Geometric optimization via composite majorization

Abstract

Many algorithms on meshes require the minimization of composite objectives, i.e., energies that are compositions of simpler parts. Canonical examples include mesh parameterization and deformation. We propose a second order optimization approach that exploits this composite structure to efficiently converge to a local minimum.Our main observation is that a convex-concave decomposition of the energy constituents is simple and readily available in many cases of practical relevance in graphics. We utilize such convex-concave decompositions to define a tight convex majorizer of the energy, which we employ as a convex second order approximation of the objective function. In contrast to existing approaches that largely use only local convexification, our method is able to take advantage of a more global view on the energy landscape. Our experiments on triangular meshes demonstrate that our approach outperforms the state of the art on standard problems in geometry processing, and potentially provide a unified framework for developing efficient geometric optimization algorithms.

Accelerating ADMM for efficient simulation and optimization

“…Claici et al [2017] proposed a preconditioner for fast minimization of distortion energies. Shtengel et al [2017] applied the idea of majorizationminimization [Lange 2004] to iteratively update and minimize a convex majorizer of the target energy in geometric optimization. Zhu et al [2018] proposed a fast solver for distortion energy minimization, using a blended quadratic energy proxy together with improved line-search strategy and termination criteria.…”

“…Such tasks are often formulated as unconstrained optimization, where the target function penalizes the violation of certain conditions so that they are satisfied as much as possible by the solution. It has been an active research topic to develop fast numerical solvers for such problems, with various methods proposed in the past [Sorkine and Alexa 2007;Liu et al 2008;Bouaziz et al 2012;Wang 2015;Kovalsky et al 2016;Liu et al 2017;Shtengel et al 2017;Rabinovich et al 2017].…”

Anderson acceleration for geometry optimization and physics simulation

“…[2017] presented a scalable approach for optimizing flip-preventing energies, using a reweighted proxy function in each iteration. Shtengel et al [2017] derived convex majorizers of composite energies via convex-concave decompositions, which are repeatedly updated and minimized to obtain a minimum of the target energy.…”

A Subspace Method for Fast Locally Injective Harmonic Mapping

“…Further timing comparisons to composite majorization (CM) [SPSH∗17] on selected group B models are in Table . Here, our method also possesses a significant speedup, and can process large models with millions of triangles in several seconds.…”