Publications List

Articles published in ISI Journals

• Alvarez, Felipe; Flores, Salvador.
  Existence and Approximation for Variational Problems Under Uniform Constraints on the Gradient by Power Penalty (preprint here)
  
  SIAM J. Math. Anal., 47(5), 3466--3487, 2015. [+ bibtex citation]

  @Article{ AlvarezFlores2015,
  author = {Alvarez, Felipe and Flores, Salvador},
  title = {Existence and Approximation for Variational Problems Under Uniform Constraints on the Gradient by Power Penalty},
  journal = {SIAM J. Math. Anal.},
  year = {2015},
  volume={47},
  number={5},
  pages={3466--3487},
  doi = {DOI:10.1137/140988619},
  }

  show abstract [+]

  Variational problems under uniform quasi-convex constraints on the gradient are studied. Our technique consists in approximating the original problem by a one-parameter family of smooth unconstrained optimization problems. Existence of solutions to the problems under consideration is proved as well as existence of Lagrange multipliers associated to the uniform constraint; no constraint qualification condition is required. The solution-multiplier pairs are shown to satisfy an Euler--Lagrange equation and a complementarity property. Numerical experiments confirm the ability of our method to accurately compute solutions and Lagrange multipliers. [-]
• Flores, Salvador.
  SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression (preprint here)
  
  European Journal of Operational Research 246 (2015) pp. 44-50, 2015
  [+ bibtex citation]

  @Article{ FloresEJOR2015,
  author = "Salvador Flores",
  title = "SOCP relaxation bounds for the optimal subset selection problem applied to robust linear regression",
  journal = "EJOR",
  year = "2015",
  volume="246",
  number="1",
  pages="44--50" doi = "DOI: 10.1016/j.ejor.2015.04.024", }

  show abstract [+]

  This paper deals with the problem of finding the globally optimal subset of $h$ elements from a larger set of $n$ elements in $d$ space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, tipically $d \leq 5$, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem. [-]
• Flores, Salvador.
  Sharp non-asymptotic performance bounds for $\ell_1$ and Huber robust regression estimators (preprint here)
  
  TEST 24(4), pp. 796-812, 2015. [+ bibtex citation]

  @Article{ Flores2015,
  author = "Salvador Flores",
  title = "Sharp non-asymptotic performance bounds for $\ell_1$ and Huber robust regression estimators",
  journal = "Test",
  year = "2015",
  number = "4",
  volume = "24",
  pages = "796 -- 812",
  doi = "DOI: 10.1007/s11749-015-0435-5", }

  show abstract [+]

  A quantitative study of the robustness properties of the ℓ1 and the Huber M-estimator on finite samples is presented. The focus is on the linear model involving a fixed design matrix and additive errors restricted to the dependent variables consisting of noise and sparse outliers. We derive sharp error bounds for the ℓ1 estimator in terms of the leverage constants of a design matrix introduced here. A similar analysis is performed for Huber’s estimator using an equivalent problem formulation of independent interest. Our analysis considers outliers of arbitrary magnitude, and we recover breakdown point results as particular cases when outliers diverge. The practical implications of the theoretical analysis are discussed on two real datasets [-]
• Flores, Salvador.
  On the efficient computation of robust regression estimators. (preprint here)
  
  Computational Statistics & Data Analysis, 54 (12),pp. 3044--3056, 2010. [+ bibtex citation]

  @Article{ Flores2010, author = "Salvador Flores",
  title = "On the efficient computation of robust regression estimators",
  journal = "Computational Statistics \& Data Analysis",
  volume = "54",
  number = "12",
  pages = "3044--3056",
  year = "2010",
  doi = "DOI: 10.1016/j.csda.2010.03.020", }

  show abstract [+]

  The problem of providing efficient and reliable robust regression algorithms is considered. The impact of global optimization methods, such as stopping conditions and clustering techniques, in the calculation of robust regression estimators is investigated. The use of stopping conditions permits us to devise new algorithms that perform as well as the existing algorithms in less time and with adaptive algorithm parameters. Clustering global optimization is shown to be a general framework encompassing many of the existing algorithms. [-]
• Alvarez, Felipe; Flores, Salvador.
  Remarks on Lipschitz solutions to measurable differential inclusions and an existence result for some nonconvex variational problems. (preprint here)
  
  J. Convex Anal. 15 (2), pp. 349-363, 2008. [bibtex citation]

  @article {Alvarez-Flores2008,
  AUTHOR = {Alvarez, Felipe and Flores, Salvador},
  TITLE = {Remarks on {L}ipschitz solutions to measurable differential inclusions and an
  existence result for some nonconvex variational problems},
  JOURNAL = {J. Convex Anal.},
  VOLUME = {15},
  YEAR = {2008},
  NUMBER = {2},
  PAGES = {349--363}
  }

  show abstract [+]

  In this paper we examine the problem of finding a Lipschitz function on an open domain with prescribed boundary values and whose gradient is required to satisfy some nonhomogeneous pointwise constraints a.e. in the domain, $$\nabla u(x) \in C(x)\ \text{ a.e. in }\Omega,\quad u|_{\partial \Omega}=g \in {\rm Lip}(\partial \Omega).$$ These constraints are supposed to be given by a measurable set-valued mapping $C:\mathbb{R}^n\rightrightarrows \mathbb{R}^n$   with convex, uniformly compact and nonempty-interior values. We discuss existence and metric properties of maximal solutions of such a problem. We exploit some connections with weak solutions to discontinuous Hamilton-Jacobi equations, and we provide a variational principle that characterizes maximal solutions. We investigate the case where the original problem is supplemented with bilateral obstacle constraints on the function values. Finally, as an application of these results, we prove existence for a specific class of nonconvex problems from the calculus of variations, with and without obstacle constraints, under mild regularity hypotheses on the data. [-]

Working Papers

• Flores, Salvador.
  Does Compressed Sensing Have Applications in Robust Statistics? (preprint here)
  

  show abstract [+]

  The connections between robust linear regression and sparse reconstruction are brought to light. We show that in the context of fixed design, the notion of breakdown point coincides with exact recovery of sparse signals from highly incomplete information. The main consequence of this connection in robust regression is that there exists, for any dimension, ``many'' designs on which the $\ell_1$ estimator has a positive breakdown point. This result clarifies a common misunderstanding on the robustness of M-estimators. [-]

PhD. Thesis

Subject : "Global optimization problems in robust statistics "
Advisors : Anne Ruiz-Gazen, Université Toulouse 1 Capitole
     Marcel Mongeau, Université Paul Sabatier
Defended : february 25, 2011
Committee : Jean-Baptiste Hiriart-Urruty, Université Paul Sabatier
       Marcel Mongeau, Université Paul Sabatier
       Fabio Schoen, Università di Firenze, Italy
       Michael Schyns, Université de Liège, Belgium
       Roy Welsch, Massachusetts Institute of Technology, USA





show abstract [+]

Robust statistics is a branch of statistics dealing with the analysis of data containing contaminated observations. The robustness of an estimator is measured notably by means of the breakdown point. High-breakdown point estimators are usually defined as global minima of a non-convex scale of the errors, hence their computation is a challenging global optimization problem. The objective of this dissertation is to investigate the potential contributions of modern global optimization methods to this class of problems.
The first part of this thesis is devoted to the $\tau$-estimator for linear regression, which is defined as a global minimum of a nonconvex differentiable function. We investigate the impact of incorporating clustering techniques and stopping conditions in existing stochastic algorithms. The consequences of some phenomena involving the nearest neighbor in high dimension on clustering global optimization algorithms is thoroughly discussed as well.
The second part is devoted to deterministic algorithms for computing the least trimmed squares regression estimator, which is defined through a nonlinear mixed-integer program. Due to the combinatorial nature of this problem, we concentrated on obtaining lower bounds to be used in a branch-and-bound algorithm. In particular, we propose a second-order cone relaxation that can be complemented with concavity cuts that we obtain explicitly. Global optimality conditions are also provided. [-]

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Last update: 19/08/2011