The Gilbert–Steiner problem is a generalization of the Steiner tree problem and specific optimal mass transportation, which allows the use additional (branching) point in a transport plan. A specific feature of the problem is that the cost of transporting a mass m along a segment of length l is equal to l × m^p for a fixed 0 < p < 1 and segments may end at points not belonging to the supports of given measures (branching points). Main result of this paper determines all pairs of (p,d) for which the Gilbert–Steiner problem in ℝ^d admits only branching points of degree 3. Namely, it happens if and only if d = 2 or p < 1/2.
We introduce a robust classification model designed for feature selection. Support vector machine (SVM) models continue to play a crucial role in binary classification, particularly with tabular data. Their robust variants are essential for developing classifiers that remain stable despite shifts in data distribution. Additionally, sparse classifiers are highly desirable, as they offer improved performance in classification tasks and help reduce overfitting, especially when the number of features exceeds the number of samples. In this context, penalty methods for feature selection are fundamental to the development of sparse optimization models. Despite its inherent nonlinearity and nonconvexity, the l(0)-norm remains a popular choice for this task and has been effectively applied in diverse fields, such as pattern recognition and signal processing. Our objective is to evaluate the performance of two classification approaches that integrate robust classification in SVM-type models with embedded feature selection. We propose a diagonal algorithm for numerically solving these optimization models, which we validate through numerical tests on benchmark datasets. The results are compared to those obtained using models based on t2-norm regularization.
Variationally convex functions and prox-regular functions are placed in a single framework that coordinates and enhances the many connections between them. Their characterization in terms of monotonicity properties of subgradient mappings is confirmed by a more informative argument than has previously been offered, which yields more a definitive result.
S.-T. Hu [Boundedness in topological spaces, J. Math. Pures Appl. 228 (1949) 287-320] characterized those bornologies on a metrizable space that are metric bornologies, i.e., that agree with the bornology of metrically bounded subsets as determined by some metric compatible with the topology. We show that the family of such metric bornologies forms a sublattice of the complete lattice of all bornologies on the underlying set which furthermore is stable under denumerable suprema in that larger lattice, but not necessarily denumerable infima. We characterize those metrizable spaces for which the bornology of relatively compact subsets-the intersection of the family of metric bornologies in any case-is itself a metric bornology. Finally, we characterize those metrizable spaces for which each bornology on the space with a countable closed base (resp. countable open base) is already a metric bornology.
This paper has two sides: a mathematical side and a behavioral side. In the first part, we consider the unconstrained multiobjective optimization problem of finding Pareto critical points of difference of convex functions. We study a method that minimizes at each iteration a convex approximation instead of the (non-convex) objective function constrained to a possible non-convex set which imposes the method decreases, at each step, each objective function. In the second part, as an application, we propose a variational rationality perspective to multiproduct firm problem that wants to improve progressively the profits of all its sub-units by adjusting their production levels. This approach is completly new in management sciences/economics. It opens the door to a variational rational theory of the multiproduct firm.
