In this paper, we consider set optimization problems and investigate the qualitative properties of their weakly Henig efficient solutions, including solvability, the starshaped property, and stability. First, we introduce nonlinear scalarization functions and examine their properties, such as (semi)continuity and convexity. Next, using these nonlinear scalarization functions, we establish the solvability and scalar representation of the weakly Henig efficient solutions for such problems. Then, based on the aforementioned representation results and generalized convexity conditions on the objective functions, we derive the starshaped property of the solution sets and the stability in the sense of Hausdorff continuity for the weakly Henig efficient solution maps of parametric set optimization problems. Finally, to illustrate the applicability of these results, we consider a welfare economic model and apply the aforementioned findings to establish the corresponding results for this model.
The present study addresses a specific Bus Crew Scheduling problem (CSP) that arises in a public transport company in Spain. The classical CSP problem tries to create bus-driver schedules covering all the bus routes and timetables published by the company. In contrast, the problem considered here does not require all the published schedules to be covered, but rather focuses on optimizing driver assignments and on maximizing passenger collections. This relaxed constraint arises from the difficulty of acquiring new drivers and resources, and from the resulting need to design a tool capable of making intelligent use of those available. We propose an integer programming model for the problem. The real instances to be solved are clustered beforehand to allow general solvers to provide us with an optimal solution. Subsequently, we present a novel bi-objective constructive heuristic that determines a good solution for the instances that we have not solved exactly. As a result, the transportation company can have valuable solutions within a reasonable execution time. This is demonstrated in a final computational experiment.
This study establishes optimality conditions for two classes of solutions in vector optimization problems: Ch -quasi proper non-dominated points and Ch -quasi proper minimal points. These solutions are analysed within a variable ordering structure induced by polyhedral cones, which play a crucial role in modelling constrained multiobjective problems. The proposed approach employs a linear scalarization technique under convexity-type assumptions, including cone-convexity of the vector objective and convexity of the constraint functions. A key contribution of this work is the explicit formulation of the scalarizing functional, which is constructed using a polyhedral dilating cone derived from the ordering cone. This formulation extends classical scalarization theory to the setting of point-dependent polyhedral domination, offering a theoretically tractable pathway for characterizing quasi-proper solutions. The results lay the analytical foundation for future algorithmic development leveraging the facial structure of polyhedral cones.
This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable - the Lipschitz constant for gradients and a concept of the local variance in search directions. Our findings yield a nearly hyperparameter-free algorithm for stochastic optimization, which has provable convergence properties and exhibits truly problem adaptive behaviour on classical image classification tasks. Our framework is set in a general Hilbert space and thus enables the potential inclusion of a preconditioner through the choice of the inner product.
