{"id":"3a43c8c6-5ff1-44b7-b052-ad507cfba531","arxiv_id":"2507.18363","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A proximal quasi-Newton method built on local model functions converges to stationary points without assuming bounded variable metrics, and under the KL property the whole sequence converges.","lead":"This math paper proposes a proximal quasi-Newton optimization algorithm for nonsmooth and nonconvex problems that does not require the Hessian approximation to be bounded in advance. It proves convergence to stationary points under a Kurdyka-Łojasiewicz condition and tests the method on feasibility and quadratic inverse problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3 and Assumption 4.2(H2) are the load-bearing junction: the bound is stated locally in (3.6) but used globally, and the motivating examples only satisfy it locally.","rationale":"The reader's weakest assumption identifies exactly the same junction: Assumption 4.2(H2) and the unproved, locally stated Proposition 3.3. My stress-test confirms that this is the most load-bearing point. If H2 is global, Theorem 4.11 can go through only under an assumption much stronger than the local model definition, weakening the paper's claimed generality but not disproving the theorem. If H2 is only local, the proofs of Proposition 4.6 and Lemma 4.10 lack a uniform-radius argument, so the central convergence result is incomplete. The x^4 example shows the global statement is not automatic. This does not refute the main theorem for problems satisfying global H2; it means the manuscript must either prove Proposition 3.3 with a clear global/local specification, add an explicit uniform-local H2 hypothesis, or restrict the claims accordingly. Since the reader already issued CONDITIONAL for missing proofs and overstatement, and this concern reinforces that conditional status without demonstrating a false central theorem, the appropriate verdict is unchanged.","tokens_in":26806,"tokens_out":31926,"duration_ms":324576,"concrete_test":"For f(x)=x^4 and f_xbar(x)=max{0, xbar^4+4xbar^3(x−xbar)} with fixed xbar≠0, compute sup_{x≠xbar} |g_xbar'(x)|/|x−xbar|, where g_xbar=f_xbar−f. If this quantity is infinite, the global form of Assumption 4.2(H2) fails for the paper's own motivating local model. This check settles whether H2 must be treated as a genuinely global assumption or replaced by an explicit uniform-local condition with a radius that is uniform along the convergent subsequence, which the current proofs do not supply.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence claims (Theorems 4.7 and 4.11) depend on Proposition 3.3, which transfers subgradient information from the model function back to f. Its hypothesis (3.6) is explicitly local: it holds only in a neighborhood of the model center xbar. However, Assumption 4.2(H2) restates it as a global bound, ∂g_xbar(·) ⊂ L B_{||·−xbar||}(0), with a single L for all x and all model centers. The proofs in Proposition 4.6, Proposition 4.9, and Lemma 4.10 apply H2 at points x_{k+1} or the rejected candidate x̂_k when only ||x_{k+1}−x_k||→0 is known. If the neighborhood in (3.6) can shrink along the convergent subsequence, the key estimate |g_xk(x̂_k)| ≤ L||x̂_k−x_k||² + ε||x̂_k−x_k|| can fail even though the points converge to one another. Thus the stationarity transfer is not secured by the stated local model definition. Conversely, if H2 is genuinely global, it is a substantial additional assumption and is not implied by Definition 3.2 or by the paper's own illustrations: for f(x)=x^4 with model f_xbar(x)=max{0, xbar^4 + 4xbar^3(x−xbar)} from Section 2, one has |g_xbar'(x)|/|x−xbar| → ∞ as |x|→∞ for fixed xbar≠0, so global H2 fails for the motivating example. The abstract's claim that boundedness of the variable metric is a consequence rather than a prerequisite therefore rests on an unstated strengthening of the local model condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a model-based proximal quasi-Newton method (Algorithm 1) for minimizing proper lower semicontinuous functions. The key algorithmic ingredients are a local model function f_xbar, a variable metric generated by a continuous matrix-valued map H(x), and a backtracking line search based on the model error. Under an assumption on the subdifferential of the model error (Assumption 4.2(H2)), the paper claims subsequential stationarity of all accumulation points (Theorem 4.7), and, adding a Kurdyka-Łojasiewicz property, convergence of the whole sequence to a stationary point (Theorem 4.11) together with convergence rates (Theorem 4.12). The stated goal is to avoid assuming uniform boundedness of the variable metric a priori. Numerical experiments on polytope feasibility and quadratic inverse problems are reported.","tokens_in":27172,"tokens_out":20355,"duration_ms":190262,"significance":"If the gaps listed below are repaired, the framework would be a valuable contribution: it targets a genuine restriction in nonsmooth proximal quasi-Newton analysis (the uniform boundedness of the Hessian approximation) and combines it with a flexible local-model framework. The continuity of the metric generator and the local model definition are natural and the worked examples in Section 5 are useful. The paper also ships a large set of detailed auxiliary proofs and numerical experiments, which is a strength. At present, however, the central results are not supported as written: the key subdifferential transfer result is unproved and stated too broadly, the global formulation of the model-error assumption is not matched by the local examples, and the KL-based proof and the rate theorem rely on omitted arguments.","major_comments":[{"comment":"Proposition 3.3 is stated without proof and is false as stated. Its hypothesis (3.6) controls ∂g_xbar(x) only for x in a neighborhood of xbar, yet the conclusion (3.4) is asserted for an arbitrary fixed x~ in dom f. For f(x)=x^4, xbar=1, L=1, and the model f_xbar(x)=max{0,1+4(x−1)}, the point x~=100 satisfies (3.5) and (3.6) (the latter on a neighborhood of 1) but violates (3.4): ∂f_xbar(100)={4} is not contained in ∂f(100)+B_{99}(0)={40000}+[−99,99]. The proposition must be restated with x~ lying in the neighborhood where (3.6) is valid, and a proof must be supplied. The subsequent uses in Theorem 4.7 (via (4.19)), Lemma 4.10, and Theorem 4.11 all depend on this correction.","section":"Section 3, Proposition 3.3"},{"comment":"Assumption 4.2(H2) is a global condition: ∂g_xbar(·)⊂L B_{||·−xbar||}(0) is required for every x and every model center xbar. This is not a consequence of the local model definition (3.3), and the appendix only establishes the local version: Propositions 8.1 and 8.2 verify (3.6) 'for x close to xbar', not globally. The motivating example in Section 2, f(x)=x^4 with f_xbar(x)=max{0,xbar^4+4xbar^3(x−xbar)}, fails the global bound: for fixed xbar≠0, |∇g_xbar(x)|/|x−xbar|→∞ as |x|→∞. The proofs of Proposition 4.6, Proposition 4.9, and Lemma 4.10 apply H2 at points x_{k+1} or xhat_k for which only ||x_{k+1}−x_k||→0 is known. The author should either replace H2 by a localized version with a uniform L on a relevant compact set (with a complete argument), or give examples that genuinely satisfy the global assumption.","section":"Section 4, Assumption 4.2(H2)"},{"comment":"The proof of Lemma 4.10 is omitted with the justification that it is 'highly similar' to Theorem 4.7. This lemma is load-bearing for Theorem 4.11: the relative error estimate (4.29) is exactly the inequality that feeds the KL inequality in the finite-length induction. Theorem 4.7 alone does not provide the uniform bound γ_k≤γbar_ρ on I_ρ, nor the uniform constant L over the compact set Cρ, so the claimed similarity is not immediate. A complete proof must be written out.","section":"Section 4.3, Lemma 4.10"},{"comment":"The four rate statements (i)–(iv) — finite termination, superlinear convergence, Q-/R-linear convergence, and sublinear rates — are asserted without proof. Since Sections 2 and 7 explicitly claim rate-of-convergence results, these statements are unsupported in the present version. The author should supply at least a complete proof sketch or a precise derivation of the standard KL-based inequalities in the current setting, including the dependence of the constants on the stepsize bounds and the metric bound M.","section":"Section 4.3, Theorem 4.12"},{"comment":"The induction argument in Theorem 4.11 contains a constant mismatch. The quantity α in (4.26) is defined with the term 2γbar_ρ M φ(...)/((1−δ)μγmin), but in the induction step for statement (a) the proof arrives at the bound 2(γbar_ρ M + L) φ(...)/((1−δ)μγmin) and then asserts equality with α. This equality is false unless L=0. The definition of α must be corrected to incorporate L (or the proof adjusted accordingly); otherwise the claimed induction that x_k stays in B_α(x*) is not established. This is a central step in the proof of the main sequential convergence result.","section":"Section 4.3, Theorem 4.11"},{"comment":"The proof of Proposition 4.6 needs a regular subgradient η_k of g_xk at xhat_k with ||η_k||≤L||xhat_k−x_k||, and it then uses the two displayed lim-inf inequalities with this η_k. The existence of such η_k is not justified from the stated assumptions: the optimality of xhat_k for the subproblem yields a limiting subgradient of g_xk, not necessarily a regular subgradient, and the regular subdifferential can be empty at a point while the limiting subdifferential is nonempty. Without η_k, the bound |g_xk(xhat_k)|≤L||xhat_k−x_k||^2+ε||xhat_k−x_k|| is not derived, and hence the conclusion γ_k||x_{k+1}−x_k||→0 is not supported. This gap affects Theorem 4.7, which uses Proposition 4.6 directly.","section":"Section 4.2, Proposition 4.6"}],"minor_comments":[{"comment":"The phrase 'the sequence of the variable metric is proved not to be bounded' misstates the results: Theorem 4.7 proves convergence without assuming boundedness, and along the convergent subsequence H(x_k)→H(x*) by continuity; Theorem 4.11 then proves uniform boundedness under KL. The intended meaning is 'not required to be bounded a priori', and the wording should be corrected.","section":"Abstract and Section 2"},{"comment":"In the proof of Theorem 4.7, the sentence 'Hence, by Proposition 4.9, we have dist(0,∂f(x_{k+1}))≤...' should refer to Proposition 4.6 (or directly to the preceding inclusion (4.19)), since it is the convergence γ_k||x_{k+1}−x_k||→0 along K that is needed.","section":"Theorem 4.7 proof"},{"comment":"The proof of Proposition 4.6 references '(4.20)' before that equation is defined (it appears only in Section 4.3) and invokes 'Definition 3.1' where the relevant model error bound comes from Definition 3.2. These cross-reference errors should be fixed.","section":"Proposition 4.6 proof"},{"comment":"There are numerous typos and grammar issues: 'matric' for 'metric', 'disingularization' for 'desingularization', 'respective' for 'perspective', 'terninate' for 'terminate', 'closed to' for 'close to', and 'decent' for 'descent'. A thorough editorial pass is needed.","section":"Various sections"},{"comment":"Algorithm 1 requires an exact global minimizer of each subproblem (4.1). The numerical experiments, however, solve the subproblems with ADMM or PDHG, which are inexact heuristics. The gap between the theoretical exact-solver oracle and the practical implementation should be acknowledged, and ideally the theory should be extended to inexact solves.","section":"Section 4, Algorithm 1"},{"comment":"In the polytope feasibility experiments, only the results of Algorithm 1 are reported because the gradient method returned 'NaN' for all tested p. Since the comparison is claimed as a demonstration of effectiveness, omitting the gradient method's failure trajectories makes the empirical comparison less informative.","section":"Section 6.1"}],"recommendation":"major_revision","confidential_remarks":"The central idea — that boundedness of the variable metric can be a consequence of convergence rather than a prerequisite — is plausible and potentially publishable, but the current version does not support it rigorously. The most urgent items are a corrected and proved Proposition 3.3, a coherent statement of Assumption 4.2 that matches the local model definition and the examples, a full proof of Lemma 4.10, and a repair of the induction in Theorem 4.11. I would recommend a careful revision and a re-review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s the bottom line. The paper attacks a real problem — the a priori boundedness assumption on variable metrics in proximal quasi-Newton methods — and the local model-function idea is genuinely new. But the main claim is not actually reached by the stated assumptions. Proposition 3.3, which converts model subgradients into objective subgradients, is stated without proof and is used as load-bearing in Theorem 4.7 and Theorem 4.11. More importantly, the proof needs Assumption 4.2(H2), a global first-order bound on the model error, while Definition 3.2 only gives local control. For the paper’s own example f(x)=x^4 with f_xbar(x)=max{0, xbar^4+4xbar^3(x−xbar)}, H2 fails globally: the ratio |g_xbar'(x)|/|x−xbar| is unbounded as |x|→∞ for fixed xbar≠0. So the boundedness-free convergence theorem depends on an unstated strengthening of the local model condition. That is the place to focus referee attention.\n\nWhat the paper does well: the local model framework is a sensible weakening of the Taylor-like model condition from Drusvyatskiy et al. and Ochs et al., and the continuous-matrix-generator perspective is worth having. The KL analysis is mostly standard but honestly laid out, and the composite examples in Section 5 are useful. The appendix propositions do provide legitimate local verifications of (3.6).\n\nSoft spots, in order. (1) Proposition 3.3 has no proof; it needs one, or a citation to an existing result. (2) H2's globality is not justified. If it is meant to be global, the paper should say so and the abstract should not claim boundedness is \"proved not to be bounded\" — the theorem only removes a prerequisite. If it is meant to be local, the convergence proofs need a mechanism to control the constant along the sequence. (3) Lemma 4.10 is omitted as \"highly similar\" but it is part of the KL argument; omitting it is risky. (4) Theorem 4.12 rates are stated without proof. (5) Numerics: the polytope example has no comparison baseline, and there is no code or error bars; still, the quadratic inverse problem table is informative.\n\nMy overall read: this is a serious attempt at a worthwhile extension, not a throwaway. But the central theorem is not supported as written. A referee round should push for a proof of Proposition 3.3, a clarification of H2, and a rewrite of the abstract. I would not cite it yet; I would bring it to a reading group to decide whether the local-to-global step is repairable.","headline":"The local-model setup is a real contribution, but the theorem that drops boundedness of the variable metric relies on an unstated global version of the model-error bound, so the paper is not yet ready as written.","tokens_in":27668,"tokens_out":3385,"would_cite":false,"duration_ms":33765,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J52","65K05","90C26","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A proximal quasi-Newton algorithm for nonsmooth nonconvex problems converges to a stationary point without assuming the Hessian approximations stay bounded, deriving their boundedness from the Kurdyka–Łojasiewicz property.","keywords":["proximal quasi-Newton methods","local model function","nonsmooth nonconvex optimization","Kurdyka-Łojasiewicz property","variable metric boundedness","backtracking line search","composite optimization","quadratic inverse problems"],"falsifier":"Run Algorithm 1 on $f(x)=|x|^{3/2}$ with the first-order Taylor model $f_{\\bar x}(x)=|\\bar x|^{3/2}+\\tfrac32\\operatorname{sgn}(\\bar x)|\\bar x|^{1/2}(x-\\bar x)$ and starting points approaching 0. Along accepted steps $x_{k+1}\\to 0$, compute the ratio $|f_{x_k}(x_{k+1})-f(x_{k+1})|/\\|x_{k+1}-x_k\\|^2$; because the model-error derivative near 0 behaves like $|\\bar x|^{-1/2}\\|x-\\bar x\\|$, this ratio should be unbounded even when the acceptance test passes. A nonstationary accumulation point in such a run would refute the claimed reach of the convergence theorem without the fixed-$L$ control, while a stationary one would show that the local model alone is doing the work attributed to Assumption 4.2(H2).","tokens_in":26507,"feed_emoji":"📉","tokens_out":11406,"duration_ms":116882,"temperature":0.7,"pith_summary":"This paper proposes a proximal quasi-Newton algorithm for minimizing proper, lower semicontinuous functions that may be both nonsmooth and nonconvex. In place of a fixed global model of the objective, each iteration uses a local model function centered at the current iterate, and the next iterate is obtained by minimizing that model plus a quadratic proximity term weighted by a variable metric $H_k$ generated by a continuous matrix-valued map. The central claim is that the Hessian approximations $H_k$ need not be assumed bounded: under a mild first-order control on the model error, every accumulation point of the generated sequence is stationary even while $\\|H_k\\|$ may grow, and if the objective has the Kurdyka–Łojasiewicz property at an accumulation point, the whole sequence converges to a stationary point and the variable metrics become uniformly bounded as a consequence. If this is right, a standard boundedness assumption in proximal quasi-Newton theory is not a prerequisite but a byproduct of convergence, which matters for problems where Hessian approximations legitimately blow up near sharp minima.","feed_headline":"Hessian bounds follow from convergence, not assumptions","feed_subtitle":"Proximal quasi-Newton convergence for nonsmooth, nonconvex problems where boundedness of the metric is a byproduct.","key_machinery":"The load-bearing object is the local model function (Definition 3.2): a proper lsc function $f_{\\bar x}$ with domain equal to that of $f$, whose approximation error to $f$ is controlled near the model center by a growth function, $|f(x)-f_{\\bar x}(x)|\\le \\omega_{\\bar x}(\\|x-\\bar x\\|)$ for $x$ approaching $\\bar x$. Together with Assumption 4.2(H2), $\\partial g_{\\bar x}(\\cdot)\\subset L B_{\\|\\cdot-\\bar x\\|}(0)$, Proposition 3.3 converts this into the subdifferential inclusion $\\partial f_{\\bar x}(\\tilde x)\\subset \\partial f(\\tilde x)+L B_{\\|\\bar x-\\tilde x\\|}(0)$, which is what transfers stationarity from the subproblem solution to the objective. The algorithm's acceptance criterion $|f(x_{k,i})-f_{x_k}(x_{k,i})|\\le \\delta\\gamma_{k,i}\\|x_{k,i}-x_k\\|^2_{H_k}/2$ guarantees sufficient descent, and the continuity of $H$ plus the quadratic proximity term makes the subproblems coercive.","core_discovery":"The paper's core result is Theorem 4.11: under Assumption 4.1 (lower boundedness, affine lower bounds on model functions, a continuous uniformly positive definite matrix generator $H$, and existence of an accumulation point), Assumption 4.2 with smooth model errors $g_{\\bar x}=f_{\\bar x}-f$, and the Kurdyka–Łojasiewicz property at an accumulation point, the sequence $\\{x_k\\}$ generated by Algorithm 1 converges to a stationary point $x^*$ of $f$, and the variable metric sequence $\\{H(x_k)\\}$ is uniformly bounded. Before invoking the KL property, Theorem 4.7 already establishes that every accumulation point is stationary while $\\{H(x_k)\\}$ need not be bounded. The paper reads this as evidence that the boundedness of the variable metric should be a problem-tailored consequence of convergence rather than an assumption imposed a priori, and it backs the claim with convergence-rate statements (Theorem 4.12) under standard desingularization functions and with experiments on polytope feasibility and (sparse) quadratic inverse problems.","pith_inferences":["The same proof skeleton suggests a design principle for other quasi-Newton schemes: any Hessian update that can be written as a continuous matrix-valued function of the current iterate, plus a model-error control like Assumption 4.2(H2), should admit convergence analysis without a priori metric bounds; this points toward testing limited-memory and learned Hessian generators in the same framework.","Since only the KL property at a single accumulation point is required, the theory should apply to functions definable in o-minimal structures, including semialgebraic objectives like polytope feasibility, where KL holds automatically; a natural extension is to constraints or Bregman proximities using the same local-model acceptance test.","A testable consequence of the paper's philosophy: on objectives with sharp minima where second derivatives blow up, allowing $H_k$ to grow along the nonsmooth directions should improve conditioning and speed rather than hurt, exactly because the theory tolerates unbounded metrics; this could be benchmarked against fixed-metric proximal gradient on functions like $|x|^p$ with $p\\in(1,2)$."],"forward_implications":["Boundedness of the Hessian approximations becomes a conclusion reachable from convergence: if the objective has the KL property at an accumulation point, $\\{H(x_k)\\}$ is uniformly bounded without being assumed so.","Subsequential stationarity does not require bounded iterates or bounded metrics; it follows from the first-order control on the model error alone.","The rate theorem gives concrete behavior depending on the desingularization exponent: finite termination, superlinear, linear, or polynomial convergence, so the method inherits the classical KL landscape of rates.","Because $H_k=I$ is an allowed generator, the framework covers proximal gradient and proximal descent methods, and the convergence results apply to additive composite problems and nonlinear composite problems of the form $q(x)+h(A(x))$.","On the tested quadratic inverse problems, the model quasi-Newton variant used fewer outer and inner iterations and produced sparser solutions than the model gradient variant, and the gradient method failed on some instances."],"supporting_citations":[{"why":"Defines Taylor-like model functions and growth-function error control, which the local model definition relaxes to a neighborhood.","marker":"[24]"},{"why":"Supplies the model-function/Bregman proximal framework whose local version this paper extends, including the lemma for quadratic error bounds.","marker":"[54]"},{"why":"Provides the model-based Bregman proximal gradient method that is compared against and whose model-function setup is generalized.","marker":"[48]"},{"why":"Gives the inclusion $\\partial f_{\\bar x}(\\bar x)\\subset\\partial f(\\bar x)$ at the model center used to detect stationarity in the inner-loop termination proof.","marker":"[55]"},{"why":"Introduces the Kurdyka–Łojasiewicz inequality for nonsmooth nonconvex functions, the tool behind whole-sequence convergence.","marker":"[2]"},{"why":"Establishes the KL property for semialgebraic and tame functions, justifying the assumption in typical applications.","marker":"[8]"},{"why":"A forward-backward quasi-Newton analysis that treats boundedness of the Hessian approximation as a prerequisite, which the paper's theorems are positioned against.","marker":"[59]"},{"why":"Supplies the variational-analysis subdifferential calculus used throughout, including sum rules and limiting subdifferentials.","marker":"[57]"}],"fun_headline_variants":["Proximal quasi-Newton: metric bounds emerge from convergence","No a priori Hessian bounds: they follow from convergence","Boundedness of the metric: consequence, not assumption","Nonsmooth nonconvex: convergence yields bounded Hessian approximations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 4.2(H2): the first-order variation of the model error must be bounded by a fixed constant $L$ times the distance to the model center at every iterate, and without that control the quadratic model-error bound and the transfer of stationarity to the objective both break.","fun_headline_variants_meta":{"raw":{"variants":["Proximal quasi-Newton: metric bounds emerge from convergence","No a priori Hessian bounds: they follow from convergence","Boundedness of the metric: consequence, not assumption","Nonsmooth nonconvex: convergence yields bounded Hessian approximations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000548,"raw_usage":{"total_tokens":2663,"prompt_tokens":1034,"completion_tokens":1629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":1559}},"tokens_in":650,"tokens_out":1629,"duration_ms":12299,"temperature":1.0,"reasoning_tokens":1559,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:15:13.649636+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 1 on $f(x)=|x|^{3/2}$ with the first-order Taylor model $f_{\\bar x}(x)=|\\bar x|^{3/2}+\\tfrac32\\operatorname{sgn}(\\bar x)|\\bar x|^{1/2}(x-\\bar x)$ and starting points approaching 0. Along accepted steps $x_{k+1}\\to 0$, compute the ratio $|f_{x_k}(x_{k+1})-f(x_{k+1})|/\\|x_{k+1}-x_k\\|^2$; because the model-error derivative near 0 behaves like $|\\bar x|^{-1/2}\\|x-\\bar x\\|$, this ratio should be unbounded even when the acceptance test passes. A nonstationary accumulation point in such a run would refute the claimed reach of the convergence theorem without the fixed-$L$ control, while a stationary one would show that the local model alone is doing the work attributed to Assumption 4.2(H2).","supporting_citations":[{"cited_title":"Drusvyatskiy, A","cited_arxiv_id":null,"evidence_quote":"Defines Taylor-like model functions and growth-function error control, which the local model definition relaxes to a neighborhood."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the model-function/Bregman proximal framework whose local version this paper extends, including the lemma for quadratic error bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the model-based Bregman proximal gradient method that is compared against and whose model-function setup is generalized."},{"cited_title":"Ochs and Y","cited_arxiv_id":null,"evidence_quote":"Gives the inclusion $\\partial f_{\\bar x}(\\bar x)\\subset\\partial f(\\bar x)$ at the model center used to detect stationarity in the inner-loop termination proof."},{"cited_title":"Stella, A","cited_arxiv_id":null,"evidence_quote":"A forward-backward quasi-Newton analysis that treats boundedness of the Hessian approximation as a prerequisite, which the paper's theorems are positioned against."}],"review_version":2}