TY - JOUR A1 - Heitsch, Holger A1 - Henrion, René T1 - On the Lipschitz continuity of the spherical cap discrepancy around generic point sets JF - Unif. Distrib. Theory N2 - The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - Lipschitz continuity KW - necessary optimality conditions Y1 - 2025 U6 - https://doi.org/10.2478/udt-2025-0011 VL - 20 IS - 1 SP - 35 EP - 63 ER - TY - JOUR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations JF - Control and Cybernetics N2 - We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the maximum step size for discretizing the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks. KW - Optimization with differential equations KW - Optimal value function KW - Lipschitz continuity KW - Parametric optimization KW - Mixed-integer nonlinear programming Y1 - 2018 ER -