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.

• 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 the hub location problem under interhub link failures, a hub location problem in which activated interhub links may fail with a given probability. Two different optimization models are studied, which construct hub backbone networks protected under interhub link disruptions by imposing that, for each commodity, an additional routing path exists besides its original routing path. Both models consider the minimization of the fixed costs of the activated hubs and interhub links plus the expected value of the routing costs of the original and alternative paths. The first model builds explicitly the alternative routing paths, whereas the second model guarantees that, for each commodity, at least one alternative path exists using a large set of connectivity constraints although the alternative paths are not built explicitly. The results of extensive computational testing allow us to analyze the performance of the two proposed models and to evaluate the extra cost required to design a robust backbone network under interhub link failures. The obtained results support the validity of the proposal. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Application. Funding: The authors of this research acknowledge financial support by the Spanish Ministerio de Ciencia y Tecnología, Agencia Estatal de Investigación and Fondos Europeos de Desarrollo Regional (FEDER) via projects PID2020-114594GB-C21 and MTM2019-105824GB-I00. The authors also acknowledge partial support from projects FEDER-US-1256951, Junta de Andalucía P18-FR-422, P18-FR-2369, B-FQM-322-UGR20 (COXMOS), and NetmeetData: Ayudas Fundación BBVA a equipos de investigación científica 2019. The first author was partially supported by the IMAG-Maria de Maeztu grant [CEX2020-001105-M/AEI/10.13039/501100011033] and UE-NextGenerationEU (ayudas de movilidad para la recualificación del profesorado universitario). Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1296 .

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.

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 $p>1$.

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.

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.