arXiv subjects

Lukas Hager

Publications and source records attributed to Lukas Hager.

3 recordsLinked to original sources

arXiv2026

Optimal triangulations for piecewise linear approximations of non-convex variable products

We show optimal triangulations for piecewise linear (PWL) approximations of indefinite quadratic functions over the plane. Optimal triangulations have minimum triangle density while allowing a PWL approximation that fulfills a prescribed error bound measured in the L-infinity norm. In 2000, Pottmann et al. proved optimal triangulations for PWL interpolations and conjectured that these are also optimal for general PWL approximations. This conjecture was refuted in 2018 by Atariah et al., who allowed a constant deviation at the vertices of the triangles and decreased the triangle density by roughly 3%, though they left open whether their construction was optimal. In this paper, we resolve this open question: allowing varying deviations at vertices reduces the optimal triangle density by 25% compared to Atariah et al., and we prove this is globally optimal. We thus show that the potential of general PWL approximations is significantly lower for indefinite than for definite quadratic functions, where the triangle density can be halved when allowing general approximations compared to interpolations. Furthermore, we prove that among parallelogram tilings -- triangulations built from translated copies of a triangle and its point-reflection -- the constant-deviation construction of Atariah et al. is optimal when continuity of the PWL approximation is required. We conjecture that this optimality extends to all continuous triangulations, not just those based on parallelograms.

math.OC↗

arXiv2023

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part I

We study mixed-integer programming (MIP) relaxation techniques for the solution of non convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present MIP relaxation methods for non convex continuous variable products. In Part I, we consider MIP relaxations based on separable reformulation. The main focus is the introduction of the enhanced separable MIP relaxation for nonconvex quadratic products of the form z=xy, called hybrid separable (HybS). Additionally, we introduce a logarithmic MIP relaxation for univariate quadratic terms, called sawtooth relaxation. We combine the latter with HybS and existing separable reformulations to derive MIP relaxations of MIQCQPs. We provide a comprehensive theoretical analysis of these techniques, underlining the theoretical advantages of HybS compared to its predecessors. We perform a broad computational study to demonstrate the effectiveness of the enhanced MIP relaxation in terms of producing tight dual bounds for MIQCQPs. In Part II, we study MIP relaxations that extend the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT) and present further theoretical and computational analyses.

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arXiv2023

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part II

This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multi-parametric disaggregation technique (NMDT), applying a sophisticated discretization to both variables. We refer to this approach as doubly discretized normalized multiparametric disaggregation technique (D-NMDT). In a comprehensive theoretical analysis, we underline the theoretical advantages of the enhanced method D-NMDT compared to NMDT. Furthermore, we perform a broad computational study to demonstrate its effectiveness in terms of producing tight dual bounds for MIQCQPs. Finally, we compare D-NMDT to the separable MIP relaxations from Part I and a state-of-the-art MIQCQP solver.

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