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A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A projected variable-smoothing algorithm reaches epsilon-approximate solutions in O(epsilon-cubed) iterations for weakly convex composite problems, and it supplies proximity formulas for supremum functions.

desk verdict Solid prox formulas for weakly convex suprema, but Algorithm 2's stopping rule tests the full gradient while the theory only bounds the projected gradient, so the O(ε^{-3}) termination guarantee is false as written; easy fix. read the letter →

arxiv 2502.00525 v1 pith:GYUZV3Z5 submitted 2025-02-01 math.OC

Sergio López-Rivera , Pedro Pérez-Aros , Emilio Vilches This is my paper · ORCID
classification math.OC MSC 90C2649J5265K05
keywords variablesmoothingweaklyconvexoptimizationproximalmappingsupremumfunctionprojectedsplittingalgorithmMoreauenvelopedistributionallyrobustmaxdispersionproblem
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 addresses the problem of minimizing $h(x)+g(Ax)$ over a closed vector subspace, where $h$ is smooth and $g$ is weakly convex. Its proposed method, projected variable smoothing, replaces $g$ by its Moreau envelope with a smoothing parameter that decays as $Ck^{-\alpha}$ and projects each step onto the subspace. The main complexity theorem states that, when $g$ is Lipschitz, an $\epsilon$-approximate critical point appears within $O(\epsilon^{-2/(1-\alpha)})$ iterations, which becomes $O(\epsilon^{-3})$ for $\alpha=1/3$; an epoch version returns a single iterate satisfying both residual tolerances. The paper also proves that the Moreau envelope of a supremum of weakly convex functions is the supremum of their Moreau envelopes, and from this derives computable proximity operators for quadratic and affine-quadratic supremum functions. These tools are applied to distributionally robust optimization, constrained LASSO, and the max dispersion problem, with numerical experiments on the last of these.

What carries the argument

The machinery is the Moreau envelope $g_\mu(x)=\inf_y\{g(y)+\frac{1}{2\mu}\|x-y\|^2\}$ and its proximity operator $\operatorname{prox}_{\mu g}(x)$, which smooths a weakly convex $g$ into a function with gradient Lipschitz constant $L_{\nabla h}+\|A\|^2/\mu_k$ when composed with $A$. Iterating $x_{k+1}=P_V(x_k-\gamma_k\nabla(h+g_{\mu_k}\circ A)(x_k))$ with $\mu_k=Ck^{-\alpha}$ and $\gamma_k=1/L_k$ is the projected variable smoothing step; the projection $P_V$ enforces the subspace constraint and enters the optimality residual through $N_V(x)$. For supremum functions, the load-bearing identity is Theorem 5.1, $f_\mu(x)=\sup_{c\in C}(f_c)_\mu(x)$, which allows one to compute $\operatorname{prox}_{\mu f}$ by selecting an index attaining that supremum and applying $\operatorname{prox}_{\mu f_c}$.

What would settle it

Run Algorithm 2 with $\alpha=1/3$, $C=1$, $h=0$, $g(x)=|x|$, $A=1$, and $V=H$, and check whether stopping occurs by the stated iteration bound $2\max\{\tilde C^3, C^3L_g^3\}\epsilon^{-3}$ for $\epsilon=10^{-3}$; if either residual tolerance is exceeded, the $O(\epsilon^{-3})$ complexity bound is false. Separately, for the max-dispersion instance with $R=(1\,1\,1)$ and $B$ the unit ball, record $\sup_{k\le 10^6}\|x_k\|$ under Algorithm 4; an unbounded sequence would confirm that Theorem 3.2's boundedness hypothesis is not automatic.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the composite weakly convex problem (2), minimize $h(x)+g(Ax)$ over $x$ in a closed vector subspace $V$, is solved by the projected variable smoothing iteration: set $\mu_k=Ck^{-\alpha}$, form $F_k=h+g_{\mu_k}\circ A$, and update $x_{k+1}=P_V(x_k-\gamma_k\nabla F_k(x_k))$. Theorems 3.1 and 4.1 prove that under $L_g$-Lipschitz $g$ the sequence satisfies $\min_{1\le j\le k} d(-\nabla h(x_j), A^*\partial g(\operatorname{prox}_{\mu_j}g(Ax_j))+N_V(x_j)) \le k^{(\alpha-1)/2}\tilde C$ and $\|Ax_k-\operatorname{prox}_{\mu_k}g(Ax_k)\|\le C L_g k^{-\alpha}$, so an $\epsilon$-approximate solution costs $O(\epsilon^{-2/(1-\alpha)})$ iterations, or $O(\epsilon^{-3})$ at $\alpha=1/3$. Theorem 3.2 extends the rates to non-Lipschitz $g$ in finite-dimensional codomain under a boundedness assumption on the iterates. For supremum functions $f=\sup_{c\in C} f_c$ with weakly convex data, Theorem 5.1 gives $f_\mu(x)=\sup_c(f_c)_\mu(x)$, and Theorem 5.2 identifies $\operatorname{prox}_{\mu f}(x)$ with $\operatorname{prox}_{\mu f_c}(x)$ for any $c$ attaining the supremum; Proposition 5.2 gives a closed-form prox for the maximum of quadratics, and Proposition 5.3 gives a convergent iterative prox for affine-quadratic suprema. Penalty versions of DRO, constrained LASSO, and max dispersion fit into problem (2), and the numerical section compares two subspace formulations for max dispersion.

Load-bearing premise

The load-bearing premise is that the weakly convex function $g$ is Lipschitz; for the non-Lipschitz applications the paper instead assumes the generated sequence stays bounded, and that boundedness is not verified for the DRO and max-dispersion examples.

Editorial extensions

If this is right

  • For Lipschitz weakly convex composites with subspace constraints, an $\epsilon$-approximate point is guaranteed in $O(\epsilon^{-2/(1-\alpha)})$ iterations, and the user can set $\alpha=1/3$ to get the $O(\epsilon^{-3})$ rate.
  • When the generated sequence stays bounded, the same rate holds without any Lipschitz assumption on $g$, expanding the class of usable objectives to functions like negative quadratics.
  • The supremum-envelope identity reduces proximity evaluation for supremum functions to a maximization over the index set $C$; for the maximum of finitely many quadratic terms, the maximizer has a KKT closed form, making the prox computation explicit.
  • The method is immediately applicable to penalty reformulations of DRO, constrained LASSO with weakly convex regularizers, and max dispersion, with the max-dispersion experiments indicating which of two subspace formulations converges faster.

Reading between the lines

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

  • A natural next step is to verify the boundedness hypothesis of Theorem 3.2 numerically for Algorithms 3 and 4; if the iterates stay bounded on the reported instances, the $O(\epsilon^{-3})$ guarantee would cover the non-Lipschitz DRO and max-dispersion problems as well.
  • The closed-form proximity for the maximum of quadratics should transfer to other max-type weakly convex losses, such as robust estimation or adversarial training, wherever the active index follows the same sorted-$\alpha_i$ KKT pattern.
  • An implicit testable consequence is that Algorithm 4's advantage in wall-clock time over Algorithm 3 should widen with larger $N$, because Algorithm 4 uses a closed-form proximity operator while Algorithm 3 runs an inner convergent iteration for the same task.
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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

3 major / 4 minor

Summary. The paper studies the composite minimization problem min_{x in V} h(x) + g(Ax), where h has a Lipschitz gradient, g is weakly convex, A is a bounded linear operator, and V is a closed vector subspace. The authors propose a projected variable smoothing method that interleaves decreasing smoothing parameters with projection onto V, and they prove convergence and complexity results: under a Lipschitz assumption on g they obtain an O(epsilon^{-2/(1-alpha)}) iteration bound (O(epsilon^{-3}) for alpha=1/3) for an epsilon-approximate optimality certificate, and under a boundedness assumption on the generated sequence they obtain analogous guarantees without the Lipschitz assumption. The paper also develops tools for suprema of weakly convex functions, showing that the Moreau envelope of a supremum is the supremum of Moreau envelopes and deriving formulas and algorithms for proximity operators in two cases (quadratic and affine data). These results are applied to distributionally robust optimization, constrained LASSO, and max dispersion problems, with numerical experiments for max dispersion.

Significance. The paper extends the variable smoothing framework of Boehm and Wright to the projected setting and to suprema of weakly convex functions, and the technical results in Section 5 appear useful and correct. The explicit formulas and convergent algorithms for proximity operators of weakly convex supremum functions are a genuine contribution. The main complexity theorem, however, is not correct as written: the stopping criterion used in Algorithm 2 tests the full gradient norm while the proof controls only the projected gradient norm. This is a load-bearing flaw because Theorem 4.1 is the paper's headline guarantee. The flaw is local and repairable by changing the stopping statistic to the projected gradient norm (or the exact distance in equation (10)), and with that correction the epoch argument supports the stated O(epsilon^{-3}) bound. The non-Lipschitz applications in Sections 6 and 7 are also not covered by the stated theorems, because boundedness of the generated sequence is assumed, not verified. If the stopping criterion is corrected and the application claims are qualified appropriately, the paper would be a solid contribution to weakly convex and nonsmooth optimization.

major comments (3)
  1. [Section 4, Algorithm 2 and Theorem 4.1] The stopping criterion in Algorithm 2 is incompatible with the proof of Theorem 4.1. Algorithm 2 stores S_l = min ||nabla F_{k+1}(x_{k+1})|| over the epoch and stops when S_l <= epsilon, but the proof of Theorem 4.1 establishes a bound only on the projected gradient: equation (14) bounds min ||P_V nabla F_j(x_j)||, and equation (10) bounds the optimality distance by ||P_V nabla F_j(x_j)||. Since ||P_V y|| <= ||y||, a small projected gradient does not imply a small full gradient. Concretely, take H = R^3, V = ker(1,1,1), h(x) = x_1+x_2+x_3, g = 0, and A = I. Then every x in V is a critical point because nabla h(x) = (1,1,1) belongs to V^perp = N_V(x). Algorithm 1 keeps x_{k+1} = P_V(x_k - gamma_k v) = x_k, and Algorithm 2 computes S_l = ||v|| = sqrt(3) in every epoch, so it never terminates for any epsilon < sqrt(3), contradicting the claimed O(epsilon^{-2/(1-alpha)}) termination guarantee. The fix is to store and test ||P_V nabla F_{k+1}(x_{k+1})||, or the exact distance in (10), in place of the full gradient norm; with that change the epoch argument in the proof of Theorem 4.1 goes through, since (14) and (15) then apply to the iterate on which the epoch minimum is attained.
  2. [Sections 6.3 and 7; Theorems 3.1 and 3.2] The max dispersion application is not covered by the theoretical guarantees as presented. The function rg in (40) and g in (42) are not globally Lipschitz on R^n or R^{nN}, so Theorem 3.1 and Theorem 4.1, which both assume g is L_g-Lipschitz, do not apply. Theorem 3.2 removes the Lipschitz assumption only under the additional hypothesis that the sequence (x_k) generated by Algorithm 1 is bounded, and no boundedness proof is supplied for Algorithms 3 and 4 in Section 7. Coercivity of the penalized objective for lambda > 2 does not by itself imply boundedness of the iterates of a variable-smoothing scheme with changing objectives. Thus the numerical experiments in Section 7 demonstrate heuristic performance, but the paper does not establish the advertised convergence or O(epsilon^{-3}) complexity for the showcased non-Lipschitz problems. I recommend either proving the required boundedness for these instances or clearly stating that the complexity guarantees apply only to the Lipschitz case and that the non-Lipschitz applications are supported by the conditional Theorem 3.2 when boundedness can be verified.
  3. [Section 3, Theorem 3.2 and Proposition A.2] The proof of Theorem 3.2 depends on Proposition A.2, which provides a uniform bound on ||nabla g_mu(y)|| for y in a bounded set S, but the argument in Proposition A.2 uses Lipschitz continuity of g on the bounded set rS. The claim that g is Lipschitz on rS follows from weak convexity and boundedness, but the proof as written omits the verification that g is finite and continuous on rS; g is assumed lower semicontinuous and weakly convex on R^m, so it is locally Lipschitz on the interior of its domain, but the domain is not stated to be all of R^m. If g is real-valued, this is fine; if g is extended-valued, the passage from weak convexity on R^m to Lipschitzness on rS needs an explicit domain assumption or a localization argument. This point should be clarified, since Theorem 3.2 is the only result covering non-Lipschitz applications.
minor comments (4)
  1. [Section 2, first paragraph] There is a typo in 'the set of extended valued functions ohH'; it should read 'on H'.
  2. [Section 4, Algorithm 2] The initialization j_l <- 2^l is never used before being overwritten; consider initializing j_l to 2^l and documenting that the recorded index is updated only when S_l changes.
  3. [Section 6.3, equations (39)-(41)] The equivalence between (39) and (41) is asserted but not proved; a brief argument that x = (x, ..., x) solves (41) if and only if x solves (39) would improve readability.
  4. [Section 7, numerical experiments] The stopping criterion based on ||x_{k+1} - x_k|| <= 10^{-5} is not justified by the theory, which provides stopping rules based on (projected) gradient norms or optimality distances; the experiments should either use a theory-compatible stopping rule or explicitly describe the stopping rule as a heuristic.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the complexity proof and the supremum-function prox results reduce to prior independent theorems rather than to the paper's own conclusions.

full rationale

The central complexity result (Theorem 4.1) is proved by the epoch-summing argument adapted from [6], using descent inequalities and Lipschitz/prox estimates whose proofs are cited from [6] and not from the target theorem. The genuinely new element, projection onto the subspace V, enters through the inequality d(-z, N_V(x)) <= ||P_V z||, which is proved in the paper. No parameter is fitted and no optimality measure is defined in terms of the complexity bound being claimed. The supremum-function section reduces the weakly convex case to the convex case via Lemma 5.1 and then invokes [22, Theorem 3.1] and [22, Theorem 3.5]. These are prior published results by two of the authors, but they are not restatements of the present target: they apply to convex data functions, whereas the present theorems concern weakly convex supremum functions. The reduction via [1, Theorem 3.4] and Lemma 5.1 is an external mathematical step, so the self-citation is real evidence rather than a circular premise. There is a genuine correctness gap in Section 4: Algorithm 2 stops on the full gradient norm ||∇F_{k+1}(x_{k+1})||, whereas the proof of Theorem 4.1 bounds only the projected norm ||P_V∇F_j(x_j)||. That mismatch is a bug in the written termination argument, not a circularity, because it does not make the claimed result equivalent to its inputs. Similarly, Theorem 3.2 assumes boundedness of the generated sequence without verifying it in the applications; this is an unproven hypothesis, not a circular derivation. Overall, the paper does not derive its conclusions from those same conclusions, and the central claims have independent content.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The core results introduce no constants fitted to data. The algorithm hyperparameters alpha, C, and the penalty lambda are hand-chosen but the complexity bound is uniform in them. The proofs depend on standard convex analysis and on the cited external results [1,3,6,22].

free parameters (3)
  • alpha (smoothing exponent) = 1/3 in experiments
    Algorithm parameter in (0,1); complexity exponent 2/(1-alpha) is minimized at alpha=1/3. Chosen by hand.
  • C (initial smoothing scale) = 1/4 in experiments
    Satisfies 2*rho*C <= 1 for rho=2; controls the initial smoothing parameter mu_1=C. Chosen by hand.
  • lambda (penalty weight) = 100 and 200 in experiments
    Penalty parameter in the quadratic penalty method; theory requires lambda > 2*sigma (sigma=1) for coercivity in the max dispersion application. Chosen by hand.
assumptions (6)
  • standard math Projection onto closed subspace is linear and N_V(x)=V^perp for x in V
    Used in Theorem 3.1 proof to relate distance to normal cone with projected gradient (eq. (9)-(10)).
  • domain assumption Bohm-Wright lemmas [6, Lemma 3.1-3.3, 4.1]
    Provide Lipschitz constant of nabla F_k, subdifferential inclusion for smoothed composite, prox residual bound, and inter-iteration function decrease.
  • domain assumption Attouch-Aze envelope relations [1, Theorem 3.4]
    Used in Theorem 5.1 and 5.2 to connect Moreau envelopes of f and h=f+(rho_bar/2)||x||^2.
  • domain assumption Convex supremum envelope theorems [22, Theorem 3.1 and 3.5]
    Used in Theorems 5.1 and 5.2 to transfer the envelope and prox formulas from convex to weakly convex data.
  • standard math KKT conditions for simplex-constrained convex problem (21)
    Used to prove closed-form solution p_bar in Proposition 5.2.
  • standard math Krasnoselskii-Mann convergence [3, Theorem 5.15]
    Used in Proposition 5.3 to prove convergence of the fixed-point iteration for computing the prox.

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Cite this review

Pith. "Pith review of A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions." pith.science (2026). https://pith.science/paper/GYUZV3Z5

@misc{pith2026250200525,
  author       = {Pith},
  title        = {Pith review of: A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYUZV3Z5}},
  note         = {Machine review of arXiv:2502.00525}
}
abstract

In this paper, we address two main topics. First, we study the problem of minimizing the sum of a smooth function and the composition of a weakly convex function with a linear operator on a closed vector subspace. For this problem, we propose a projected variable smoothing algorithm and establish a complexity bound of $\mathcal{O}(\epsilon^{-3})$ to achieve an $\epsilon$-approximate solution. Second, we investigate the Moreau envelope and the proximity operator of functions defined as the supremum of weakly convex functions, and we compute the proximity operator in two important cases. In addition, we apply the proposed algorithm for solving a distributionally robust optimization problem, the LASSO with linear constraints, and the max dispersion problem. We illustrate numerical results for the max dispersion problem.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Finite-Max Minimization and MIMO Applications

    math.OC 2025-06 accept novelty 6.0 of 10

    A proximal variable smoothing method with backtracking stepsizes finds stationary points for nonlinearly composite nonsmooth optimization with O(epsilon^-3) iteration complexity.

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