Theoretical Guarantees for HexOpt Optimization Performance

Establish rigorous theoretical proofs that guarantee the optimization performance of the HexOpt algorithm, which maximizes the Rectified Hybrid Quadratic Jacobian objective for hexahedral meshes under surface projection equality constraints using the augmented Lagrangian method, L-BFGS updates, and Armijo line search.

Background

HexOpt improves all-hexahedral mesh quality by maximizing a hybrid Jacobian-based objective while enforcing exact surface fitting through augmented Lagrangian constraints. The optimization employs L-BFGS for search directions and Armijo line search for step sizes. Although empirical results show robust performance across diverse models, the paper notes the absence of formal theoretical guarantees.

The authors highlight that prior approaches often only prove monotonic quality improvement under restrictive settings and do not provide lower bounds on mesh quality. In contrast, HexOpt seeks broader guarantees, motivating the need for formal proofs of optimization performance (e.g., convergence and quality guarantees) for the specific AL + L-BFGS + Armijo framework applied to hexahedral mesh optimization with surface constraints.