Laboratoire de Mathématiques de l'INSA de Rouen

This paper is devoted to difference of convex functions (d.c.) optimization: d.c. duality, local and global optimality conditions in d.c. programming, the d.c. algorithm (DCA), and its application to solving the trust-region problem. The DCA is an iterative method that is quite different from well-known related algorithms. Thanks to the particular structure of the trust-region problem, the DCA is very simple (requiring only matrix-vector products) and, in practice, converges to the global solution. The inexpensive implicitly restarted Lanczos method of Sorensen is used to check the optimality of solutions provided by the DCA. When a nonglobal solution is found, a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the DCA at this point. It is shown that in the nonconvex case, the DCA converges to the global solution of the trust-region problem, using only matrix-vector products and requiring at most 2m+2 restarts, where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem. Numerical simulations establish the robustness and efficiency of the DCA compared to standard related methods, especially for large-scale problems.

Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an ℓ1-perturbed algorithm/reweighted-ℓ1 algorithm / reweighted-ℓ2 algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions.

A so-called DCA method based on a d.c.\ (difference of convex functions) optimization approach (algorithm) for solving large-scale distance geometry problems is developed. Different formulations of equivalent d.c.\ programs in the $l_{1}$-approach are stated via the Lagrangian duality without gap relative to d.c.\ programming, and new nonstandard nonsmooth reformulations in the $l_{\infty }$-approach (resp., the $l_{1}-l_{\infty }$-approach) are introduced. Substantial subdifferential calculations permit us to compute sequences of iterations in the DCA quite simply. The computations actually require matrix-vector products and only one Cholesky factorization (resp., with an additional solution of a convex program) in the $l_{1}$-approach (resp., the $l_{1}-l_{\infty }$-approach) and allow the exploitation of sparsity in the large-scale setting. Two techniques---respectively, using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure---are investigated in order to compute a good starting point for the DCA. Finally, many numerical simulations of the molecular optimization problems with up to 12567 variables are reported, which prove the practical usefulness of the nonstandard nonsmooth reformulations, the globality of found solutions, and the robustness and efficiency of our algorithms.

We present an algorithm to solve: Find $( x,y ) \in A \times A^ \bot $ such that $y \in Tx$, where A is a subspace and T is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn’s decomposition method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.

We study flatness of multi-input control-affine systems. We give a geometric characterization of systems that become static feedback linearizable after an invertible one-fold prolongation of a suitably chosen control. They form a particular class of flat systems. Namely, they are of differential weight $n + m+1$, where $n$ is the dimension of the state-space and $m$ is the number of controls. We propose conditions (verifiable by differentiation and algebraic operations) describing that class and provide a system of PDEs giving all minimal flat outputs. We illustrate our results by several examples, in particular, an example of the quadrotor helicopter.

To study how nonlinear waves propagate across Y- and T-type junctions, we consider the two-dimensional (2D) sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a one-dimensional effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when v>1-ω, where v and ω are, respectively, its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscillatory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.

This technical note deals with switched linear system identification and more particularly aims at solving switched linear regression problems in a large-scale setting with both numerous data and many parameters to learn. We consider the recent minimum-of-error framework with a quadratic loss function, in which an objective function based on a sum of minimum errors with respect to multiple submodels is to be minimized. The technical note proposes a new approach to the optimization of this nonsmooth and nonconvex objective function, which relies on Difference of Convex (DC) functions programming. In particular, we formulate a proper DC decomposition of the objective function, which allows us to derive a computationally efficient DC algorithm. Numerical experiments show that the method can efficiently and accurately learn switching models in large dimensions and from many data points.

We consider the task of design optimization, where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver, and based on the resulting approximate reduced derivatives, also an optimization iteration that updates the design variables simultaneously. To coordinate the three iterative processes, we use an exact penalty function of a doubly augmented Lagrangian type that should be consistently reduced. Some numerical experiments on a variant of the Bratu problem are presented.

Recently, the industries converge to the integration of the industry 4.0 paradigm to keep responding to the variable market demands. This integration is realized by the adoption of several components of the industry 4.0 such as IoT, Big Data and Cloud Computing, etc. Several difficulties concerning the integration of data management were encountered during first level of Industry 4.0 integration because of the unexpected quantity of data generated by IoT devices. The Fog computing can be considered as a new component of Industry 4.0 to resolve this kind of problem. However its implementation in the industrial field faces several challenges from different natures. This paper explains the role of Fog Computing solution to enhance the Cloud layer (distribution, low latency, real-time,. . . ) and studies its ability to be implemented in manufacturing systems. The Fog Manufacturing is introduced as the new industrial Fog vision. The challenges preventing the Fog Manufacturing implementation are studied and the links between each other are justified. A future use case is described to carry out the solutions given to satisfy the Fog Manufacturing challenges.

International audience

We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistributions. We also distinguish regular points, at which the system is equivalent to the canonical contact system, and singular points, at which we propose a new normal form that generalizes the canonical contact system on Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes the canonical contact system on Jn(R,R), which is also called Goursat normal form.

A three-terminal Josephson junction consists of three superconductors coupled coherently to a small nonsuperconducting island, such as a diffusive metal, a single or double quantum dot. A specific resonant single quantum dot three-terminal Josephson junction $({S}_{a},{S}_{b},{S}_{c})$ biased with voltages $(V,\ensuremath{-}V,0)$ is considered, but the conclusions hold more generally for resonant semiconducting quantum wire setups. A simple physical picture of the steady state is developed, using Floquet theory. It is shown that the equilibrium Andreev bound states (for $V=0$) evolve into nonequilibrium Floquet-Wannier-Stark-Andreev (FWS-Andreev) ladders of resonances (for $V\ensuremath{\ne}0$). These resonances acquire a finite width due to multiple Andreev reflection (MAR) processes. We also consider the effect of an extrinsic linewidth broadening on the quantum dot, introduced through a Dynes phenomenological parameter. The dc-quartet current manifests a crossover between the extrinsic relaxation dominated regime at low voltage to an intrinsic relaxation due to MAR processes at higher voltage. Finally, we study the coupling between the two FWS-Andreev ladders due to Landau-Zener-St\"uckelberg transitions, and its effect on the crossover in the relaxation mechanism. Three important low-energy scales are identified, and a perspective is to relate those low-energy scales to a recent noise cross-correlation experiment (Y. Cohen et al., arXiv:1606.08436).

We report on a study of the nontrivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov--de Gennes equations, we obtain a tight-binding model in Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a nontrivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values ${\ensuremath{\varphi}}_{B}=0$ or ${\ensuremath{\varphi}}_{B}=\ensuremath{\pi}$. We demonstrate that this nontrivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if ${\ensuremath{\varphi}}_{B}=\ensuremath{\pi}$. Our numerical calculations based on the Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.

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minimization problems by decomposition branch and bound method

Segmentation and registration are cornerstone steps of many imaging situations: while segmentation aims to identify relevant constituents of an image for visualization or quantitative analysis, registration consists of mapping salient features of an image onto the corresponding ones in another. Instead of treating these tasks linearly one after another, so without correlating them, we propose a unified variational model, in a hyperelasticity setting, processing these two operations simultaneously. The dissimilarity measure relates local and global (or region-based) information, since it relies on weighted total variation and nonlocal shape descriptors inspired by the piecewise constant Mumford--Shah model. Theoretical results emphasizing the mathematical and practical soundness of the model are provided, including existence of minimizers, connection with the segmentation step, nonlocal characterization of weighted seminorms, asymptotic results, and $\Gamma$-convergence properties. A preliminary version of this work appeared in [N. Debroux and C. Le Guyader, “A unified hyperelastic joint segmentation/registration model based on weighted total variation and nonlocal shape descriptors,” in Sixth International Conference on Scale Space and Variational Methods in Computer Vision, F. Lauze, Y. Dong, and A. B. Dahl, eds., Springer International, Cham, 2017, pp. 614--625], but it contains neither proofs of the proposed material nor details on the numerical treatment (developed nonlocal algorithm and extensive comparisons with related works).

We investigate the sparse eigenvalue problem which arises in various fields such as machine learning and statistics. Unlike standard approaches relying on approximation of the l0norm, we work with an equivalent reformulation of the problem at hand as a DC program. Our starting point is the eigenvalue problem to which a constraint for sparsity requirement is added. The obtained problem is first formulated as a mixed integer program, and exact penalty techniques are used to equivalently transform the resulting problem into a DC program, whose solution is assumed by a customized DCA. Computational results for sparse principal component analysis are reported, which show the usefulness of our approach that compares favourably with some related standard methods using approximation of the l0-norm. 1.

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Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology, and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables us to easily update the estimates when the data arrive sequentially and to have research steps in all directions. We establish the almost sure convergence of our stochastic Newton algorithm as well as its asymptotic normality. All our theoretical results are illustrated by numerical experiments.