Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality

Anshika; Debdas Ghosh; Ravi Raushan; V. Vetrivel

REVIEW 3 major objections 6 minor 29 references

Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Set optimization with finite-cardinality set-valued maps is tackled by two proximal gradient methods—one with an Armijo-type line search, one without—that converge globally to stationary points under a mild image-space boundedness assumptio

desk verdict The proximal-gradient framework for finite-cardinality set optimization is worth engaging, but the stationarity characterization has a quantifier-reversal error that breaks the claimed link to weak minimality as stated. read the letter →

arxiv 2607.25252 v1 pith:GTFSYQBR submitted 2026-07-28 math.OC

Ravi Raushan , Debdas Ghosh , Anshika , V. Vetrivel This is my paper · ORCID

classification math.OC MSC 90C4790C2949J5390C46

keywords setoptimizationlowerset-lessorderingproximalgradientmethodArmijolinesearchstationarityconditionset-valuedmapsoffinitecardinalityglobalconvergenceO(1/√k)rate

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

This paper introduces the first proximal gradient methods for unconstrained set-valued optimization, where the objective map returns a finite set of vectors ordered by a cone. The authors establish a stationarity condition for these problems—a point is stationary precisely when a certain proximal subproblem over the partition set of active indices returns the zero direction—and build two algorithms around it: one with an Armijo-type line search and one with a fixed unit step-size. They prove both methods converge globally to stationary points under a mild image-space boundedness assumption, and they show that the line-search variant converges without the regularity hypotheses that earlier steepest descent methods required. They also derive a best-iterate O(1/√k) convergence rate for both algorithms and demonstrate the methods on uncertain set optimization test problems. A sympathetic reader would care because this extends composite non-smooth optimization to a setting that models uncertainty and set comparisons, and it supplies the missing convergence and complexity analysis for a whole class of first-order set optimization methods.

What carries the argument

The load-bearing object is the stationarity function Θ_l and its minimizer v_l. For a point ȳ, the partition set P̄_ȳ collects, for each enumeration of the minimal set M(H(ȳ),C), one choice of index for each minimal value; the scalarizing function φ(w)=sup_{z∈Z} zᵀw converts cone inequalities into real inequalities. Θ_l(ȳ) is the minimum over partition elements a and directions v of max_j φ(∇f_{a_j}(ȳ)ᵀv + g_{a_j}(ȳ+v) – g_{a_j}(ȳ)) + (l/2)||v||². This single function does three jobs: it characterizes stationarity (Θ_l=0 iff stationary), it produces a C-descent direction v_l(ȳ) at non-stationary points, and its value bounds the per-iteration decrease of the scalar gap ξ∘H, which is what tele

What would settle it

Run Algorithm 1 on a smooth set optimization problem (g_j=0) whose solution point is non-regular—e.g., a problem where the active minimal-set indices change at the optimum—and check whether ∥v_l^k∥ tends to 0 from a starting point in its domain. If the iterates consistently fail to converge or converge to a non-stationary point, the claim that regularity can be dropped would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that Algorithms 1 and 2 generate iterates whose stationarity measure Θ_l(x^k) tends to zero (equivalently, the proximal descent directions v_l^k vanish) for any starting point in dom(H), under Assumption 5.1: every ⪯ℓ-decreasing sequence of images has a bounded lower bound set. The stationarity measure is defined as the minimum over the partition set P_{x^k} and directions v of the maximum, over the active minimal-set indices, of the scalarized composite proximal term plus (l/2)||v||². The line-search variant achieves this without requiring the regularity of the solution point that the steepest descent method needed; the no-line-search variant instead requires Lipschitz

Load-bearing premise

The convergence and complexity theorems all depend on Assumption 5.1: every decreasing sequence of image sets below the starting image must have some bounded set lying below all of them—a kind of image-space lower boundedness that the paper assumes rather than derives.

Editorial extensions

If this is right

When all non-smooth terms vanish (g_j=0), the line-search algorithm reduces to the steepest descent method for set optimization studied in the literature, and the new convergence proof removes the regularity assumption on the solution point that the earlier analysis imposed.
For p=1 the problem reduces to composite vector optimization with an arbitrary solid cone; the paper thereby supplies the first proximal gradient method for that vector setting, covering objectives not handled by descent methods relying on gradient Lipschitz continuity.
Both methods achieve a best-iterate O(1/√k) rate on the stationarity measure, giving a first complexity benchmark for first-order set optimization methods.
The robust minimax test problems used in the numerical section show the methods can be applied directly to uncertain set-valued optimization, with the image sets shrinking to an optimal set in the lower set-less order.
Because the partition set is finite, the subproblem is a finite min-max, so existing scalar convex optimization solvers can be used inside each iteration.

Reading between the lines

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

The partition-set reduction is a template: any vector optimization method equipped with a scalar stationarity measure could be lifted to set optimization by minimizing that measure over the partition set at each iterate, suggesting that accelerated or inexact proximal variants may inherit the same or better rates.
The image-space boundedness Assumption 5.1 is likely to hold whenever ξ∘H is coercive on the sublevel set L; practitioners could verify convergence by checking coercivity of the scalarized objective, a condition easier to test than the assumption's stated form.
The authors' observation that step-size collapses below 10^-15 in the line-search variant points to a concrete next test: replacing the Armijo rule with a non-monotone or spectral step could avoid stagnation and preserve the convergence guarantee.
The best-iterate rate suggests that recording the smallest observed |Θ_l(x^k)|, rather than the last iterate, is the right way to use these algorithms in practice; an averaging scheme might convert the bound to a last-iterate rate under convexity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: the stress-test note is correct, and it hits a load-bearing part of the paper. Definition 3.4 defines stationarity as, for each partition element a, ∃j such that ∀v, D_v h_{a_j}(y)∉-int(C). But the proof of Proposition 3.2 only establishes the weaker ∀v∃j, and the proof of Theorem 3.1 (the direction (ii)=> (i)) also gives ∀v∃j, not ∃j∀v. The affine example in the stress test is clean: H(x)={(x1,x2),(1−x1,−1−x2)} at 0 is locally weakly minimal, fails Definition 3.4, yet Θ_l(0)=0. So Theorem 3.1 and Proposition 3.2 are false as written. The good news: redefining stationarity as ∀a∀v∃j fixes the connection, and the algorithms’ stopping criterion Θ_l=0 converges exactly to that corrected condition. This looks like a genuine error, not a shallow one, but it is also a repairable one.

What is new and useful: this is the first proximal-gradient treatment of set-valued maps of finite cardinality with nonsmooth composite terms; the Θ_l/v_l subproblem is a natural descent-direction scheme; the claimed removal of Bouza et al.’s regularity assumption is interesting; and the O(1/√k) best-iterate complexity rate is plausible. The numerical experiments are extensive, though there is no code or data.

Soft spots, in order: 1. The quantifier error above. It is load-bearing and must be fixed before the results can be trusted. 2. Theorem 4.1’s proof has sign/inequality problems: the contradiction assumption seems to assume the Armijo condition holds when it should fail, and the subsequent ⪯_C chain goes in the wrong direction. The existence of an Armijo step-size is likely recoverable by a standard argument, but the written proof is not correct. 3. The O(1/√k) conclusion is asserted from a best-iterate bound on Θ_l; the short step |Θ_l|≥(l/2)||v_l||² that gives ||v_l||=O(1/√k) is not written down. 4. The introduction claims there is no proximal gradient method for the vector problem (VOPC), while citing [2], [4], and [27] that do exactly that. That overclaim should be removed. 5. Assumption 5.1 is doing real work (image-space lower boundedness). It is an assumption, not a consequence, but it is a standard kind of level-boundedness and not itself a red flag.

Recommendation: send to peer review. The core idea is sound, the flaws are specific and repairable, and the topic is timely for the set-optimization community. I would not cite the current version in my own work, but I would engage with a revision.

Referee Report

3 major / 6 minor

Summary. The paper studies the unconstrained set-valued optimization problem (SOP_l) with objective H(x)={h_1(x),...,h_p(x)}, each h_j=f_j+g_j, under the lower set-less order induced by a closed convex pointed solid cone. It introduces a stationarity notion based on a scalarization of the minimal set and a partition set of active indices, then proposes two proximal gradient methods: Algorithm 1 with an Armijo-type line search and Algorithm 2 with a fixed step size and adjusted proximal parameter. The paper claims global convergence of both methods to stationary points under an image-space level-boundedness assumption (Assumption 5.1) and an O(1/sqrt(k)) best-iterate complexity rate, followed by numerical experiments on constructed set-valued test problems with robust uncertainty terms.

Significance. If the theoretical claims are correct, this would be a valuable contribution: it appears to be the first proximal gradient framework for non-smooth set-valued optimization problems with finitely many image points, it removes the regularity assumptions required by the steepest descent method of Bouza et al., and it provides the first complexity analysis for this class. The numerical implementation is reasonably extensive and demonstrates practical viability. However, the central optimality characterization is false as stated: the quantifier order in Definition 3.4 is interchanged, which breaks the claimed equivalence with weak minimality and therefore undermines the interpretation of the algorithms' stopping criteria. The paper also contains serious proof errors in the line-search existence argument and an unsupported rate claim. These issues are repairable, but the current version is not publishable without major revision.

major comments (3)

[Section 3 (Definition 3.4, Proposition 3.2, Theorem 3.1)] The quantifier order in Definition 3.4 is invalid. From Proposition 3.1 applied to the vector subproblem (VOP_a) one obtains for all a in P_y and all v in R^n that there exists j with D_v h_{a_j}(y) not in -int(C). Definition 3.4 instead requires for all a that there exists j such that this holds for all v. This is strictly stronger. The proof of Theorem 3.1(ii)=>(i) actually derives max_j phi(D_v h_{a_j}(y)) >= 0 for all a,v, which is the correct forall-v-exists-j condition, not the stated definition. The difference is material: take C=R_+^2, H(x)={(x_1,x_2),(1-x_1,-1-x_2)}, y=0. The point y is locally weakly minimal, yet no single index j blocks both v=(-1,-1) and v=(1,1); meanwhile Theta_l(0)=0 and v_l(0)=0 because max(phi(v),phi(-v))=||v||_infty. Thus Theorem 3.1 and Proposition 3.2 are false as written, and the claimed connection between the algorithms' stationarity test and weak mi
[Section 4.1 (Theorem 4.1 and Algorithm 1)] The proof of Theorem 4.1 is logically misdirected. After assuming that no non-singleton interval satisfies the Armijo condition (17), the proof must exhibit a sequence alpha_k -> 0 for which (17) fails for some index j. Instead, the proof writes the Armijo inequality as holding and then derives a contradiction from (19). This does not establish the existence of a step-size, so the well-definedness of Step 4 of Algorithm 1 is not proved. The existence statement is true and can be proved by a direct expansion using convexity of g and the fact that the vector (1-alpha)D + small terms lies in -int(C) for D in -int(C), but the proof as presented is invalid.
[Section 5 (Theorems 5.3, 5.4 and following paragraph)] The claimed O(1/sqrt(k)) complexity does not follow from the displayed results. Theorems 5.3 and 5.4 only bound |Theta_l(x^k)| < C/k for some best iterate k. Since Theta_l(x^k) = F_k(v_l^k) + (l/2)||v_l^k||^2 with F_k(v_l^k) <= 0, a bound on |Theta| does not by itself bound ||v_l^k||. The missing step is the strong convexity of v -> F_k(v)+(l/2)||v||^2: evaluating the strong convexity inequality at v=0 yields (l/2)||v_l^k||^2 <= -Theta_l(x^k) = |Theta_l(x^k)|. This step should be stated explicitly; without it, the abstract's O(1/sqrt(k)) claim is unsupported.

minor comments (6)

[Equation (11) and surrounding text] The function phi_l is introduced as phi_l(a,v; y) but later referred to as phi_l(v;a,y); make the argument order consistent throughout.
[Lemma 3.1] The lemma assumes lower semi-continuity of h_j in its proof, but this assumption is not stated in the lemma. Please add it to the hypotheses.
[Remark 3.1] The 'if and only if' claim for weak minimality is stated without proof. Either prove it or indicate precisely where it follows from Lemma 3.1.
[Algorithm 1, Step 4] The notation h_{a_k,j} is used before it is defined. Define the indexing map (k,j) -> a_{k,j} explicitly.
[Table 2] Several entries in the performance table appear misaligned, making it hard to associate min/mean/max columns with Algo 1 and Algo 2. Please reformat.
[End of Section 5] Theorems 5.3 and 5.4 provide best-iterate bounds; the text should explicitly state that the O(1/sqrt(k)) rate is a best-iterate rate, not a rate for the last iterate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: convergence is measured by the algorithm's own stationarity subproblem, but no parameter is fitted and the proofs are self-contained; the main concern is a correctness gap, not circularity.

full rationale

The convergence and complexity results (Theorems 5.1–5.4) are not circular. They explicitly assume Assumption 5.1 (image-space level boundedness) and Lipschitz continuity of ∇f_j, then telescope the scalarization ξ∘H along the iterates using the descent lemma (Lemma 2.2) and the three-point property. No data are fitted and later renamed as predictions. The stationarity measure Θ_l is defined as the minimum of the same proximal subproblem (13) that the algorithms solve, so proving Θ_l(x^k)→0 is the intended convergence target; the identity v_l=0 ⇔ Θ_l=0 is explicitly displayed in (14), and Theorem 3.1 is an asserted equivalence between that condition and the independent derivative-based Definition 3.4. Thus stationarity is not defined in terms of Θ_l by construction; an equivalence is argued, not assumed. The self-citations [12], [13], [14] appear only as literature context and are not load-bearing; the main proofs rely on [3], [5], [10] and on the paper's own explicit assumptions. The most serious issue is a proof-level quantifier exchange in Proposition 3.2/Theorem 3.1: the proof goes from '∀a ∀v: D_v h_a ∉ −int(C^ω)' to '∀a ∃j ∀v: D_v h_{a_j} ∉ −int(C)', which is a ∀v∃j to ∃j∀v reversal. This is a correctness/rigor concern (potentially making the stationarity equivalence false as stated), not a circular reduction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theory introduces no fitted parameters and no new entities. The result depends on standard convex-analysis facts plus explicit domain assumptions: differentiability/Lipschitz structure on f_j, convexity of g_j, and an image-space level-boundedness condition (Assumption 5.1). Algorithmic constants such as l, ρ, µ, δ and matrices A_j^i are user-chosen or adaptively adjusted, not fitted to data.

assumptions (6)

domain assumption Each h_j = f_j + g_j with f_j continuously differentiable and g_j proper, closed, convex; ordering cone C solid, closed, convex and pointed.
Defines the problem class in Section 1; used throughout the optimality and convergence analysis.
domain assumption ∇f_j is Lipschitz continuous for each j, with L := max L_j.
Assumed for Algorithm 2 (Section 4.2), Proposition 4.1, Lemma 5.1 and the complexity theorems; not assumed for the basic line-search convergence (Theorem 5.1).
domain assumption Assumption 5.1: image-space level sets are bounded from below in the lower set-less order.
Used in Theorems 5.1–5.4 to bound telescoped merit sums; without it the convergence and complexity proofs fail.
standard math Drummond–Svaiter scalarization φ(w) = sup_{z∈Z} z·w with Z = C* ∩ {||z||=1}; properties in Lemma 2.1 and the vector descent lemma (Lemma 2.2).
Adapted from [10] and [3, Proposition A.24]; underpins the descent arguments.
standard math Three-point property for proximal operators (inequality (22)).
Used in Section 4.2 to derive F_k(v_l^k) ≤ -l||v_l^k||², which is essential for the no-line-search descent and the O(1/√k) rate.
domain assumption Lower semi-continuity of h_j, j∈[p].
Invoked in Lemma 3.1 for the characterization of local weak minimality via partition problems; follows from f continuous and g lower semi-continuous.

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Cite this review

Pith. "Pith review of Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality." pith.science (2026). https://pith.science/paper/GTFSYQBR

@misc{pith2026260725252,
  author       = {Pith},
  title        = {Pith review of: Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTFSYQBR}},
  note         = {Machine review of arXiv:2607.25252}
}

abstract

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.

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Optimizatio n 71(1), 33–53 (2022)

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