On the mathematical structure and numerical solution of discrete-continuous optimization problems in DDCM

In this work, we investigate data-driven elasticity problems defined on a closed interval of the real line that are spatially discretized by means of the finite element method. This one-dimensional setting allows us to gain a deeper understanding of the underlying discrete-continuous quadratic optimization problems. We provide an in-depth analysis of their structural properties and prove their global solvability. Based on this analysis, we propose a new structure-specific initialization for a solution strategy relying on an alternating direction method, and we prove that it is globally optimal in certain symmetric cases. Finally and to support our formal mathematical analysis, we also provide a series of examples that show the benefits of this kind of approach and briefly illustrate the challenges when dealing with real experimental data.