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.

ABSTRACT This paper introduces a novel combinatorial optimization problem with ordering constraints, termed the Ordered Median Traveling Salesman Problem (OMTSP). The OMTSP integrates key elements from both the classic Traveling Salesman Problem (TSP) and the Ordered Median Location Problem. Specifically, the objective is to identify a tour that minimizes a weighted sum of the sorted arc lengths within the tour. This flexible framework enables the modeling of a wide range of combinatorial optimization problems related to the traditional TSP, like the bottleneck TSP, the balanced TSP, or other TSP variants in which fairness measures are applied to the arcs of the tour. In this work, we present new mathematical formulations for the OMTSP across different variable spaces, which are solved using a branch‐and‐cut approach. Leveraging several newly derived structural properties, we enhance these formulations through advanced preprocessing strategies, variable bounding and variable fixing techniques, as well as through the introduction of new valid inequalities. A comprehensive computational study is conducted to evaluate the effectiveness of the proposed formulations and their associated improvements.

• Optimization-based method for individualized reclassification under tree ensembles. • Mathematical model to build feasible shifts with bounded modification effort. • Robust FS variants using min-probability and CVaR formulations. • Application to obesity data enabling feature-importance-based reclassification. • Validation across UCI datasets confirming model generality and robustness. In this paper we provide a novel mathematical optimization based methodology to perturb the features of a given observation to be re-classified, by a tree ensemble classification rule, to a certain desired class. The method is based on these facts: the most viable changes for an observation to reach the desired class do not always coincide with the closest distance point (in the feature space) of the target class; individuals put effort on a few number of features to reach the desired class; and each individual is endowed with a probability to change each of its features to a given value, which determines the overall probability of changing to the target class. Putting all together, we provide different methods to find the features where the individuals must exert effort to maximize the probability to reach the target class. Our method also allows us to rank the most important features in the tree-ensemble. The proposed methodology is tested on different real datasets, validating the proposal.

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.

Regression analysis is an important instrument to determine the effect of the explanatory variables on response variables. When outliers and bias errors are present, the standard weighted least squares estimator may perform poorly. For this reason, many alternative robust techniques have been studied in literature. In these terms, the Least Squares Quantile (LQS), and in particular the Least Squares Median, are among the regression estimators that exhibit better robustness properties. However, the accurate computation of this estimators is computationally demanding, resulting in a difficult estimator to obtain. In this paper, new novel approaches to compute a global optimal solution for the LQS estimator based on single-level and bilevel optimization methods are proposed. An extensive computational study is provided to support the efficiency of the methods considered, and an ad hoc procedure to address the scalability of the problem to larger instances is proposed.

Recent developments in the interplay between Operational Research and Statistics allowed us to exploit advances in Mixed-Integer Optimisation (MIO) solvers to improve the quality of statistical analysis. In this work, we tackle Canonical Correlation Analysis (CCA), a dimensionality reduction method that jointly summarises multiple data sources while retaining their dependency structure. We propose a new technique for encoding sparsity in CCA by means of a mathematical programming formulation that allows one to obtain an exact solution using readily available solvers (such as Gurobi) or design solution algorithmic procedures based on it. Finally, we evaluate the performance of alternative solution strategies presented on multiple datasets from the literature. The results of the extensive comparison study highlight that the proposed approach is capable of finding the optimal correlation or finding good quality solutions, better than those provided by other conventional methods.

Abstract In this paper we present a novel mathematical optimization-based methodology to construct tree-shaped classification rules for multiclass instances. Our approach consists of building Classification Trees in which, except for the leaf nodes, the labels are temporarily left out and grouped into two classes by means of a SVM separating hyperplane. We provide a Mixed Integer Non Linear Programming formulation for the problem and report the results of an extended battery of computational experiments to assess the performance of our proposal with respect to other benchmarking classification methods.

Abstract This article deals with the problem of partitioning a graph into connected components by optimizing some balancing objective functions related to the vertex weights. Objective functions based on the gap or range of the partition's components, that is, the difference between the maximum and minimum weight of a vertex in the component, have been already introduced in the literature. Here we introduce the notion of aggregated gap, defined as the sum of the differences between the weights of the vertices and the minimum weight of a vertex in the component. We study new connected ‐partitioning problems whose objective is a function of the components' aggregated gap, and give NP‐hardness results for these problems on general graphs. Mathematical programming formulations are proposed for these problems adopting flow‐based constraints for modeling connectivity in a partition. Even if they are introduced for the new aggregated gap problems, such formulations are rather general and apply also to the classical non‐aggregated gap problems. Extensive computational tests, both for aggregated and non‐aggregated gap problems, are performed on a set of squared grids and randomly generated graphs with up to 120 vertices, and a number of components ranging from 2 to 9. In our experiments, we test several alternative formulations for our problems providing a comparative analysis of their performance.

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).

The discrete ordered median problem (DOMP) is formulated as a set-partitioning problem using an exponential number of variables. Each variable corresponds to a set of demand points allocated to the same facility with the information of the sorting position of their corresponding costs. We develop a column generation approach to solve the continuous relaxation of this model. Then we apply a branch-price-and-cut algorithm to solve small- to large-sized instances of DOMP in competitive computational time.

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 a connected graph G with n vertices, p of which are centers, while the remaining ones are units. For each unit‐center pair, there is a fixed assignment cost and for each vertex there is a nonnegative weight. In this article, we study the problem of partitioning G into p connected components such that each component contains exactly one center (p‐centered partition). We analyze different optimization problems of this type by defining different objective functions based on the assignment costs, or on the vertices' weights, or on both of them. For these problems, we show that they are NP‐hard on very special classes of graphs, and for some of them we provide polynomial time algorithms when G is a tree. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(1), 69–81 2016

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

This paper addresses the problem of decomposing a numerical semigroup into $m$-irreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so-called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal $m$-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we prove that there is a suitable transformation that reduces the original problem to find an optimal solution of a compact integer linear problem. This result ensures a polynomial time algorithm for each given multiplicity $m$. We have implemented the different algorithms and have performed some computational experiments to show the efficiency of our methodology.

Abstract This work studies the problem of finding optimal paths with respect to the center, median and centdian objective functions, on networks with uncertain vertex weights that are given as intervals. Our approach looks for minimax regret paths which minimize the worst‐case opportunity loss in the corresponding objective function. These problems are NP‐hard on general graphs, therefore we study them on trees. We show that a discrete optimal path always exists for each of them, and provide polynomial time solution algorithms. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Flexible discrete location problems are a generalization of most classical discrete locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer, which depends on the position of that cost relative to the costs of fulfilling the demand of other customers. In this paper a covering type of model for the discrete ordered median problem is presented. For the solution of this model two sets of valid inequalities, which reduces the number of binary variables tremendously, and several variable fixing strategies are identified. Based on these concepts a specialized branch & cut procedure is proposed and \nextensive computational results are reported.

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 In this article we consider the problem of locating path‐shaped facilities on a tree network, minimizing the variance objective function. This type of objective is generally adopted in location problems arising in public sector applications, such as the location of evacuation or mass transit routes. We consider a weighted tree, in which a positive weight is assigned to each vertex of the tree, and positive real lengths are associated with its edges. We study the general case in which the path is continuous, that is, the end points of the optimal path can be either vertices, or points along an edge, and there is an upper bound on the length of the path. Given a tree with n vertices, for this problem we provide an O(n2) algorithm, and we show how it can be applied, with the same complexity, to the discrete case, that is, when the end points of the optimal path are vertices of the tree. We improve the previous best complexity bound in (Cáceres et al., Discr Appl Math 145 (2004), 72–79), for the unrestricted length continuous path‐variance problem, by a factor of log n. We also show that the optimal point for the variance objective function does not satisfy any nestedness property with respect to the optimal path in the unconstrained (discrete or continuous) version of the problem. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gröbner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gröbner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “á la Buchberger” is developed to compute partial Gröbner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.

Abstract This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit‐center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP‐hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

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

Abstract In this paper, we address the ordered p‐median problem, which includes as special cases most of the classical multifacility location problems discussed in the literature. Finite dominating sets (FDS) are known for particular instances of this problem: p‐median, p‐center, and p‐centdian. We find an FDS for the ordered p‐median problem. This set allows us to gain a better insight into the general FDS structure of network location problems. This FDS is later used to present the first polynomial time algorithm for p‐facility ordered median problems on tree networks. This result is combined with some approximation algorithms to give an O(log M log log M) approximate solution of these problems on general networks, where M is the number of vertices. © 2002 Wiley Periodicals, Inc.

In this paper, we introduce a new type of single-facility location problem on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominating set for the optimal locations are developed. Also, the extension to the multifacility case is discussed. The frontiers for finding easy finite dominating sets are shown by a counterexample. © 1999 John Wiley & Sons, Inc. Networks 34: 283–290, 1999