Electron tomography (ET) is a technique for 3D nanoscale characterization whose practical application is often hampered by severe artifacts arising from an experimentally limited number of projections and a restricted tilt range. While methods like Compressed Sensing (CS) have been developed to address this data scarcity, their performance degrades significantly under highly constrained conditions. This paper introduces a novel deep learning methodology for image restoration that overcomes these limitations. We propose a supervised Convolutional Neural Network (CNN) architecture based on conditional GAN and RIDNet to eliminate artifacts from initial reconstructions. The central innovation lies in our training strategy: the network is trained exclusively on simple geometric primitives, such as circles and squares, thereby circumventing the need for large, complex, and sample-specific training datasets. We demonstrate that a network trained on this simple basis can remarkably generalize to restore complex, irregular nanomaterials that it has never seen. Quantitative and qualitative comparisons demonstrate that our method significantly outperforms traditional CS, producing high-fidelity 3D reconstructions free of common artifacts. This work establishes a broadly applicable and data-efficient restoration framework that presents a robust and accessible tool for improving the reliability of electron tomography in materials science.

We present a novel mathematical optimization framework for outlier detection in multimodal datasets, extending Support Vector Data Description approaches. We provide a primal formulation, in the shape of a Mixed Integer Second Order Cone model, that constructs Euclidean hyperspheres to identify anomalous observations. Building on this, we develop a dual model that enables the application of the kernel trick, thus allowing for the detection of outliers within complex, non-linear data structures. An extensive computational study demonstrates the effectiveness of our exact method, showing clear advantages over existing heuristic techniques in terms of accuracy and robustness.

This paper introduces a novel Support Vector Regression (SVR) model that incorporates Ordered Weighted Average (OWA) operators to differently penalize deviations of observations from the ɛ -strip. The penalty is determined based on the position of each deviation in the ordered vector of all deviations. A key contribution of this work is the development of two nonlinear formulations: a continuous formulation for non-decreasing monotone weight vectors and a mixed-integer formulation for general weight vectors. By leveraging dual approaches associated with these formulations, the model accommodates nonlinear kernel functions. To enhance computational efficiency, two heuristic approaches are proposed for deriving suitable regression hyperplanes in reduced computation times, as compared to exact methods. Additionally, a third heuristic approach is developed to handle nonlinear kernel functions effectively. Computational experiments conducted on both real and synthetic datasets demonstrate that the exact formulations yield remarkable regression functions with respect to the following standard metrics: mean absolute errors (MAE) and mean squared errors (MSE). The heuristic approaches are shown to be particularly efficient for larger datasets, striking a balance between solution quality and computational effort.

Abstract In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP‐hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in time for uniform weights and NP‐hard for non‐uniform weights. On paths, the single facility problem is solvable in time, while the ‐facility problem is NP‐hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo‐polynomial algorithm is developed for the single facility problem on trees with integer parameters.

This paper introduces an ℓ1-norm model based on Total Variation Minimization for tomographic reconstruction. The reconstructions produced by the proposed model are more accurate than those obtained with classical reconstruction models based on the ℓ2-norm. This model can be linearized and solved by linear programming techniques. Furthermore, the complementary slackness conditions can be exploited to reduce the dimension of the resulting formulation by removing unnecessary variables and constraints. Since the efficacy of the reduced formulation strongly depends on the quality of the dual-multipliers used when applying the reduction method, Lagrangian relaxation is used to obtain near-optimal multipliers. This allows solving larger instances in an efficient way.

This paper deals with a cost sensitive extension of the standard Support Vector Machine (SVM) using an ordered weighted sum of the deviations of misclassified individuals with respect to their corresponding supporting hyperplanes. In contrast with previous heuristic approaches, an exact method that applies the ordered weighted average operator in the classical SVM model is proposed. Specifically, when weights are sorted in non-decreasing order, a quadratic continuous formulation is developed. For general weights, a mixed integer quadratic formulation is proposed. In addition, our results prove that nonlinear kernel functions can be also applied to these new models extending its applicability beyond the linear case. Extensive computational results reported in the paper show that the predictive performance provided by the proposed exact solution approaches are better than the ones provided by the classical models (linear and nonlinear kernel) and similar or better than the previous ones provided by the heuristic solution by Maldonado et al. (2018).

In this paper, we extend the methodology developed for Support Vector Machines (SVM) using the $\ell_2$-norm ($\ell_2$-SVM) to the more general case of $\ell_p$-norms with $p>1$ ($\ell_p$-SVM). We derive second order cone formulations for the resulting dual and primal problems. The concept of kernel function, widely applied in $\ell_2$-SVM, is extended to the more general case of $\ell_p$-norms with $p>1$ by defining a new operator called multidimensional kernel. This object gives rise to reformulations of dual problems, in a transformed space of the original data, where the dependence on the original data always appear as homogeneous polynomials. We adapt known solution algorithms to efficiently solve the primal and dual resulting problems and some computational experiments on real-world datasets are presented showing rather good behavior in terms of the accuracy of $\ell_p$-SVM with $p>1$.

We consider 2‐facility location problems with equity measures, defined on networks. The models discussed are, the variance, the mean of absolute weighted deviations, the maximum weighted absolute deviation, the sum of absolute weighted differences, and the range. We give new algorithmic results for these models in the 2‐facility case. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 1–9. 2015

Abstract This article deals with incorporating the mean absolute deviation objective function in several robust single facility location models on networks with dynamic evolution of node weights, which are modeled by means of linear functions of a parameter. Specifically, we have considered two robustness criteria applied to the mean absolute deviation problem: the MinMax criterion, and the MinMax regret criterion. For solving the corresponding optimization problems, exact algorithms have been proposed and their complexities have been also analyzed. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

We consider the single-facility ordered median location problem with uncertainty in the parameters (weights) defining the objective function. We study two cases. In the first case, the uncertain weights belong to a region with a finite number of extreme points, and in the second case, they must satisfy some order constraints and belong to some box (convex case). To deal with the uncertainty, we apply the minimax regret approach, providing strongly polynomial time algorithms to solve these problems. Finally, we also extend the proposed methodology to other problems with order constraints, which are not necessarily convex.

Abstract Given a tree network T with n nodes, let 𝒫L be the subset of all discrete paths whose length is bounded above by a prespecified value L. We consider the location of a path‐shaped facility P ∈ 𝒫L, where customers are represented by the nodes of the tree. We use a bi‐criteria model to represent the total transportation cost of the customers to the facility. Each node is associated with a pair of nonnegative weights: the center‐weight and the median‐weight. In this doubly weighted model, a path P is assigned a pair of values (MAX(P),SUM(P)), which are, respectively, the maximum center‐weighted distance and the sum of the median‐weighted distances from P to the nodes of the tree. Viewing 𝒫L and the planar set {(MAX(P),SUM(P)) : P ∈ 𝒫L} as the decision space and the bi‐criteria or outcome space respectively, we focus on finding all the nondominated points of the bi‐criteria space. We prove that there are at most 2n nondominated outcomes, even though the total number of efficient paths can be Ω(n2), and they can all be generated in O(n log n) optimal time. We apply this result to solve the cent‐dian model, whose objective is a convex combination of the weighted center and weighted median functions. We also solve the restricted models, where the goal is to minimize one of the two functions MAX or SUM, subject to an upper bound on the other one, both with and without a constraint on the length of the path. All these problems are solved in linear time, once the set of nondominated outcomes has been obtained, which in turn, results in an overall complexity of O(n log n). The latter bounds improve upon the best known results by a factor of O(log n). © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(4), 237–247 2006