Fejér and Fejér* Monotonicity: New Results and Limiting Examples

Many algorithms in convex optimization and variational analysis can be analyzed using Fejér monotone sequences. In 2024, Behling, Bello-Cruz, Iusem, Alves Ribeiro, and Santos introduced a new, more general, notion: Fejér* monotonicity. They obtained basic results and discussed applications in optimization. In this work, we complement Behling et al.’s work by presenting a thorough study of Fejér* monotonicity. We reveal striking similarities and differences between these notions, including descriptions of the maximal Fejér* set. Moreover, we also touch upon Opial sequences and quasi-Fejér monotonicity. Throughout this paper, we provide numerous limiting examples and counterexamples.

💡 Research Summary

This paper offers a comprehensive study of the recently introduced notion of Fejér* monotonicity, which generalizes the classical Fejér monotonicity used to analyze many convex‑optimization and variational‑analysis algorithms. The authors begin by recalling the definitions: a sequence (xₙ) is Fejér monotone with respect to a non‑empty set C if the distance to every point y∈C never increases, i.e., ‖xₙ₊₁−y‖ ≤ ‖xₙ−y‖ for all n. Fejér* monotonicity relaxes this requirement: for each y∈M there exists an index N(y) such that the inequality holds for all n ≥ N(y).

A central construction is the family of closed convex half‑spaces Cₙ = {y | ‖xₙ₊₁−y‖ ≤ ‖xₙ−y‖}. The pointwise limit inferior lim Cₙ = ⋂ₙ ⋃_{k≥n} C_k is shown (Proposition 2.9) to be the maximal Fejér set*: a sequence is Fejér* monotone with respect to M iff M ⊆ lim Cₙ. Unlike the classical case where the maximal Fejér set is the intersection ⋂ₙ Cₙ (always closed), lim Cₙ may be non‑closed (Example 3.2), highlighting a fundamental difference between the two notions.

The paper proves that every Fejér* sequence is bounded and Opial (Proposition 2.2), so Opial’s Lemma and its strong‑convergence counterpart apply. Nevertheless, many familiar properties of Fejér monotone sequences fail for Fejér* sequences. The authors present a series of counter‑examples: distances d_M(xₙ) can eventually increase (Example 3.14); the shadow sequence (P_M xₙ) may not converge weakly (Example 3.19(iv)); and even when M is an affine subspace and (xₙ) converges weakly to a point in M, the limit need not be the projection of the initial point (Example 3.5). Thus, Fejér* monotonicity is strictly weaker than Fejér monotonicity.

A striking positive result emerges when the relative interior of M is non‑empty. Theorem 4.1 shows that for any non‑empty compact K ⊂ ri(M), the sequence eventually becomes Fejér monotone with respect to K. This fails if K is only weakly compact (Example 4.3). Moreover, if C is a non‑empty closed convex subset of aff(M), the shadow sequence (P_C xₙ) has a finite‑length trajectory and therefore converges strongly (Corollary 4.7). This corollary is new even for the classical Fejér setting. Further, when the limit point z lies in ri(M), the distance sequence (d_M(xₙ)) eventually becomes decreasing (Corollary 4.11), and if z∈aff(M) the iterates either eventually equal z or never equal z (Corollary 4.14).

Section 5 provides quantitative directional results. Assuming xₙ → z and defining K = cone(M−z), Theorem 5.5 proves that the set of weak cluster points of the normalized differences (xₙ−xₙ₊₁)/‖xₙ−xₙ₊₁‖ is contained in the negative polar cone K⁻, extending a known Fejér result. In finite dimensions, Corollary 5.7 strengthens this to d_K(xₙ−z)/‖xₙ−z‖ → 1, showing that the iterates approach z essentially along the cone K. When ri(M)=∅, Theorem 5.10 guarantees a constant Γ>0 such that ‖xₙ−z‖ ≤ Γ d_K(xₙ−z) eventually, a bound shown to be sharp by Example 5.12. An “exclusion” result (Corollary 5.13) states that if consecutive terms are distinct, ri(M)=∅ and z∈aff(M), then eventually xₙ lies outside the translated cone z+K. These estimates are useful for establishing linear convergence rates (Remark 5.18) and for clarifying a previously obscure proof in the literature (Remark 5.19).

Section 6 characterizes the maximal Fejér* set and the maximal Opial set when int(M)=∅. Theorem 6.3 describes lim Cₙ (up to closure and interior) via the polar cone of all weak limits of the normalized difference vectors (xₙ−xₙ₊₁)/‖·‖. An analogous description is given for the maximal Opial set (Theorem 6.13).

Section 7 investigates the relationship between Fejér*, Fejér, and quasi‑Fejér monotonicity (types I, II, III). It is known that any Fejér* sequence is automatically quasi‑Fejér of type III. The authors prove that if M is bounded and ri(M)=∅, then the sequence is also quasi‑Fejér of type II (Proposition 7.8), and this is sharp (Examples 7.10, 7.11). If int(M)=∅, the sequence is quasi‑Fejér of type I (Corollary 7.12). More generally, Theorem 7.14 shows that Fejér* monotonicity implies quasi‑Fejér type I under very mild conditions, leaving only a narrow window where the implication can fail when ri(M)≠∅. Finally, Corollary 7.15 notes that for closed M with ri(M)=∅, every Fejér* sequence is quasi‑Fejér type I.

The paper is organized as follows: Section 2 reviews Opial sequences, proves the maximality of lim Cₙ, discusses relative interiors, and recalls cone properties. Section 3 supplies a battery of limiting examples illustrating the stark contrast between Fejér and Fejér* behavior. Section 4 shows that a non‑empty relative interior restores many Fejér‑like properties. Section 5 develops quantitative directional estimates. Section 6 gives structural descriptions of maximal Fejér* and Opial sets. Section 7 connects Fejér* to quasi‑Fejér monotonicity. Appendices contain detailed constructions of the examples.

In summary, the authors systematically develop the theory of Fejér* monotonicity, delineate its similarities and crucial differences from classical Fejér monotonicity, and provide a rich collection of examples, counter‑examples, and quantitative tools. The results deepen our understanding of convergence mechanisms in modern splitting algorithms and open new avenues for designing algorithms that exploit the relaxed Fejér* framework, especially when the underlying Fejér* set possesses a non‑empty relative interior.