## A Nonsmooth Trust-Region Method for Locally Lipschitz Functions with Application to Optimization Problems Constrained by Variational Inequalities

Abstract: We propose a general trust-region method for the minimization of nonsmooth and nonconvex, locally Lipschitz continuous functions that can be applied, e.g., to optimization problems constrained by elliptic variational inequalities. The convergence of the considered algorithm to C-stationary points is verified in an abstract setting and under suitable assumptions on the involved model functions. For a special instance of a variational inequality constrained problem, we are able to properly characterize the Bouligand subdifferential of the reduced cost function, and, based on this characterization result, we construct a computable trust-region model which satisfies all hypotheses of our general convergence analysis. The article concludes with numerical experiments that illustrate the main properties of the proposed algorithm.

## An exact approach for the multi-constraint graph partitioning problem

Abstract: In this work, a multi-constraint graph partitioning problem is introduced. The input is an undirected graph with costs on the edges and multiple weights on the nodes. The problem calls for a partition of the node set into a fixed number of clusters, such that each cluster satisfies a collection of node weight constraints, and the total cost of the edges whose end nodes are in the same cluster is minimized. It arises as a sub-problem 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 with the respective polyhedra are proved. An exact algorithm based on Branch & Bound and cutting planes is proposed, and it is tested on real-world instances.

## An exact approach for the multi-constraint graph partitioning problem

Abstract: In this work, a multi-constraint graph partitioning problem is introduced. The input is an undirected graph with costs on the edges and multiple weights on the nodes. The problem calls for a partition of the node set into a fixed number of clusters, such that each cluster satisfies a collection of node weight constraints, and the total cost of the edges whose end nodes are in the same cluster is minimized. It arises as a sub-problem 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 with the respective polyhedra are proved. An exact algorithm based on Branch & Bound and cutting planes is proposed, and it is tested on real-world instances.

## Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and 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 a 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 on 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 for the state equation.

## Modeling multiple wave systems in the eastern equatorial Pacific

Abstract: While moderate wind and wave conditions prevail in the eastern equatorial Pacific, modeling waves in this area remains challenging due to the presence of multiple wave systems converging from different parts of the ocean. This area is covered by swells originated far away including the storm belts of both hemispheres, coexisting with local generation due to the regular action of both the southern trade winds and the wind jets from Central America. In this context, our ability to predict waves in the area depends on the overall quality (i.e., at Pacific scale) of the meteorological input, and also on the skills of the wave model itself. Clearly any error at the remote generation areas translates into larger errors the further waves go, especially if attention is focused on coastal areas. A relevant aspect is that the traditional integral parameters do not offer the possibility to properly assess the errors associated with the different parts of the spectrum (e.g., wind sea and swell). To gain insight in this direction, we make use of partitioning techniques, which enables us to neatly cross-assign and evaluate three spectral components. Not surprisingly, the performance for the swell part is lower than that of the corresponding wind sea. This is further explored with a couple of tests modifying both the wind input and the wave model physics. We find that although at first sight the initial scheme (i.e., ST4) seems to provide the better estimate, the spectral analysis reveals a substantial underestimation of wind sea, compensated with a substantial overestimation of swell. This suggests a problem with too high winds and wave generation in the storm belts together with a likely lack of dissipation or dispersion of swell. In turn, local waves are generally underestimated due to a corresponding underestimation of the local winds. This insight emphasizes the need and advantages of evaluation methods able to look at the different sectors of the wave spectrum.

## A BDF2-semismooth Newton algorithm for the numerical solution of the Bingham flow with temperature dependent parameters

Abstract: This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we discuss the existence of solutions for the problem, and focus on a multiplier formulation which leads us to a coupled system of PDEs involving a Navier–Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of the regularized problem. A detailed finite element discretization, based on the so called (cross-grid ) - elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton (SSN) algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiments that exhibit the main properties of the numerical approach

## Covariate Extreme Value Analysis Using Wave Spectral Partitioning

Abstract: Covariate Extreme Value Analysis Using Wave Spectral PartitioningJESÚSPORTILLA-YANDÚNResearch Center of Mathematical Modelling (MODEMAT), and Department of MechanicalEngineering, Escuela Politécnica Nacional, Quito, EcuadorEDWINJÁCOMEDepartment of Mechanical Engineering, Escuela Politécnica Nacional, Quito, Ecuador(Manuscript received 5 December 2019, in final form 16 March 2020)ABSTRACTAn important requirement in extreme value analysis (EVA) is for the working variable to be identicallydistributed. However, this is typically not the case in wind waves, because energy components with differentorigins belong to separate data populations, with different statistical properties. Although this information isavailable in the wave spectrum, the working variable in EVA is typically the total significant wave heightHs,aparameter that does not contain information of the spectral energy distribution, and therefore does not fulfillthis requirement. To gain insight in this aspect, we develop here a covariate EVA application based onspectral partitioning. We observe that in general the totalHsis inappropriate for EVA, leading to potentialover- or underestimation of the projected extremes. This is illustrated with three representative cases undersignificantly different wave climate conditions. It is shown that the covariate analysis provides a meaningfulunderstanding of the individual behavior of the wave components, in regard to the consequences for pro-jecting extreme values.

## Can we extrapolate climate in an inner basin? The case of the Red Sea

Abstract: We examine the possibility of making useful climate extrapolations in inner basins. Stressing the role of the local geographic features, for a practical example we focus our attention on the Red Sea. We observe that in spite of being an enclosed and relatively small Sea, its climate conditions are heavily affected by those of the larger neighboring regions, in particular the Mediterranean and the Arabian Seas. Using existing high-resolution information of the recent decades, we use both reasoned extrapolation and knowledge of, past and future, longer term general climatic information to frame what is presently possible to assess for the Red Sea. Specifically, the northern part, influenced by the Mediterranean Sea, shows a clear decreasing trend of both the meteorological and wave conditions in the recent decades.