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

Abstract

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 <i>discrete-continuous quadratic optimization problems</i>. 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 <i>alternating direction method</i>, 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.