Optimizing Oblique Projections for Nonlinear Systems using Trajectories

The paper’s Appendix E proves exactly the claims at issue: for the scaled Riemannian Dai–Yuan (sRDY) CG method with the associated retraction/transport on Gn,r × Gn,r, one can always choose a step satisfying weak Wolfe, which yields monotone decrease J(Vk+1,Wk+1) ≤ J(Vk,Wk) and, moreover, a sufficiently small step keeps the entire tested segment within the initial sublevel set; under a closed sublevel-set hypothesis Dc ⊂ D, the method satisfies lim inf ||∇J|| = 0 (Theorem E.1 and its proof) . The algorithmic ingredients (retraction, transport, sRDY coefficient, Wolfe conditions) are defined in Section 5.3 and built from a quotient construction on the product Grassmannian . The candidate solution reproduces the same structure: descent via Sato’s sRDY, existence of a Wolfe step and monotonicity via the standard Wolfe existence lemma, a small-step sublevel-set argument, and global convergence under Dc using Lipschitz bounds on D(J∘R). A minor nuance is that the paper avoids assuming the search curve is globally defined by constructing αk inside D through a sublevel-set inclusion (their (E.12)–(E.2)), whereas the model invokes the classical Wolfe existence theorem before establishing the same inclusion. Otherwise, the proofs are the same in substance.