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A Steepest Gradient Method with Nonmonotone Adaptive Step-sizes for the Nonconvex Minimax and Multi-Objective Optimization Problems

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.02010 v1 pith:DVFY5ZWT submitted 2025-02-04 math.OC

Nguyen Duc Anh , Tran Ngoc Thang This is my paper · ORCID
classification math.OC MSC 90C2690C2990C3090C47
keywords SteepestdescentNonmonotoneadaptivestepsizeNonconvexminimaxproblemMultiobjectiveoptimizationQuasiconvexfunctionsPseudoconvexParetostationarity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Section 3.3, Lemma 3.2] 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.
  2. [Section 3.3, Lemma 3.3] 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.
  3. [Section 3.3, Theorem 3.5] 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.
  4. [Sections 3.3 and 4.5, Theorems 3.6, 3.7, 4.6] 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.
minor comments (5)
  1. [Algorithm 1, lines 3-8] 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.
  2. [Algorithm 1, Step 2] 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.
  3. [Notation in Lemmas 3.2 and 3.3] 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.
  4. [Theorem 4.6 proof] The text 'From Lemma 3.7' should be 'From Theorem 3.7', since there is no Lemma 3.7 in the manuscript.
  5. [Section 5, Numerical Results] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence claims do not reduce to fitted constants or to self-citations; the main defect is a non-sequitur in Lemma 3.2, which is a correctness gap, not circular reasoning.

full rationale

The paper's derivation chain is input-driven rather than output-equivalent. Assumptions (A1)-(A3) are stated up front: existence of a solution, bounded level set, and Lipschitz gradients; these are not disguised forms of the stationarity conclusion. The step-size rule in Algorithm 1 is adaptive, but no parameter is fitted to the data used in the numerical experiments, and the examples do not back-propagate into the proofs. The only self-citation is [23], used in a background sentence about Tchebycheff scalarization with directional concessions; it is not invoked in Lemma 3.2, Theorem 3.5, Theorem 3.6, or Theorem 3.7, so it is not load-bearing. The serious internal problem is a logical gap in Lemma 3.2: after proving a positive infimum for the allowable bound alpha_k := min{1, delta/(||p_k||(K+||p_k||/2)), (1/2-epsilon)/L}, the proof asserts 'alpha_k is not converge to 0, which means ||p_k||^2 goes to 0'. A positive lower bound on an admissible step-size sequence does not imply the actual search directions tend to zero, so the stationarity conclusion is unsupported. Similarly, Theorem 3.5's Cauchy-type argument does not by itself establish 0 in the generalized subdifferential of G at theta*. These are non-sequiturs and correctness risks, but they are not circular steps: the theorems do not define stationarity in terms of alpha_k or p_k, nor do they fit a parameter to the target conclusion. Consequently, no step in the paper reduces by construction to its own inputs, and the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The proof relies on standard subdifferential calculus and on the stated assumptions. The only unusual item is the implicit assumption about the meaning of the counter s, which is an internal inconsistency rather than an invented entity.

free parameters (4)
  • epsilon = 0.4 in experiments; in (0,1/2) in theory
    Sufficient decrease parameter; chosen by hand, not fitted to data.
  • sigma = 0.9 in experiments; in (0,1) in theory
    Step-size shrinkage factor; chosen by hand.
  • eta_k = 0.01^k in experiments; positive summable sequence in theory
    Additive step-size increment; chosen by hand.
  • delta = not specified numerically
    Active set width parameter in J_delta; affects the stationarity characterization.
assumptions (5)
  • domain assumption Assumption A1: Problem (MP) has a solution; A2: the level set Omega is bounded; A3: gradients are Lipschitz on Omega.
    Stated in Assumption 3.1; needed for bounded iterates, Lipschitz expansions, and existence of limit points.
  • standard math Theorem 2.6 from Mifflin: the Clarke subdifferential of a max of locally Lipschitz functions is contained in, or equal to, the convex hull of active subdifferentials under semiconvexity or quasidifferentiability.
    Used to connect p_k to the Clarke subdifferential of G and to justify equality in the pseudoconvex case.
  • standard math Proposition 2.3: quasiconvex locally Lipschitz functions are Clarke-subdifferentially quasiconvex.
    Used in Theorem 3.6 to bound inner products involving the quasiconvex objective G.
  • domain assumption The subsidiary problem (SP) is solvable at every iteration and its KKT conditions hold with multipliers u_i.
    A Slater-type condition holds because p=0 and beta > G(theta_k) is strictly feasible, but this is not stated; the proof assumes it.
  • ad hoc to paper The proof requires that s in Algorithm 1 counts failed iterations only.
    Lemma 3.3's induction uses sigma raised to the number of failures, but the pseudocode increments s on every iteration, so the theorem's premise does not match the algorithm as printed.

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Cite this review

Pith. "Pith review of A Steepest Gradient Method with Nonmonotone Adaptive Step-sizes for the Nonconvex Minimax and Multi-Objective Optimization Problems." pith.science (2026). https://pith.science/paper/DVFY5ZWT

@misc{pith2026250202010,
  author       = {Pith},
  title        = {Pith review of: A Steepest Gradient Method with Nonmonotone Adaptive Step-sizes for the Nonconvex Minimax and Multi-Objective Optimization Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVFY5ZWT}},
  note         = {Machine review of arXiv:2502.02010}
}
read the original abstract

This paper proposes a new steepest gradient descent method for solving nonconvex finite minimax problems using non-monotone adaptive step sizes and providing proof of convergence results in cases of the nonconvex, quasiconvex, and pseudoconvex differentiate component functions. The proposed method is applied using a referenced-based approach to solve the nonconvex multiobjective programming problems. The convergence to weakly efficient or Pareto stationary solutions is proved for pseudoconvex or quasiconvex multiobjective optimization problems, respectively. A variety of numerical experiments are provided for each scenario to verify the correctness of the theoretical results corresponding to the algorithms proposed for the minimax and multiobjective optimization problems.

Figures

Figures reproduced from arXiv: 2502.02010 by the authors.

Figure 1
Figure 1. Results from Algorithm 2 applied to Example 5.1 16 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Results from Algorithm 2 applied to Example 5.2 By setting the dimensionality n to 20 and using nine reference points, we demonstrate the algorithm’s ability to effectively handle and solve high-dimensional, nonconvex prob￾lems. The results, as shown in [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Results from Algorithm 2 applied to Example 5.3 These results are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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Cited by 1 Pith paper

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