## Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

Abstract: 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.

## A SECOND-ORDER METHOD WITH ENRICHED HESSIAN INFORMATION FOR IMAGING COMPOSITE SPARSE OPTIMIZATION PROBLEMS

Abstract: 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.

## NONSMOOTH EXACT PENALIZATION SECOND–ORDER METHODS FOR INCOMPRESSIBLE BINGHAM FLOWS

Abstract: 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.

## An integer programming model to assign patients based on experiential avoidance for tele-psychotherapy intervention during the Covid–19 emergency

Abstract: 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

## estimacion de parametros para modelos del sars-cov-2 en ecuador en presencia de incertidumbre

Abstract: 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

## modelizacion y simulacion de la propagacion del virus sars-cov-2 en ecuador

Abstract: Ante la inminente llegada de la pandemia de la enfermedad Covid 19, conocida como Coronavirus, al territorio ecuatoriano, el Centro de Modelización Matemática de la Escuela Politécnica Nacional conformó un equipo de trabajo especializado para modelizar y simular la propagación del SARS-CoV-2 (virus causante de la enfermedad), bajo varios escenarios de política pública aplicados a la contención del mismo. En este reporte explicamos los modelos utilizados, las simulaciones obtenidas y las recomendaciones para mejorar los modelos y, sobre todo, para el diseño de políticas públicas que ayuden a minimizar los efectos negativos de la pandemia en la población, mientras se desarrollan las actividades productivas esenciales para el país.

## Bilevel parameter optimization for nonlocal image denoising models

Abstract: 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.

## Multi-weight graph partitioning problem

Abstract: 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.

## University timetabling with heterogeneous lectures and compactness constraints

Abstract: 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.

## Integrated Vehicle and Pollster Routing

Abstract: 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.

## Primal-dual proximal splitting and generalized conjugation in non-smooth non-convex optimization

Abstract: 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.

## Block-proximal methods with spatially adapted acceleration

Abstract: 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.

## Inertial, corrected, primal--dual proximal splitting

Abstract: 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.

## A bilevel learning approach for optimal observation placement in variational data assimilation

Abstract: 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.

## Testing and non-linear preconditioning of the proximal point method

Abstract: 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.

## Interior-proximal primal-dual methods

Abstract: 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.

## ERROR ESTIMATES FOR THE FEM APPROXIMATION OF STATE–CONSTRAINED ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH SPARSITY IN A FINITE–DIMENSIONAL CONTROL SPACE

Abstract: 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.

## Acceleration and global convergence of a first-order primal--dual method for nonconvex problems

Abstract: 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.

## Preconditioned proximal point methods and notions of partial subregularity

Abstract: 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.

## Total Generalized Variation Regularization in Variational Data Assimilation for Burgers' Equation

Abstract: 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.

## The set covering polyhedron of circular matrices: Minor vs. row family inequalities

Abstract: 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 $C_{4k}^k$, $k\ge 2$.

## On the Chvátal-rank of facets for the set covering polyhedron of circular matrices

Abstract: 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.

## On dominating set polyhedra of circular interval graphs

Abstract: 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.

## On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

Abstract: 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.

## A non-smooth trust-region method for B-differentiable functions with application to optimization problems constrained by variational inequalities

Abstract: 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.

## A DC–PROGRAMMING APPROACH FOR SPARSE PDE OPTIMAL CONTROL PROBLEMS WITH NONCONVEX FRACTIONAL COSTS

Abstract: 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.

## An Exact Approach for the Balanced k-Way Partitioning Problem with Weight Constraints and its Application to Sports Team Realignment

Abstract: 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.

## A locking-free optimal control problem with $L^1$ cost for optimal placement of control devices in Timoshenko beam

Abstract: 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.

## A MULTIGRID OPTIMIZATION APPROACH FOR THE NUMERICAL SOLUTION OF A CLASS OF VARIATIONAL INEQUALITIES OF THE SECOND KIND

Abstract: In this work we introduce a Multigrid Optimization Algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind, which involve the p-Laplacian operator and the $L^1$-norm of the gradient. The solution of the variational inequality is given by the solution of an optimization algorithm.We proposed a Huber regularization of the functional for the $L^1$-norm of the gradient and a finite element discretization for the problem. Finally, we analyze the performance of the MG/OPT algorithm when used to simulate the visco-plastic flows in a pipe. Several numerical experiments are carried out to show the main features of the proposed method.

## A MULTIGRID OPTIMIZATION APPROACH FOR THE NUMERICAL SOLUTION OF A CLASS OF VARIATIONAL INEQUALITIES OF THE SECOND KIND

Abstract: In this work we introduce a Multigrid Optimization Algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind, which involve the p-Laplacian operator and the $L^1$-norm of the gradient. The solution of the variational inequality is given by the solution of an optimization algorithm.We proposed a Huber regularization of the functional for the $L^1$-norm of the gradient and a finite element discretization for the problem. Finally, we analyze the performance of the MG/OPT algorithm when used to simulate the visco-plastic flows in a pipe. Several numerical experiments are carried out to show the main features of the proposed method.

## OPTIMIZACIÓN BINIVEL DEL PARÁMETRO DE REGULARIZACIÓN CON DEPENDENCIA ESPACIAL DEL MODELO DE VARIACIÓN TOTAL GENERALIZADA PARA EL FILTRADO DE RUIDO EN IMÁGENES

Abstract: In this thesis, we study and solve a nonlinear bilevel optimization problem in function spaces. The goal is to determine the optimal spatially dependent regularization parameters in total variation (TV) and generalized total variation (TGV) image denoising models. Considering the spatial dependence of the parameters allows us to filter non-uniform noise in an image, which brings us closer to real situations where the type and distribution of noise are not known. We present some analytical results like the existence of solutions of the problem of parameter optimization, the Fréchet differentiability of the solution operator, which allows to prove the existence of Lagrange multipliers. The multipliers associated with the positivity constraints are regular Borel measured which are very difficult to compute. In order to overcome this issue, we proposed a Moreau-Yosida regularization, where the optimality system associated with the regularized problem was established and we prove that the solutions of regularized problems converge to the solution of the original one.

## SOLUCIÓN DE UN PROBLEMA DE ASIMILACIÓN DE DATOS Y UN PROBLEMA DE LOCALIZACIÓN ÓPTIMA MEDIANTE MÉTODOS DE OPTIMIZACIÓN BINIVEL

Abstract: Data assimilation problems have been widely studied in numerical weather prediction as a technique for the reconstruction of the atmosphere’s initial condition. Taking this problem as motivation, our goal in this project is finding the optimal placements of the locations of a data assimilation problem, represented by a parabolic equation. We worked in a bilevel optimization problem where the inner level solves the data assimilation problem, and the upper level solves the optimal placement problem. To solve the data assimilation problem, we used a variational approach 4D-VAR. The existence and uniqueness of the data assimilation problem were demonstrated using usual techniques in optimization. By using the Lagrangian approach, we derive the optimality system of the inner problem. As result, we got an adjoint equation with measures on the right-hand side. This was due to the structure of the objective functional. We proved the existence of a unique very weak solution of the adjoint equation by using the transposition method. The numerical solution was also worked in two levels. The inner problem was solved by using the BFGS method while the upper level used the BFGS projected method through an estimation of $\epsilon$-active sets jointly with a modified Armijo line search and a stop criterio which takes full steps.

## PHOTOGRAPHIC DATASET: PLAYING CARDS

Abstract: A photographic dataset is collected for testing image processing algorithms.

## Balanced Partition of a Graph for Football Team Realignment in Ecuador

Abstract: In the second category of the Ecuadorian football league, a set of football teams must be grouped in k geographical zones 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 zone is minimized. This problem can be modeled as a k clique partitioing problem with constraints on the sizes and weights of the cliques. An integer programming formulation and a heuristic approach were developed to provide a solution to the problem which has been implemented in the 2015 edition of this football championship.

## VEHICLE AND DUTY SCHEDULING FOR A PUBLICTRANSPORTATION SYSTEM IN QUITO

Abstract: We address the vehicle and duty scheduling problems in the context of the Trolebus public transportation system of Quito. An integer programming model for the vehicle scheduling problem based on a multi-commodity flow formulation is presented. The model aims at minimizing both the size of the fleet and the total idle time of the vehicles at the terminals. A primal heuristic for solving this model is proposed. This heuristic is later extended to an algorithm for solving the integrated vehicle and duty scheduling problem under the particular circumstances of the studied system. The method is based on the idea of aggregating sequences of trips that minimize the idle time and at the same time conform to legal and administrative constraints which are inherent to the duty scheduling problem. Computational results for real-world data with up to 1,400 timetabled trips are reported.

## LEARNING OPTIMAL SPATIALLY-DEPENDENT REGULARIZATION PARAMETERS IN TOTAL VARIATION IMAGE RESTORATION

Abstract: We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image restoration models. First- and second-order optimality conditions for the bilevel problem are studied, when the spatially-dependent parameter belongs to the Sobolev space $H1( )$. A combined Schwarz domain decomposition semismooth Newton method is proposed for the solution of the full optimality system and local superlinear convergence of the semismooth Newton method is analyzed. Exhaustive numerical computations are finally carried out to show the suitability of the approach.

## Métodos de optimización para la segmentación numérica de imágenes usando el modelo de Chan-Vese

Abstract: La segmentación de imágenes consiste en dividir una imagen en subconjuntos donde cada uno de ellos corresponda a un objeto que la constituye. Durante los últimos años se ha propuesto una gran variedad de modelos de segmentación vinculados a diferentes áreas de la Matemática. En nuestro caso, nos enfocamos en la formulación variacional del problema propuesta por Chan y Vese. El modelo se plantea utilizando conjuntos de nivel con el objetivo de minimizar el funcional de energía asociado al problema de segmentación. El método consiste en la evolución de una curva de nivel que, bajo ciertos criterios, se detiene en el contorno de los objetos que forman la imagen. Resolver numéricamente este problema es, en general, muy costoso. La utilización de métodos de optimización numérica permite dar una solución eficiente a este inconveniente y garantiza la convergencia de la solución hacia un mínimo. Se analizan métodos tales como: el método de descenso explícito, un método de tipo proximal y LBFGS, que combinados con el método del momento, permiten hacer uso de la información de primer y segundo orden del funcional para acelerar los métodos utilizados tradicionalmente.

## Bilevel parameter learning for Higher-order Total Variation Regularisation Models

Abstract: We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost, based on a Huber regularised TV-seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a quasi-Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

## The structure of optimal parameters for image restoration problems

Abstract: We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalized variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original – in image restoration typically non-smooth variational problem – as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it $\Gamma$ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.

## Bilevel approaches for learning of variational imaging models

Abstract: We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.

## A Preconditioned Descent Algorithm for Variational Inequalities of the Second Kind Involving the p-Laplacian Operator

Abstract: This paper is concerned with the numerical solution of a class of variational inequalities of the second kind, involving the p-Laplacian operator. This kind of problems arise, for instance, in the mathematical modelling of non-Newtonian fluids. We study these problems by using a regularization approach, based on a Huber smoothing process. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to the solution of the original problem is verified. We propose a preconditioned descent method for the numerical solution of these problems and analyze the convergence of this method in function spaces. The existence of admissible descent directions is established by variational methods and admissible steps are obtained by a line search algorithm, which combines the classical Armijo’s rule with an interpolation technique. Finally, several numerical experiments are carried out to show the efficiency of the methodology here introduced.

## A bus scheduling model for a public transportation system in Quito

Abstract: We propose an integer programming model for the bus scheduling problem in the METROBUS-Q public transportation system of Quito. The model is based on a multi-commodity flow formulation of the problem, and aims at minimizing both the size of the fleet and the total idle time of the vehicles in the terminals. Two primal solution heuristics and an algorithm for obtaining lower bounds from a mincost relaxation of the problem are presented. Finally, computational results for real-world data with up to 1,400 timetabled trips are reported.

## An Integer Programming Model for Scheduling Classes at the Escuela Politécnica Nacional in Quito

Abstract: We present an integer programming model for scheduling lectures at a university, taking into account restrictions imposed by the availability of lecturers and classrooms, as well as by the students’ curricula. We propose a constructive heuristic and an local improvement heuristic to obtain feasible solutions for this model, and report computational results for real instances concerning the timetable planning at the Escuela Politécnica Nacional in Quito.

## Generalized minor inequalities for the set covering polyhedron related to circulant matrices

Abstract: We study the set covering polyhedron related to circulant matrices. In particular, our goal is to characterize the first Chv´atal closure of the usual fractional relaxation. We present a family of valid inequalities that generalizes the family of minor inequalities previously reported in the literature and includes new facet-defining inequalities. Furthermore, we propose a polynomial time separation algorithm for a particular subfamily of these inequalities.

## Finite element error estimates for an optimal control problem governed by the burgers equation.

Abstract: We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady onedimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, an L2 superlinear order of convergence for the control is obtained; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to h3/2. The theoretical findings are tested experimentally by means of numerical examples.

## Imaging with Kantorovich-Rubinstein discrepancy

Abstract: We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this ap-proach to several other recently proposed methods such as total general-ized variation and norms capturing oscillating patterns. We also show thatthe respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence, can be solved by standard tools. Numerical examples exhibit interesting features and favourable performance for denoising and cartoon-texture decomposition.

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