{"id":"0a09c0ed-c668-427c-a615-9055bd284b65","arxiv_id":"2502.02010","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A nonmonotone adaptive steepest descent algorithm is claimed to converge for nonconvex minimax and multiobjective problems, but the supporting proofs are invalid as written.","lead":"This paper proposes a line-search-free steepest descent method with an adaptive step size for nonconvex minimax and multiobjective problems, and claims convergence to stationary or efficient points. The proofs as written contain load-bearing gaps, so the main convergence result is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2 derives p_k→0 from a positive infimum of allowable step sizes, a non-sequitur; Theorem 3.5 adds no valid stationarity argument. The central convergence claim is unsupported.","rationale":"In good faith, the paper's main contribution is the claimed convergence of a line-search-free nonmonotone adaptive steepest descent method for nonconvex minimax and multiobjective problems. The reader's weakest_assumption correctly identifies the step where the proof must show that the search directions vanish and that failures occur only finitely often. My independent reading confirms this is the load-bearing point: Lemma 3.2 conflates the allowable step-size bound with the generated sequence and draws a conclusion that does not follow; the proof even makes the same non-sequitur as the reader notes. Lemma 3.3's induction is also invalid for the algorithm as written, because the exponent of σ in the success update is the iteration index, not the number of failures, and the asserted upper bound is algebraically wrong. Theorem 3.5 does not provide a stationarity argument; it asserts a Cauchy-like property and declares the limit stationary without using the subproblem's optimality conditions. The numerical experiments are illustrative but do not compensate for the absence of a valid proof. Therefore the reader's REJECT verdict is appropriate; I see no reason to change it. The paper may be repairable, but the submitted version does not support its central claims.","tokens_in":13163,"tokens_out":7076,"duration_ms":66292,"concrete_test":"Test the inference of Lemma 3.2 in isolation: fix any constants K,L>0, δ>0, ε∈(0,1/2), and choose the constant sequence p_k=(1,0,…,0) for all k. Then α1_k=δ/(‖p_k‖(K+½‖p_k‖)) = δ/(K+½) is constant and positive, α'_k=(½−ε)/L is constant and positive, so inf_k min{1,α1_k,α'_k} > 0 while ‖p_k‖=1 does not converge to 0. This directly disproves the claimed implication in Lemma 3.2. To test the algorithmic claim rather than the proof step, implement Algorithm 1 with the stated parameters on the scalar nonconvex minimax G(θ)=max{θ²,(θ−1)²} and verify whether ‖p_k‖ actually tends to 0; the proof provides no support for this, and a failure of this property would falsify Theorem 3.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central convergence claims rest on two assertions: (i) the generated directions satisfy p_k→0, and (ii) the adaptive step size α_k is eventually bounded below so that the sufficient-decrease condition yields a summable decrease. Lemma 3.2 attempts (i) but proves something different. It shows (assuming existence of a solution) that the allowable upper-bound sequence α_k := min{1, δ/(‖p_k‖(K+½‖p_k‖)), (½−ε)/L} has a positive infimum. From this it concludes 'α_k is not converge to 0, which means ‖p_k‖²→0'. This is a non-sequitur: a positive lower bound on an admissible step size imposes no bound on the actual step sizes, and it certainly does not imply the search directions vanish. In fact, any bounded sequence p_k with ‖p_k‖=c>0 yields a positive infimum for this allowable bound, while p_k never tends to 0. Lemma 3.3, meant to control failures, also has an induction error: with the algorithm's updates α_{k+1}=α_k+η_k σ^k on success and α_{k+1}=σ α_k on failure, the claimed bound α_k≤σ^k(α0+Ση_i) is not implied by the success case, since σ^l(α0+Σ_{i=0}^{l-1}η_i)+η_l σ^l is not ≤σ^{l+1}(α0+Σ_{i=0}^l η_i) for σ<1. Thus the proof that failures occur only finitely often collapses. Theorem 3.5 then asserts the iterates are Cauchy and 'therefore' θ* is stationary, without connecting the limit to the subproblem's KKT conditions. Removing these gaps may be possible, but as written the paper does not establish its main theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a line-search-free steepest descent method with a nonmonotone adaptive step size (Algorithm 1) for the finite minimax problem (MP), and a reference-based extension (Algorithm 2) to multiobjective optimization via Tchebycheff scalarization. It claims convergence to a stationary point under nonconvex and quasiconvex assumptions, to a global minimum under pseudoconvexity, and correspondingly to Pareto critical or weakly efficient points of (MOP). Assumptions A1-A3 are stated, and numerical experiments on three small problems are reported.","tokens_in":13518,"tokens_out":14665,"duration_ms":139156,"significance":"If valid, this would be a useful contribution: a line-search-free adaptive step-size rule with convergence guarantees is attractive for large-scale and machine-learning settings, and approximating the Pareto front by solving a series of minimax problems is practical. The authors state their assumptions up front, do not fit parameters to the numerical examples, and the experiments illustrate the intended behavior. However, the proof of the central convergence theorem is not sound, so the advertised guarantees are not established.","major_comments":[{"comment":"This lemma is load-bearing, and its proof conflates two different objects. The proof defines the allowable upper bound alpha_k = min{1, delta/(||p_k||(K + 1/2 ||p_k||)), (1/2 - epsilon)/L} and shows that inf_k alpha_k > 0 under the stated assumptions. That is a fact about an admissible step-size bound, not about the step-size sequence actually generated by Algorithm 1. The concluding sentence \"alpha_k is not converge to 0, which means ||p_k||^2 goes to 0\" is a non-sequitur: a positive lower bound on the allowable step does not force the actual step sizes away from zero, since a failure halves alpha, and it does not force p_k to vanish, since a bounded sequence p_k with constant nonzero norm yields a positive infimum for this allowable bound. The assertion that existence of a solution to (MP) implies ||p_k|| not -> infinity is also not justified. Consequently, the basis for Lemma 3.3 and for the conclusion p_k -> 0 in Theorems 3.5-3.7 is missing.","section":"Section 3.3, Lemma 3.2"},{"comment":"The induction proving that only finitely many iterations fail is algebraically wrong. In the success case the proof derives alpha_{l+1} <= sigma^l(alpha_0 + sum_{i=0}^l eta_i) but then claims the right-hand side is bounded by sigma^{l+1}(alpha_0 + sum_{i=0}^l eta_i); since sigma in (0,1), the latter is strictly smaller, so the inequality does not follow. If, instead, the index l is meant to be the number of failures rather than an iteration counter, then the induction step does not handle a success at all, because a success does not increase the failure count. Under either reading, the lemma's conclusion that sum I(...) < infinity is unsupported. The proof also uses s both as an iteration count and as a failure count without clarification.","section":"Section 3.3, Lemma 3.3"},{"comment":"The proof does not establish stationarity. Even assuming the previously unproved fact alpha_k ||p_k|| -> 0, the claim that {theta_k} is Cauchy does not follow; consecutive increments tending to zero do not imply that ||theta_m - theta_k|| < epsilon for all large m,k. The proof then simply asserts that the limit point theta* is stationary without using the KKT conditions of the subsidiary problem (SP) from Lemma 3.4 or any stationarity criterion. This is the main nonconvex convergence claim, and it is not proven.","section":"Section 3.3, Theorem 3.5"},{"comment":"The quasiconvex, pseudoconvex, and multiobjective results all inherit the dependence on the unproved p_k -> 0 and on Lemma 3.3's finite-failure conclusion. Theorem 3.6, for instance, uses Lemma 3.3 to infer sum alpha_k epsilon ||p_k||^2 < infinity; since Lemma 3.3 is not established, the summability argument collapses. Theorem 4.6 then transfers these unsupported conclusions to Pareto criticality and weak efficiency. Thus the central claims of the paper are not supported by the written proofs.","section":"Sections 3.3 and 4.5, Theorems 3.6, 3.7, 4.6"}],"minor_comments":[{"comment":"The scope of the statement 's = s + 1' is ambiguous; if it is meant to execute on every iteration, it should be placed after 'end if', while if it belongs to the else branch, that should be made explicit. The proof's counting of s must match the pseudocode.","section":"Algorithm 1, lines 3-8"},{"comment":"The subsidiary problem (SP) is not fully specified: it should state that the minimization is over (beta, p) in R x R^n, and the variables in the constraint should be written consistently.","section":"Algorithm 1, Step 2"},{"comment":"alpha_k denotes both the actual step size and the allowable bound in Lemma 3.2, and k denotes both an iteration index and a failure count in Lemma 3.3; this overloading makes the proofs difficult to follow.","section":"Notation in Lemmas 3.2 and 3.3"},{"comment":"The text 'From Lemma 3.7' should be 'From Theorem 3.7', since there is no Lemma 3.7 in the manuscript.","section":"Theorem 4.6 proof"},{"comment":"Figures 1-3 alone do not verify the convergence claims; the section should report objective values over iterations, stopping criteria, and comparisons with existing methods.","section":"Section 5, Numerical Results"}],"recommendation":"reject","confidential_remarks":"The central convergence theorems rest on a non-sequitur in Lemma 3.2 and an algebraic error in Lemma 3.3, and Theorem 3.5 adds no valid stationarity argument. The numerical experiments are illustrative only and do not compensate. A substantially reworked convergence analysis would be needed, rather than a local revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central convergence claims are not backed by the proofs. Lemma 3.2 shows that the admissible step-size bound has a positive infimum, then concludes the search directions p_k tend to zero; that is a non-sequitur, since a bounded sequence with ||p_k||=c>0 also gives a positive infimum. Theorem 3.5's stationarity proof is just a statement that consecutive iterates get close, with no link to the subproblem's KKT conditions. The paper's main results are therefore unsupported.\n\nCredit where it is due: the algorithm is a reasonable combination of Pshenichny's subproblem with a nonmonotone adaptive step (increase by η_k σ^s after success, contract by σ after failure), and the goal of covering nonconvex, quasiconvex, and pseudoconvex components is legitimate. The exposition is clear.\n\nTwo notes on the internal review: Lemma 3.3's counter s only increments on failure (Algorithm 1 line 7 is in the else branch), so the reader and stress-test are wrong to say it counts every iteration. The induction in Lemma 3.3 also works if the exponent is read as the failure count, not the iteration index. The real gap is Lemma 3.2 and the lack of any valid argument that the actual step sizes stay bounded below; without that, the sufficient-decrease summability gives no control on p_k.\n\nThe numerical section is illustrative only: three examples, no baselines, no metrics, no code. That is secondary given the proof gaps.\n\nWho this is for: researchers working on line-search-free minimax and multiobjective methods, who might find the step-size rule worth picking up if the convergence analysis is repaired.\n\nRecommendation: send to peer review, but with a clear expectation of major revision. The flaw is identifiable and possibly fixable (e.g., prove failures are finite and the step-size sequence converges to a positive limit, then derive p_k→0 from summability). As submitted, the paper should not be accepted.","headline":"A nicely motivated adaptive step-size scheme for minimax problems, but the convergence proof conflates the admissible step-size bound with the actual step sizes; the main theorems are unproven as written.","tokens_in":14069,"tokens_out":7749,"would_cite":false,"duration_ms":70106,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C29","90C30","90C47"],"pacs":[],"model":"deepseek-v4-flash","headline":"A line-search-free steepest-gradient method with nonmonotone adaptive step sizes is claimed to reach stationary points for nonconvex minimax, and global minima or weakly efficient points under stronger convexity.","keywords":["Steepest descent","Nonmonotone adaptive step size","Nonconvex minimax problem","Multiobjective optimization","Quasiconvex functions","Pseudoconvex functions","Pareto stationarity"],"falsifier":"Check Lemma 3.2 directly: the proof shows that the infimum of the allowable step sizes $\\alpha_k$ is positive, which only gives an upper bound on $\\|p_k\\|$, and then asserts this means $\\|p_k\\|^2\\to 0$. Exhibit any sequence of iterates satisfying the subsidiary problem and the step rule with $\\|p_k\\|$ constant, for instance directions alternating between $e_1$ and $-e_1$, while $\\alpha_k$ stays bounded below; then the lemma's asserted implication fails and the stationarity theorem lacks its key premise.","tokens_in":12911,"feed_emoji":"🎯","tokens_out":11122,"duration_ms":92412,"temperature":0.7,"pith_summary":"The paper sets out to prove that a steepest-descent method with a nonmonotone adaptive step size—never performing a line search—can solve nonconvex finite minimax problems and, through a reference-based Tchebycheff reformulation, multiobjective problems. It claims that every accumulation point of the generated sequence is stationary when the component functions are merely differentiable and nonconvex, that under quasiconvexity the whole sequence converges to a stationary point, and that under pseudoconvexity the limit is a global minimum. Because the step size is updated from a single function evaluation, each iteration is cheap, which the authors see as an advantage for settings where the component functions are loss functions in model training. If these claims hold, the same adaptive rule yields Pareto-critical or weakly efficient points for multiobjective optimization depending on the convexity class of the objectives.","feed_headline":"No line search: adaptive steps converge for nonconvex minimax","feed_subtitle":"One evaluation per step yields stationary, weakly efficient, or Pareto-critical points by convexity class.","key_machinery":"The load-bearing object is the subsidiary problem (SP): at each iterate $\\theta_k$, choose $p_k$ and $\\beta_k$ to minimize $\\beta+\\frac{1}{2}\\|p\\|^2$ subject to $\\langle \\nabla g_i(\\theta_k),p\\rangle+g_i(\\theta_k)-\\beta\\le 0$ for indices in the $\\delta$-active set $J_\\delta(\\theta_k)$. Its KKT conditions give $p_k+\\sum_i u_i \\nabla g_i(\\theta_k)=0$ with $\\sum_i u_i=1$, so if $p_k\\to 0$ then $0$ lies in the convex hull of the active gradients, which is the Clarke stationary condition for $G(\\theta)=\\max_i g_i(\\theta)$. The second mechanism is the step-size rule: a success (sufficient descent with tolerance $\\varepsilon$) increases $\\alpha_k$ by $\\eta_k\\sigma^s$, while a failure multiplies it by $\\sigma$ and the counter $s$ advances. The summability of $\\eta_k$ is used to show failures occur only finitely often, so $\\alpha_k$ cannot be driven to zero; the proof then uses this non-vanishing of the step size to conclude that the directions $p_k$ vanish and stationarity follows.","core_discovery":"The paper's central claim is that the search directions $p_k$ produced by the subsidiary quadratic program tend to zero, and that this forces every accumulation point $\\theta^*$ of the iterates to satisfy the stationarity system $\\sum_{i=1}^m u_i \\nabla g_i(\\theta^*)=0$ with $\\sum_{i=1}^m u_i=1$ and $u_i(g_i(\\theta^*)-G(\\theta^*))=0$. In the quasiconvex case the argument is meant to show the full sequence converges to a stationary point, and in the pseudoconvex case the stationarity condition upgrades to $G(\\theta^*)\\le G(\\theta)$ for all $\\theta$, i.e. a global minimum of the minimax problem. Via the Tchebycheff scalarization $\\min_\\theta \\max_i (g_i(\\theta)-v_i)/d_i$, these same conclusions translate into Pareto criticality or weak efficiency for the multiobjective problem. The paper presents the nonmonotone step-size rule not as a heuristic but as a provable replacement for line search inside a classical steepest-descent minimax method.","pith_inferences":["Editorial inference: if the step-size rule does force $p_k\\to 0$, the same rule could plausibly be adapted to nonsmooth objectives, since the Clarke calculus used in the paper already accommodates locally Lipschitz max functions; the paper lists nonsmooth extension as future work.","Editorial inference: the method's single-evaluation step update resembles heuristics used in large-scale training, but the paper's analysis is deterministic; testing the rule on stochastic or mini-batch losses would show whether the convergence guarantee survives noise.","Editorial inference: the proof gap in Lemma 3.2 suggests a concrete experiment—run the algorithm while recording $\\alpha_k$ and $\\|p_k\\|$ on a nonconvex problem; if $\\|p_k\\|$ fails to tend to zero while $\\alpha_k$ stays bounded below, the stationarity theorem needs an additional assumption or a modified step rule."],"forward_implications":["Each iteration costs one function evaluation for the step-size test, removing line-search overhead from a classical steepest-descent minimax algorithm.","For differentiable nonconvex components, any accumulation point of the iterates satisfies the Clarke stationary condition for the max function $G$.","For quasiconvex components, the full sequence converges to a stationary point rather than merely having stationary cluster points.","For pseudoconvex components, the limit is a global minimum of the minimax problem and a weakly efficient point of the corresponding multiobjective problem.","Sweeping the reference vector in the Tchebycheff formulation yields a family of such points, giving a reference-based approximation of the Pareto front."],"supporting_citations":[{"why":"The steepest-descent minimax algorithm this work extends; supplies the linearized subsidiary-problem structure and the line-search baseline that the nonmonotone rule replaces.","marker":"[6]"},{"why":"Defines semiconvex and quasidifferentiable functions and the Clarke subdifferential results used to upgrade pseudoconvex stationarity to global minimality.","marker":"[25]"},{"why":"Provides the $\\partial_C$-quasiconvexity characterization used in the quasiconvex convergence proof.","marker":"[24]"},{"why":"Supplies the result that the pointwise maximum of quasiconvex functions is quasiconvex, used to apply quasiconvex analysis to $G$.","marker":"[26]"},{"why":"Gives the convergence-of-sum criterion that bounds the number of failed iterations in Lemma 3.3.","marker":"[27]"},{"why":"Defines weak efficiency and Pareto optimality and links the minimax optimum to weakly efficient points of the multiobjective problem.","marker":"[30]"},{"why":"Provides the nonconvex multiobjective test problem and the nonmonotone projected-gradient comparison context for the numerical section.","marker":"[31]"},{"why":"Supplies the high-dimensional nonconvex multiobjective test instance used in Example 5.2.","marker":"[32]"}],"fun_headline_variants":["Adaptive steps prove nonconvex minimax convergence","No line search: adaptive steps converge for minimax","New gradient method drops line search for minimax","Nonmonotone steps replace line search in minimax"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence argument rests on the step from “the step size stays bounded away from zero” to “the search directions shrink to zero,” and the second claim does not follow from the first by itself.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive steps prove nonconvex minimax convergence","No line search: adaptive steps converge for minimax","New gradient method drops line search for minimax","Nonmonotone steps replace line search in minimax"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001862,"raw_usage":{"total_tokens":7284,"prompt_tokens":895,"completion_tokens":6389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":6326}},"tokens_in":511,"tokens_out":6389,"duration_ms":41339,"temperature":1.0,"reasoning_tokens":6326,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:41:11.812188+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Lemma 3.2 directly: the proof shows that the infimum of the allowable step sizes $\\alpha_k$ is positive, which only gives an upper bound on $\\|p_k\\|$, and then asserts this means $\\|p_k\\|^2\\to 0$. Exhibit any sequence of iterates satisfying the subsidiary problem and the step rule with $\\|p_k\\|$ constant, for instance directions alternating between $e_1$ and $-e_1$, while $\\alpha_k$ stays bounded below; then the lemma's asserted implication fails and the stationarity theorem lacks its key premise.","supporting_citations":[{"cited_title":"Numerical methods in extremal problems","cited_arxiv_id":null,"evidence_quote":"The steepest-descent minimax algorithm this work extends; supplies the linearized subsidiary-problem structure and the line-search baseline that the nonmonotone rule replaces."},{"cited_title":"Semismooth and semiconvex functions in constrained optimization","cited_arxiv_id":null,"evidence_quote":"Defines semiconvex and quasidifferentiable functions and the Clarke subdifferential results used to upgrade pseudoconvex stationarity to global minimality."},{"cited_title":"Generalized convexity of functions and generalized monotonicity of set-valued maps","cited_arxiv_id":null,"evidence_quote":"Provides the $\\partial_C$-quasiconvexity characterization used in the quasiconvex convergence proof."},{"cited_title":"Disciplined quasiconvex programming","cited_arxiv_id":null,"evidence_quote":"Supplies the result that the pointwise maximum of quasiconvex functions is quasiconvex, used to apply quasiconvex analysis to $G$."},{"cited_title":"Iterative algorithms for nonlinear operators","cited_arxiv_id":null,"evidence_quote":"Gives the convergence-of-sum criterion that bounds the number of failed iterations in Lemma 3.3."},{"cited_title":"Multicriteria optimization","cited_arxiv_id":null,"evidence_quote":"Defines weak efficiency and Pareto optimality and links the minimax optimum to weakly efficient points of the multiobjective problem."},{"cited_title":"Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization","cited_arxiv_id":null,"evidence_quote":"Provides the nonconvex multiobjective test problem and the nonmonotone projected-gradient comparison context for the numerical section."},{"cited_title":"Pareto multi-task learning","cited_arxiv_id":null,"evidence_quote":"Supplies the high-dimensional nonconvex multiobjective test instance used in Example 5.2."}],"review_version":1}