechanical and piezo-charge generation characterization of ferroelectric and piezoelectric materials.

cond-mat.mtrl-sci↗