Resumen: With a view on bilevel and PDE-constrained optimisation, we develop iterative estimates of for compositions , where is the solution mapping of the inner optimisation problem or PDE. The idea is to form a single-loop method by interweaving updates of the iterate by an outer optimisation method, with updates of the estimate by single steps of standard optimisation methods and linear system solvers. When the inner methods satisfy simple tracking inequalities, the differential estimates can almost directly be employed in standard convergence proofs for general forward-backward type methods. We adapt those proofs to a general inexact setting in normed spaces, that, besides our differential estimates, also covers mismatched adjoints and unreachable optimality conditions in measure spaces. As a side product of these efforts, we provide improved convergence results for nonconvex Primal-Dual Proximal Splitting (PDPS).

Resumen: Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal-dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.

Resumen: We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss-Seidel, conjugate gradients), always taking only on

Resumen: Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.

Resumen: We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliar variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and Strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.

Resumen: This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes an inequality for the scalar product between the symmetrized gradient of the state and the Lagrange multiplier.

Resumen: In this paper, we carry out a rigorous analysis of four-dimensional variational data assimilation problems for linear and semilinear parabolic equations with control in the initial condition, coming from a Lebesgue space. Due to the nature of the data assimilation problem, spatial pointwise observations of the state have to be considered in the cost functional, which requires continuity of the solution with respect to the spatial variable. To obtain this for an initial condition in , with , we make use of maximal parabolic regularity tools, jointly with real and complex interpolation theory. We prove that the data assimilation problems admit an optimal solution and derive a first-order necessary optimality system. For the semilinear case, the situation is more involved. Indeed, due to the differentiability properties of the control-to-state mapping, we can derive first-order necessary optimality conditions just for values of . Due to the same reason, the study of the second-order sufficient optimality conditions is carried out for and taking .

Resumen: We investigate the numerical approximation of an elliptic optimal control problem which involves a nonconvex local regularization of the -quasinorm penalization (with ) in the cost function. Our approach is based on the \emph{difference-of-convex} function formulation, which leads to first-order necessary optimality conditions, which can be regarded as the optimality system of an auxiliar convex -penalized optimal control problem. We consider piecewise-constant finite element approximation for the controls, whereas the state equation is approximated using piecewise-linear basis functions. Then, convergence results are obtained for the proposed approximation. Under certain conditions on the support's boundary of the optimal control, we deduce an order of approximation rate of convergence where is the associated discretization parameter. We illustrate our theoretical findings with numerical experiments that show the convergence behavior of the numerical approximation

Resumen: We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

Resumen: In this paper we propose a second--order method for solving \emph{linear composite sparse optimization problems} consisting of minimizing the sum of a differentiable (possibly nonconvex function) and a nondifferentiable convex term. The composite nondifferentiable convex penalizer is given by --norm of a matrix multiplied with the coefficient vector. The algorithm that we propose for the case of the linear composite problem relies on the three main ingredients that power the OESOM algorithm \cite{dlrlm07}: the minimum norm subgradient, a projection step and, in particular, the second--order information associated to the nondifferentiable term. By extending these devices, we obtain a full second--order method for solving composite sparse optimization problems which includes a wide range of applications. For instance, problems involving the minimization of a general class \emph{differential graph operators} can be solved with the proposed algorithm. We present several computational experiments to show the efficiency of our approach for different application examples.

Resumen: Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The idea is to not define the electrodes as strict geometric objects on the measurement boundary, but only to assume approximate knowledge about their whereabouts and let a boundary admittivity function determine the actual locations of the current inputs. Such an approach enables reconstructing the boundary admittivity, i.e. the locations and strengths of the contacts, at the same time and with analogous methods as the interior admittivity. The functionality of the new model is verified by two-dimensional numerical experiments based on water tank data.

Resumen: Electrical resistance tomography (ERT) -based distributed surface sensing systems, or sensing skins, offer alternative sensing techniques for structural health monitoring, providing capabilities for distributed sensing of, for example, damage, strain and temperature. Currently, however, the computational techniques utilized for sensing skins are limited to planar surfaces. In this paper, to overcome this limitation, we generalize the ERT-based surface sensing to non-planar surfaces covering arbitrarily shaped three-dimensional structures; We construct a framework in which we reformulate the image reconstruction problem of ERT using techniques of Riemannian geometry, and solve the resulting problem numerically. We test this framework in series of numerical and experimental studies. The results demonstrate that the feasibility of the proposed formulation and the applicability of ERT-based sensing skins for non-planar geometries.

Resumen: In this paper we propose a second–order method for solvinglinear compos-ite sparse optimization problemsconsisting of minimizing the sum of a differentiable(possibly nonconvex function) and a nondifferentiable convex term. The compositenondifferentiable convex penalizer is given by 1–norm of a matrix multiplied with thecoefficient vector. The algorithm that we propose for the case of the linear composite1–norm problem relies on the three main ingredients that power the OESOM algo-rithm [6]: the minimum norm subgradient, a projection step and, in particular, thesecond–order information associated to the nondifferentiable term. By extending thesedevices, we obtain a full second–order method for solving composite sparse optimiza-tion problems which includes a wide range of applications. For instance, problemsinvolving the minimization of a general class ofdifferential graph operatorscan besolved with the proposed algorithm. We present several computational experiments toshow the efficiency of our approach for different application examples.

Resumen: We consider the exact penalization of the incompressibility condition $div(u) =0$ for the velocity field of a Bingham fluid in terms of the $L1$–norm. This penalization procedure results in a nonsmooth optimization problem for which we propose an algorithm using generalized second–order information. Our method solves the resulting nonsmooth problem by considering the steepest descent direction and extra generalized second–order information associated to the nonsmooth term. This method has the advantage that the divergence-free property is enforced by the descent direction proposed by the method without the need of build-in divergence-free approximation schemes. Theinexact penalization approach, given by the $L2$–norm, is also considered in our discussionand comparison.

Resumen: This study combines statistical analysis and discrete optimization techniques to solve the problem of assigning patients to therapists for crisis intervention with a single tele-psychotherapy session. The statistical analysis showed that professionals and healthcare workers in contact with Covid–19 patients or with a confirmed diagnosis had a significant relationship with suicide risk, sadness, experiential avoidance, and perception of severity. This allowed categorizing patients according to their screening and grouping therapists according to their qualifications. A multi-periodic optimization model and a heuristic are proposed to find an adequate assignment of patients to therapists over time. The integer programming model was validated with real-world data, and its results were applied in a volunteer program in Ecuador. Keywords: Integer Programming, Logistic Regression, WLSMV, AAQ-II, Tele-Psychotherapy, Clinical Psychology, SARS–CoV2

Resumen: We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

Resumen: En este reporte presentamos el esquema variacional utilizado por el Centro de Modelización Matemática para estimar los diferentes parámetros y condiciones iniciales de los modelos de propagación del SARS-CoV-2 en Ecuador, en presencia de incertidumbre en los datos. Esta estimación permite realizar actualizaciones periódicas del número efectivo de reproducción de la epidemia, así como proyecciones a corto plazo de su propagación

Resumen: The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern that can be used to optimally balance acquisition time versus quality of the reconstructed image. We use a supervised learning approach, making the assumption that our training data is representative enough of new data acquisitions. We demonstrate that this is indeed the case, even if the training data consists of just 7 training pairs of measurements and ground-truth images; with a training set of brain images of size 192 by 192, for instance, one of the learned patterns samples only 35% of k-space, however results in reconstructions with mean SSIM 0.914 on a test set of similar images. The proposed framework is general enough to learn arbitrary sampling patterns, including common patterns such as Cartesian, spiral and radial sampling.

Resumen: We propose a bilevel optimization approach for the determination of parameters in nonlocal image denoising. We consider both spatial weights in front of the fidelity term, as well as weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we propose a second-order optimization algorithm in combination with a finite element discretization of the nonlocal denoising models and a computational strategy for the solution of the resulting dense linear systems. Several experiments are run in order to show the suitability of our approach.

Resumen: We propose a bilevel optimization approach for the determination of parameters in nonlocal image denoising. We consider both spatial weights in front of the fidelity term, as well as weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we propose a second-order optimization algorithm in combination with a finite element discretization of the nonlocal denoising models and a computational strategy for the solution of the resulting dense linear systems. Several experiments are run in order to show the suitability of our approach.

Resumen: In this work a multi-weight graph partitioning problem is introduced. This problem consists in partitioning an undirected graph in a fixed number of subgraphs such that multiple node weight constraints over each partition are satisfied and the total distance between nodes in the same subgraph induced by the partition is minimized. This problem generalizes several graph partitioning problems like $k$-way equipartition, and bisection. It arises as a subproblem of an integrated vehicle and pollster problem from a real world application. Two Integer Programming formulations are provided and several families of valid inequalities associated to the respective polyhedra are proved. An exact algorithm based on branch and bound and cutting planes is proposed and it is tested on real-world instances.

Resumen: A variant of the Curriculum-Based Course Timetabling (CB-CTT)Problem arising from a practical application is studied. This variant includes twokey features: lectures with heterogeneous durations and a compactness soft con-straint for curricula. Each course lecture has a specific duration, indicating a num-ber of consecutive time periods required for the lecture to take place. Moreover,each day in the schedule of any curriculum should not exceed a prescribed max-imum length, where the length of a working day is defined as the time elapsedbetween the start of the first lecture until the end of the last lecture on that day.Finally, some soft constraints of the standard CB-CTT, such as minimum numberof working days or room capacity, are considered as hard constraints in the variantstudied here.An integer programming model and a two-stage solution algorithm are pro-posed for this problem. At first, a reduced model obtained by discarding schedulecompactness constraints is solved to optimality. The solution is then extended toan initial feasible solution of the original model, which is then improved using astandard branch-and-cut procedure. Three classes of symmetry-breaking inequal-ities are described for this integer program and tested on real-world instances.

Resumen: In this paper we propose a bilevel optimization approach for the placement of space and time observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower-level task is the variational reconstruction of the initial condition of a semilinear system, and the upper-level problem solves the optimal placement with help of a sparsity inducing function. Due to the pointwise nature of the observations, an optimality system with regular Borel measures is obtained as necessary optimality condition for the lower-level problem. The latter is then considered as constraint for the upper-level instance, yielding an optimization problem constrained by a multi-state system with measures. We demonstrate existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solution of the bilevel problem. The numerical solution is carried out also on two levels. The lower-level problem is solved using a standard BFGS method, while the upper-level one is solved by means of a projected BFGS algorithm based on the estimation of -active sets. Finally some numerical experiments are presented to illustrate the main features of our approach.

Resumen: The National Statistics Bureau of Ecuador carries out monthly polls to monitor the evolution of an economic indicator called “Consumer Price Index”, which measures the consumer prices of basic commodities. A set of stores is selected to apply these polls. A set of vehicles transport pollsters from the headquarters of the bureau to the stores to perform this activity. Moreover, pollsters move between stores using pedestrian paths or using a vehicle to shorten the travel time between far away stores. This paper introduces an integer programming model for the integrated task of scheduling visits of pollsters to selected stores, as well as routing the vehicle fleet used to transport them. Results on the computational complexity of the aforementioned problem, a three-phase algorithm, and computational experience based on real-world instances are provided.

Resumen: We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation model, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal--dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions of the functionals -- but still with a non-bilinear coupling term -- we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems.

Resumen: We study and develop (stochastic) primal--dual block-coordinate descent methods for convex problems based on the method due to Chambolle and Pock. Our methods have known convergence rates for the iterates and the ergodic gap: O(1/N2) if each block is strongly convex, O(1/N) if no convexity is present, and more generally a mixed rate O(1/N2)+O(1/N) for strongly convex blocks, if only some blocks are strongly convex. Additional novelties of our methods include blockwise-adapted step lengths and acceleration, as well as the ability to update both the primal and dual variables randomly in blocks under a very light compatibility condition. In other words, these variants of our methods are doubly-stochastic. We test the proposed methods on various image processing problems, where we employ pixelwise-adapted acceleration.

Resumen: We study inertial versions of primal--dual proximal splitting, also known as the Chambolle--Pock method. Our starting point is the preconditioned proximal point formulation of this method. By adding correctors corresponding to the anti-symmetric part of the relevant monotone operator, using a FISTA-style gap unrolling argument, we are able to derive gap estimates instead of merely ergodic gap estimates. Moreover, based on adding a diagonal component to this corrector, we are able to combine strong convexity based acceleration with inertial acceleration. We test our proposed method on image processing and inverse problems problems, obtaining convergence improvements for sparse Fourier inversion and Positron Emission Tomography.

Resumen: In this paper we propose a bilevel optimization approach for the placement of observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower level task is the variational reconstruction of the initial condition of the system, and the upper level problem solves the optimal placement with help of a sparsity inducing norm. Due to the pointwise nature of the observations, an optimality system with regular Borel measures on the right-hand side is obtained as necessary and sufficient optimality condition for the lower level problem. The latter is then considered as constraint for the upper level instance, yielding an optimization problem constrained by time-dependent PDE's with measures. After proving some extra regularity results, we demonstrate the existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solution of the bilevel problem. The numerical solution is carried out also on two levels. The lower level problem is solved using a standard BFGS method, while the upper level one is solved by means of a projected BFGS algorithm based on the estimation of e-active sets. A penalty function is also considered for enhancing sparsity of the location weights. Finally, some numerical experiments are presented to illustrate the main features of our approach.

Resumen: Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the $\alpha$-averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper "Block-proximal methods with spatially adapted acceleration". In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward--backward splitting, Douglas--Rachford splitting, Newton's method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle--Pock method.

Resumen: We study preconditioned proximal point methods for a class of saddle point problems, where the preconditioner decouples the overall proximal point method into an alternating primal--dual method. This is akin to the Chambolle--Pock method or the ADMM. In our work, we replace the squared distance in the dual step by a barrier function on a symmetric cone, while using a standard (Euclidean) proximal step for the primal variable. We show that under non-degeneracy and simple linear constraints, such a hybrid primal--dual algorithm can achieve linear convergence on originally strongly convex problems involving the second-order cone in their saddle point form. On general symmetric cones, we are only able to show an O(1/N) rate. These results are based on estimates of strong convexity of the barrier function, extended with a penalty to the boundary of the symmetric cone.

Resumen: In this work, we derive an a–priori error estimate of order $h^2|log(h)|$ for the finite element approximation of an sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box–constrains on the control are considered and finitely many pointwise state–constrains are imposed in specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation fro the state equation.

Resumen: The primal--dual hybrid gradient method (PDHGM, also known as the Chambolle--Pock method) has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step length rules for PDE-constrained optimization problems.

Resumen: Based on the needs of convergence proofs of preconditioned proximal point methods, we introduce notions of partial strong submonotonicity and partial (metric) subregularity of set-valued maps. We study relationships between these two concepts, neither of which is generally weaker or stronger than the other one. For our algorithmic purposes, the novel submonotonicity turns out to be easier to employ than more conventional error bounds obtained from subregularity. Using strong submonotonicity, we demonstrate the a posteriori linear convergence of the Primal--Dual Hybrid Gradient Method (Modified) to some strictly complementary solutions of example problems from image processing and data science. This is without the conventional assumption that all the objective functions of the involved saddle point problem are strongly convex.

Resumen: We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and the Bayesian method for the MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize-then-optimize approach and derive first-order optimality conditions for the problem. For the numerical solution, we propose a globalized reduced Newton-type method and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where among others, the performance of TGV-regularization compared to the TV-regularization is tested.

Resumen: In the context of the study of the set covering polyhedron of circular matrices, we study the relationship between row family inequalities and a previously proposed class termed minor inequalities. Recently, a complete linear description of this polyhedron has been provided in terms of row family inequalities. In this work we prove that for the particular subclass of circulant matrices, the corresponding row family inequalities are related to circulant minors. Moreover, we extend previous results on circulant matrices $C_{sk}^k$, $s\in\{2,3\}$ and $k\ge 2$, by providing a polynomial time separation algorithm for the inequalities describing the set covering polyhedron of matrices $C_{4k}^k$, $k\ge 2$.

Resumen: We study the family of minor related row family inequalities for the set covering polyhedron related to circular matrices introduced in [8]. We provide a construction to obtain facets with arbitrarily large coefficients. Moreover, we address the issue of generating these inequalities via the Chvátal-Gomory rounding procedure and provide examples of inequalities having Chvátal-rank strictly larger than one.

Resumen: Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. take advantage of this relationship to propose a complete linear description of the stable set polytope on (fuzzy) circular interval graphs. In this paper we follow si milar ideas to obtain a complete description of the dominating set polytope on circular interval graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form $Q^*(A,b):= conv\{x\in \mathbb{Z}_+^n: Ax\ge b\}$, with $A$ a $\{0,1\}$-matrix and $b$ an integer vector.

Resumen: Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of the second kind, with a twofold purpose. First, we aim at giving a glance at some of the most prominent applications of these types of variational inequalities in mechanics, and the related analytical and numerical difficulties. Second, we consider optimal control problems constrained by these variational inequalities and provide a thorough discussion on the existence of Lagrange multipliers and the different types of optimality systems that can be derived for the characterization of local minima. The article ends with a discussion of the main challenges and future perspectives of this important problem class.

Resumen: We propose a non-smooth trust-region method for solving optimization problems with B-differentiable functions, with application to problems constrained by variational inequalities of the second kind. Under suitable assumptions on the model functions, convergence of the general algorithm to a C-stationary point is verified. For variational inequality constrained problems, we are able to properly characterize the Bouligand subdifferential of the reduced cost function and, based on that, we propose a computable trust-region model which fulfills the convergence hypotheses of the general algorithm. The article concludes with the experimental study of the main properties of the proposed method based on two different numerical instances.

Resumen: We propose a local regularization of elliptic optimal control problems which involves the nonconvex $L^q$ fractional penalizations in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization formulation as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.

Resumen: In this work a balanced k-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a Branch & Cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian Football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.

Resumen: The numerical approximation of an optimal control problem with $L^1$-control of a Timoshenko beam is considered and analyzed by using the finite element method. From the practical point of view, inclusion of the $L^1$-norm in the cost functional is interesting in the case of beam vibration model, since the sparsity enforced by the $L^1$-norm is very useful for localizing actuators or control devices. The discretization of the control variables is performed by using piecewise constant functions. The states and the adjoint states are approximated by a locking free scheme of linear finite elements. Analogously to the purely $L^2$-norm penalized optimal control, it is proved that this approximation have optimal convergence order, which do not depend on the thickness of the beam.