Practical competitiveness of incomplete interpolation sets in subspace methods

Determine whether modifying the geometry-correcting subspace algorithm to initialize with only one interpolation point and no auxiliary set of points, together with corresponding modifications to the model-improvement and Lagrange-polynomial computations for incomplete sets, can produce practically competitive performance.

Background

The paper describes a geometry-correcting trust-region method operating in randomly selected subspaces. The theoretical version initializes each newly selected subspace with interpolation sets containing multiple points, but this initialization requires recomputing function values and is reported to be computationally expensive.

The authors note that the method could instead initialize a subspace with only one interpolation point and an empty auxiliary set, provided that the model-improvement procedure and Lagrange-polynomial calculations are adapted to incomplete interpolation sets. Although the modification is described as theoretically simple, its practical competitiveness is explicitly left unresolved.

References

We can also choose ${\cal Y}_0$ to contain only 1 point and ${\cal Z}_0$ to be empty by modifying model improvement step and Lagrange polynomial computation to allow for incomplete sets. This modification is simple from the theory point of view but whether it can be practically competitive is yet unclear.

— Powell-Style Model-Based Derivative-Free Optimization with Complexity Guarantees  (2609.09441 - Chaudhry et al., 8 Sep 2026) in Section 6, subsection “Geometry-correcting algorithm in subspaces,” discussion immediately preceding Algorithm 8