{"id":"70c8c629-0c1e-49c0-a67a-ba236a1bc4a1","arxiv_id":"2502.00525","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A projected variable smoothing method finds epsilon-approximate critical points of subspace-constrained weakly convex composites in O(epsilon^-3) iterations, and the Moreau envelope and prox of suprema of weakly convex functions are characterized.","lead":"This paper adapts a variable smoothing algorithm to minimize a smooth function plus a weakly convex composition over a linear subspace, proving an O(epsilon^-3) iteration bound. It also shows that the Moreau envelope of a supremum of weakly convex functions is the supremum of the envelopes, with new proximity operator formulas.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 2's stopping rule tests the full gradient norm, but Theorem 4.1 bounds only the projected gradient; for V≠H the full gradient can remain large even when the optimality gap is zero, so the stated O(ε^{-3}) termination guarantee is false as written.","rationale":"The most load-bearing claim is Theorem 4.1, the O(ε^{-3}) complexity guarantee for Algorithm 2. The proof's final step is invalid because the object controlled by the analysis and the object tested by the algorithm differ. Equation (14) bounds min ||P_V∇F_j|| and (10) bounds the distance to the optimality set by that projected norm; Algorithm 2 stops on the full gradient norm S_l = min ||∇F_j||. Since ||P_V y|| ≤ ||y||, a bound on the projected norm gives no bound on the full norm. The counterexample with h linear and orthogonal to V, g=0, shows the algorithm never terminates even though every iterate is already a critical point, violating Theorem 4.1 under every Lipschitz hypothesis. The reader's weakest-assumption concern about boundedness for non-Lipschitz applications is real but secondary: it affects only the showcase problems, whereas the stopping-criterion mismatch invalidates the main theorem in the core Lipschitz setting and in the simplest possible problem. The fix is localized: store the projected gradient norm or the exact distance in (10) in Algorithm 2, after which the epoch proof goes through. The Moreau-envelope results in Section 5 appear sound and are not affected by this issue. Overall, the paper is promising but the main complexity theorem needs a mandatory correction; a conditional acceptance with that required change is appropriate.","tokens_in":20412,"tokens_out":7830,"duration_ms":76641,"concrete_test":"Run Algorithm 2 exactly as printed on min_{x∈V}⟨v,x⟩ with V=ker(1,1,1)⊂R^3, h(x)=⟨v,x⟩, v=(1,1,1), g≡0, A=Id, α=1/3, C=1/4, and ε=0.1. Record the iteration count; the algorithm will not stop because ||∇F_k||=√3 for all k. Then modify the stopping rule to test S_l=min ||P_V∇F_{k+1}(x_{k+1})|| and rerun the same instance; it should stop in the predicted O(ε^{-3}) iterations, confirming that the stopping-criterion mismatch is the cause.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 4, Algorithm 2 sets S_l to the minimum of the full gradient norms ||∇F_{k+1}(x_{k+1})|| over an epoch and stops when S_l ≤ ε. Theorem 4.1's proof, however, only establishes a bound on the projected gradient: equation (14) gives min ||P_V∇F_j(x_j)|| ≤ rC(2^l)^{(α−1)/2}, and inequality (10) bounds the distance to the optimality set by ||P_V∇F_j(x_j)||. The stopping criterion requires the full gradient to be small, while the theory controls only its projection onto V. Since ||P_V y|| ≤ ||y||, the bound on the projected norm gives no control on the full norm. This is not a boundedness issue: it invalidates the core Lipschitz case. Explicit counterexample: take H=R^3, V=ker(1,1,1), h(x)=x_1+x_2+x_3 so ∇h≡v=(1,1,1)∈V^⊥, g≡0, and A=Id. Then every x∈V is a critical point because ∇h(x)∈N_V(x). Algorithm 1 keeps x_{k+1}=P_V(x_k−γ_k v)=x_k. Algorithm 2 computes S_l=||v||=√3 for every epoch and never satisfies S_l≤ε for any ε<√3, so it never terminates, contradicting the claimed O(ε^{−2/(1−α)}) bound. The fix is to store and test ||P_V∇F_{k+1}(x_{k+1})|| (or the exact distance in (10)) in Algorithm 2; with that change the epoch argument goes through.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20789,"tokens_out":5328,"duration_ms":57878,"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":[{"comment":"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.","section":"Section 4, Algorithm 2 and Theorem 4.1"},{"comment":"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.","section":"Sections 6.3 and 7; Theorems 3.1 and 3.2"},{"comment":"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.","section":"Section 3, Theorem 3.2 and Proposition A.2"}],"minor_comments":[{"comment":"There is a typo in 'the set of extended valued functions ohH'; it should read 'on H'.","section":"Section 2, first paragraph"},{"comment":"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.","section":"Section 4, Algorithm 2"},{"comment":"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.","section":"Section 6.3, equations (39)-(41)"},{"comment":"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.","section":"Section 7, numerical experiments"}],"recommendation":"major_revision","confidential_remarks":"I would encourage the editor to send this back with a clear request to fix the stopping criterion in Algorithm 2 and to re-examine the coverage of the non-Lipschitz applications. The counterexample to Theorem 4.1 is simple and decisive; however, the proposed correction is local and the rest of the paper, including the supremum-function results, appears technically sound. If the authors revise carefully, the paper could be suitable for publication. There is no apparent concern about citation practice: the use of [22] is a legitimate citation of the authors' own prior published work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things you should know. The Moreau-envelope and prox formulas for suprema of weakly convex functions (Theorems 5.1, 5.2 and the closed forms in Section 5.1) are a genuine, clean extension of the authors' earlier convex results in [22], and they will be useful for DRO and max-min problems. The proofs check out. The second thing is that the headline termination guarantee for the epoch variant (Algorithm 2, Theorem 4.1) is false as written.\n\nAlgorithm 2 stops when the full gradient norm S_l = min ||∇F_{k+1}(x_{k+1})|| falls below ε, but the proof only bounds the projected gradient ||P_V ∇F_j(x_j)|| and the optimality-gap distance (10), which is controlled by the projection. Since ||P_V y|| ≤ ||y||, the bound says nothing about the full norm. The counterexample is simple: take H = R^3, V = ker(1,1,1), h(x) = x1+x2+x3, g ≡ 0, A = I. Then ∇h = (1,1,1) lies in V^⊥, every x ∈ V is optimal, the iterates never move, and Algorithm 2 tests S_l = √3 forever, never terminating for ε < √3. The fix is straightforward: store and test either ||P_V ∇F_{k+1}(x_{k+1})|| or the exact distance in (10). Then the epoch argument goes through and the O(ε^{-3}) bound recovers. This is a real flaw in the core Lipschitz case, not a minor technicality.\n\nA second, softer issue: Theorem 3.2 (non-Lipschitz g) needs the generated sequence to be bounded, and the paper never verifies boundedness for the max-dispersion or DRO applications. Coercivity of the objective does not by itself bound the iterates of a smoothing method. So the headline guarantee does not actually cover the showcase problems; the authors should either prove boundedness or state the results conditionally.\n\nThe subspace constraint is less novel than the authors suggest — since any closed subspace is a Hilbert space, Algorithm 1 is just [6] applied on V — but the specialization is practically relevant for nonanticipativity constraints, and the projection step is handled cleanly. Numerical experiments are thin (one problem, no baselines), but that is a minor issue.\n\nWho is this for? Anyone needing prox formulas for weakly convex suprema or a first-order method on a subspace. The paper deserves a serious referee: the Section 5 results are correct and useful, and the termination bug is easily fixable. I'd send it back for revision, pointing at the stopping criterion and the boundedness gap.","headline":"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.","tokens_in":21349,"tokens_out":4259,"would_cite":true,"duration_ms":38248,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","49J52","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["variable smoothing","weakly convex optimization","proximal mapping","supremum function","projected splitting algorithm","Moreau envelope","distributionally robust optimization","max dispersion problem"],"falsifier":"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.","tokens_in":20200,"feed_emoji":"🧮","tokens_out":16979,"duration_ms":145141,"temperature":0.7,"pith_summary":"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.","feed_headline":"Projected smoothing solves weakly convex problems in O(epsilon^-3)","feed_subtitle":"A subspace-projected smoothing method also handles the max-type functions behind DRO and max dispersion.","key_machinery":"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 $\\operatorname{prox}_{\\mu f_c}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the baseline variable-smoothing iteration and the lemmas whose estimates Theorems 3.1 and 4.1 adapt.","marker":"[6]"},{"why":"Provides the convex Moreau-envelope-of-supremum theorem that Theorem 5.1 generalizes to weakly convex data.","marker":"[22]"},{"why":"Gives the weakly convex Moreau-envelope and proximity-operator identities used throughout the convergence proofs.","marker":"[1]"},{"why":"Supplies the nonexpansive-mapping and Krasnoselskii–Mann convergence tools used in Proposition 5.3 and in subspace projections.","marker":"[3]"},{"why":"Provides the penalty method whose convergence justifies the penalized reformulations in Section 6.","marker":"[20]"},{"why":"Introduces the max dispersion problem and its quadratic structure, the setting of the numerical experiments.","marker":"[18]"},{"why":"Supplies the distributionally robust optimization model and ambiguity-set formulation used in Section 6.","marker":"[27]"}],"fun_headline_variants":["Projected smoothing hits O(ε^-3) for weakly convex problems","New O(ε^-3) smoothing algorithm for weakly convex sums","Supremum functions get prox formulas from weakly convex smoothing","Projected variable smoothing yields O(ε^-3) rates and new prox rules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Projected smoothing hits O(ε^-3) for weakly convex problems","New O(ε^-3) smoothing algorithm for weakly convex sums","Supremum functions get prox formulas from weakly convex smoothing","Projected variable smoothing yields O(ε^-3) rates and new prox rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000736,"raw_usage":{"total_tokens":3365,"prompt_tokens":1096,"completion_tokens":2269,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":2192}},"tokens_in":712,"tokens_out":2269,"duration_ms":16495,"temperature":1.0,"reasoning_tokens":2192,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T18:42:25.919733+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"B¨ ohm and S.J","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline variable-smoothing iteration and the lemmas whose estimates Theorems 3.1 and 4.1 adapt."},{"cited_title":"P´ erez-Aros and E","cited_arxiv_id":null,"evidence_quote":"Provides the convex Moreau-envelope-of-supremum theorem that Theorem 5.1 generalizes to weakly convex data."},{"cited_title":"Attouch and D","cited_arxiv_id":null,"evidence_quote":"Gives the weakly convex Moreau-envelope and proximity-operator identities used throughout the convergence proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonexpansive-mapping and Krasnoselskii–Mann convergence tools used in Proposition 5.3 and in subspace projections."},{"cited_title":"Luenberger and Y","cited_arxiv_id":null,"evidence_quote":"Provides the penalty method whose convergence justifies the penalized reformulations in Section 6."},{"cited_title":"Jeyakumar and G","cited_arxiv_id":null,"evidence_quote":"Introduces the max dispersion problem and its quadratic structure, the setting of the numerical experiments."},{"cited_title":"Sun, A.J","cited_arxiv_id":null,"evidence_quote":"Supplies the distributionally robust optimization model and ambiguity-set formulation used in Section 6."}],"review_version":1}