{"id":"3798cc6c-5703-4cfd-beeb-e27de0a1645f","arxiv_id":"2411.14828","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A Bregman Lagrangian with a logarithmic barrier leads to a continuous-time dynamical system and discrete accelerated methods that converge to the solution of convex inequality-constrained problems.","lead":"This paper derives accelerated optimization algorithms for convex problems with inequality constraints by combining logarithmic barrier functions with a continuous-time Bregman Lagrangian. The authors prove exponential convergence for the continuous-time system and present two discrete algorithms with polynomial rates, tested on a two-dimensional quadratic problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 silently applies a fixed-barrier Lyapunov lemma to the time-varying barrier mandated by Assumption 3.1, so the central continuous-time exponential convergence claim is not established.","rationale":"The reader's weakest assumption targets the discrete step-size mismatch in Theorem 4.2 and Algorithm 2, and that criticism is valid: a fixed positive \\eta cannot satisfy \\eta \\le 1/(kL_\\Phi) for all k. However, a more fundamental gap appears earlier in the continuous-time analysis, which the paper itself identifies as its primary contribution. Lemma 3.1 establishes exponential convergence to \\hat x^* only for a fixed log-barrier \\Phi; Assumption 3.1 then makes c and s time-dependent, so the target minimizer moves. The proof of Theorem 3.1 does not account for the resulting \\partial\\Phi/\\partial t and \\dot{\\hat x}^* terms in the Lyapunov derivative. Absent new estimates for these terms, the central claim that the dynamical system (3.4) converges to x^* with rate O(e^{-\\beta_t/2}) is unsupported. I do not see a way to dismiss this as a minor typo: the dynamics, the target, and the Lyapunov function all depend on t in the regime covered by the theorem. The fixed-barrier derivation and the discrete algorithms may still be partially salvageable, but the headline continuous-time result requires either a genuinely time-dependent Lyapunov argument or a reformulation with fixed c,s and an explicit limit procedure. Since the paper has a plausible framework and the gap is a missing proof rather than a demonstrated counterexample, keeping the verdict CONDITIONAL is appropriate, but the required revision is more substantive than the reader's step-size condition alone.","tokens_in":16837,"tokens_out":14577,"duration_ms":147636,"concrete_test":"Re-derive the proof of Theorem 3.1 with \\hat x^*_t = \\arg\\min\\Phi_t under Assumption 3.1. Differentiate E_t = e^{\\beta_t}(\\Phi_t(X_t)-\\Phi_t(\\hat x^*_t)) + V_h(\\hat x^*_t, X_t+e^{-\\alpha_t}\\dot X_t) and collect all terms not present in the fixed-barrier proof; if the extra \\partial\\Phi_t/\\partial t and \\dot{\\hat x}^*_t terms cannot be bounded, the claimed Lyapunov monotonicity is not established. For a concrete numerical check, use the 2D example (5.1) with \\beta_t=2t, c=e^{\\beta_t}, s=e^{-\\beta_t}, h=\\tfrac12\\|\\cdot\\|^2, integrate (3.4), and verify whether \\|X_t-x^*\\| \\le C_0 e^{-\\beta_t/2} actually holds; an admissible parameter set where the bound fails would falsify Theorem 3.1 as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim, Theorem 3.1, is not actually proved as stated. Assumption 3.1 makes the barrier parameter and slack time-dependent: c(t)=e^{\\beta_t} and s(t)=e^{-\\beta_t}. Hence the log-barrier problem (2.8), and its minimizer \\hat x^*, change with t. Lemma 3.1 and Lemma 2.1 are proved for a fixed \\Phi and a fixed \\hat x^*; in particular, the Lyapunov computation (3.6) contains only \\nabla\\Phi and no \\partial\\Phi/\\partial t term. Theorem 3.1 applies these lemmas to the time-varying problem without justification. If one correctly differentiates E_t=e^{\\beta_t}(\\Phi_t(X_t)-\\Phi_t(\\hat x^*_t))+V_h(\\hat x^*_t,X_t+e^{-\\alpha_t}\\dot X_t), extra terms appear: e^{\\beta_t}(\\partial\\Phi_t/\\partial t(X_t)-\\partial\\Phi_t/\\partial t(\\hat x^*_t)) and \\langle\\nabla h(\\hat x^*_t)-\\nabla h(Y_t),\\dot{\\hat x}^*_t\\rangle. With c=e^{\\beta_t}, one has \\partial\\Phi_t/\\partial t = \\dot\\beta_t[e^{-\\beta_t}\\sum_i\\log(s-g_i)+e^{-2\\beta_t}\\sum_i 1/(s-g_i)], which is not sign-definite. No control of these terms is given, so non-increase of E_t, and hence (3.5) and the stated O(e^{-\\beta_t/2}) rate, do not follow. This is an internal proof gap, not a question of disagreeing with existing consensus: the fixed-barrier variational derivation is plausible, but the time-varying extension is missing.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes continuous-time and discrete-time accelerated methods for inequality-constrained convex optimization (1.1). It approximates the constrained problem by a logarithmic-barrier objective (2.7), derives a Bregman-Lagrangian dynamical system (3.4) via Hamilton's principle, and proves exponential convergence of the continuous-time trajectory to the true optimum under Assumption 3.1. It then discretizes the dynamics to obtain three algorithms and gives convergence rates, together with numerical experiments on a quadratic program.","tokens_in":17232,"tokens_out":7827,"duration_ms":70208,"significance":"If the main results were correct, the paper would be a useful contribution to the variational/continuous-time understanding of accelerated methods for inequality-constrained problems, a setting that has received less attention than equality-constrained or unconstrained problems. The Bregman-Lagrangian derivation (Proposition 3.1) is standard and, for a fixed barrier, the Lyapunov argument in Lemma 3.1 is essentially sound. The paper also makes a concrete computational proposal, Algorithm 2, and documents numerical behavior. However, as shown below, the central time-varying barrier argument and the discrete-time step-size conditions contain load-bearing gaps, so the claims are not established in their present form.","major_comments":[{"comment":"Lemma 3.1 is proved for a fixed objective Φ and a fixed minimizer x̂*, but Assumption 3.1 makes c(t)=e^{β_t} and s(t)=e^{-β_t}, so Φ_t and x̂*_t are time-dependent. The Lyapunov function in (3.6) is E_t = e^{β_t}(Φ_t(X_t)-Φ_t(x̂*_t)) + V_h(x̂*_t, X_t+e^{-α_t}Ẋ_t), and its derivative contains no ∂Φ_t/∂t or dx̂*_t/dt terms. The proof of Lemma 3.1 therefore does not imply non-increase of E_t, and inequality (3.5) does not follow for the time-varying barrier. Extra terms such as e^{β_t}(∂Φ_t/∂t(X_t)-∂Φ_t/∂t(x̂*_t)) and <∇h(x̂*_t)-∇h(Y_t), dx̂*_t/dt> must be estimated; no such estimate is given. This gap affects Theorem 3.1 and also the discrete-time Theorems 4.1 and 4.2, which invoke Assumption 3.1 while using Lyapunov functions with an implicitly fixed x̂*.","section":"§3.2, Lemma 3.1 and Theorem 3.1"},{"comment":"Theorem 4.2 states that if A_k=C k^2, 0<C≤μ/(4L_Φ), and 0<η≤1/(kL_Φ), then ε_{k+1}≤0 and ‖x_k-x*‖=O(1/k). However, Algorithm 2 fixes η>0 in its initialization and uses the same η in every iteration. Since 1/(kL_Φ) tends to 0, no fixed positive η can satisfy the theorem's condition for all k. The stated convergence proof therefore does not apply to Algorithm 2 as written. The numerical section (Section 5) uses the constant values η=1/4 and η=1/2 and asserts that all conditions of Theorem 4.2 are satisfied, which is inconsistent with the theorem's own step-size requirement.","section":"§4.2, Theorem 4.2 and Algorithm 2"},{"comment":"The derivation of Φ(y_{k+1})-Φ(x_{k+1}) ≤ -1/(2L_Φ)‖∇Φ(x_{k+1})‖² is not justified. From smoothness and y_{k+1}-x_{k+1}=-η∇Φ(x_{k+1}) one obtains Φ(y_{k+1})-Φ(x_{k+1}) ≤ -(η - L_Φ η²/2)‖∇Φ(x_{k+1})‖². For η=1/(kL_Φ), the coefficient is (2k-1)/(2k²L_Φ), which equals 1/(2L_Φ) only at k=1 and is smaller for k>1. Consequently, the subsequent inequality δα_k²/(2μ) - A_{k+1}/(2δL_Φ) ≤ 0 does not imply ε_{k+1}≤0; with the correct k-dependent coefficient, the term δα_k²/(2μ) grows like k² and cannot be controlled for all k by the stated choice of C. This gap affects the claimed O(1/k) rate even if η were allowed to be time-varying.","section":"§4.2, proof of Theorem 4.2"},{"comment":"Corollary 4.1 is stated under Assumption 3.1, but Algorithm 3 sets A_k = Σ_{i=0}^k 1/(i+1)², which is bounded above by π²/6. Hence s_k=1/A_k does not tend to 0, violating Assumption 3.1(ii). The use of Lemma 2.1 to pass from convergence of Φ(x_k)-Φ(x̂*) to convergence of x_k to the true optimum x* is therefore not justified in this case, since the approximation error |f(x̂*)-f(x*)| need not vanish. Corollary 4.1's conclusion ‖x_k-x*‖=O(1/√k) requires an additional argument that is not supplied.","section":"§4.3, Algorithm 3 and Corollary 4.1"}],"minor_comments":[{"comment":"In the partial derivative ∂L/∂X_t, the expression should be e^{γ_t}⟨∇h(X_t), ·⟩ rather than an undefined product; there is also a missing gradient symbol in the second line. These typos make the derivation harder to follow, although the fixed-barrier calculation is plausible.","section":"§3.1, Proposition 3.1 proof"},{"comment":"The derivative computation contains the term e^{β_t}⟨Φ(X_t), Ẋ_t⟩, which should be e^{β_t}⟨∇Φ(X_t), Ẋ_t⟩. The same typo appears in the displayed formula before substitution of (3.4).","section":"§3.2, Lemma 3.1 proof"},{"comment":"The second line of (4.4) reads ‖y_{k+1}-x_{k+1}‖₂ = -η∇Φ(x_{k+1}), which equates a scalar norm to a vector. This should be y_{k+1}-x_{k+1} = -η∇Φ(x_{k+1}).","section":"§4.2, equation (4.4)"},{"comment":"In the proof of Lemma 4.2, the expression for V_h(x, x̂*) uses ∇h(y) with an undefined y; it should be ∇h(x̂*).","section":"§2, Lemma 4.2 proof"},{"comment":"The final bound contains the typo f(x̂* - f(x*)); the intended expression is f(x̂*) - f(x*). Also, the constant C₀ is defined with a parenthesis mismatch in the displayed formula.","section":"§3.2, Theorem 3.1 proof"},{"comment":"The symbol T is used both as the time interval in (3.1) and as the transpose operator in (2.4) and elsewhere. This is confusing; a different symbol for the time horizon would improve readability.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant problem and contains a potentially salvageable variational framework, but the main convergence theorems for the time-varying barrier are not proved as stated, and the discrete-time step-size condition in Theorem 4.2 does not match Algorithm 2. These are load-bearing issues rather than mere presentation defects. I recommend major revision with a request to either prove the time-varying Lyapunov estimates or state the theorems under a fixed-barrier regime, and to align Algorithm 2 with a step-size sequence that satisfies the hypotheses."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The core derivation is a clean application of Wibisono et al.'s Bregman Lagrangian to a log-barrier smoothed problem, and the fixed-barrier Lyapunov computation checks out. The genuinely new bit is the combination of a logarithmic barrier plus slack with the variational acceleration framework, together with the barrier error bound that connects the smoothed minimizer to the original x*. That part is worth taking seriously. The problem is that the paper's central continuous-time claim, Theorem 3.1, is not proved as stated.\n\nAssumption 3.1 makes c(t)=e^{βt} and s(t)=e^{-βt}, so Φ and \\hat{x}* are time-dependent. Lemma 3.1 is proved for a fixed Φ and fixed \\hat{x}*; the derivative of E_t in (3.6) contains no ∂Φ/∂t or \\dot{\\hat{x}}* terms. If you differentiate the actual time-dependent E_t, those terms appear and are not sign-controlled. So the claimed O(e^{-βt/2}) tracking of x* does not follow. The stress-test note has this right.\n\nThe discrete part has a similar-size gap. Theorem 4.2 states η ≤ 1/(kL_Φ) as sufficient, but Algorithm 2 fixes η. Worse, the algebra in the proof needs η bounded below by something like 4C/μ to make ε≤0; the stated upper bound cannot hold for all k. So the O(1/k) rate is not established for the algorithm as written.\n\nSmaller issues: 'optimal rate' is asserted without any lower bound, and the experiments are one 2D quadratic, compare only against their own Algorithm 3, and ship no code. The 'optimality' language should go.\n\nRecommendation: This deserves a serious referee, but in its current form it should be returned for major revision, not accepted. The framework is plausible and the fixed-barrier part is correct; a repair would need a corrected time-varying Lyapunov computation (or a fixed-barrier version with explicit c→∞ analysis) and a step-size condition that actually matches Algorithm 2. I would not cite it until that is done.","headline":"The Bregman-Lagrangian machinery is applied cleanly to a fixed log-barrier, but the paper's central theorem silently treats the barrier as fixed while Assumption 3.1 makes it time-dependent, so the exponential convergence claim is not actually proved.","tokens_in":17803,"tokens_out":5768,"would_cite":false,"duration_ms":68090,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C30","65K05","37N40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Bregman-Lagrangian dynamical system is proved to converge exponentially to the solution of an inequality-constrained convex program, and its discretizations give accelerated gradient methods.","keywords":["inequality constrained convex optimization","logarithmic barrier","Bregman Lagrangian","continuous-time dynamical system","Lyapunov function","accelerated gradient method","Euler discretization","convergence rate"],"falsifier":"Numerically integrate (3.4) for the quadratic example (5.1) with $\\beta_t = 2t$ and measure $\\log\\|X_t - x^*\\|_2$; Theorem 3.1 predicts a slope approaching $-1$, meaning $\\|X_t - x^*\\|_2 \\sim e^{-t}$. A slope that flattens, or any iteration at which $s - g_i(X_t)$ reaches zero and the log-barrier gradient blows up, would falsify the claim.","tokens_in":16597,"feed_emoji":"⚡","tokens_out":7683,"duration_ms":66145,"temperature":0.7,"pith_summary":"This paper tries to show that an inequality-constrained convex optimization problem can be solved by a smooth dynamical system that tracks the constrained optimum exponentially fast. The strategy is to replace the hard constraints with a logarithmic barrier whose strength and a slack variable are scheduled by time, turning the problem into an unconstrained one, and then to derive a Bregman-Lagrangian flow from Hamilton's principle. The main result states that, when the barrier parameter grows as $e^{\\beta t}$ and the slack decays as $e^{-\\beta t}$, the trajectory $X_t$ of the flow satisfies $\\|X_t - x^*\\|_2 \\le C_0 e^{-\\beta t/2}$. If the proof is right, it gives a continuous-time explanation of acceleration for constrained problems and a principled route to discrete accelerated gradient methods with provable rates.","feed_headline":"Lagrangian flow solves constrained convex problems exponentially","feed_subtitle":"A scheduled barrier and slack pull the trajectory to the optimum exponentially fast; discrete versions reach O(1/k).","key_machinery":"The load-bearing object is the Bregman Lagrangian $L(X_t,\\dot X_t,t)=e^{\\alpha t+\\gamma t}(V_h(X_t+e^{-\\alpha t}\\dot X_t,X_t)-e^{\\beta t}\\Phi(X_t))$, where $V_h$ is the Bregman divergence of a strongly convex distance-generating function $h$. Under the ideal scaling conditions $\\dot\\beta_t \\le e^{\\alpha t}$ and $\\dot\\gamma_t = e^{\\alpha t}$, its Euler-Lagrange equation collapses to the differential inclusion $d\\nabla h(X_t+e^{-\\alpha t}\\dot X_t) = -e^{\\alpha t+\\beta t}\\nabla\\Phi(X_t)\\,dt$. The convergence argument is carried by the Lyapunov function $E_t=e^{\\beta t}(\\Phi(X_t)-\\Phi(\\hat{x}^*)) + V_h(\\hat{x}^*, X_t+e^{-\\alpha t}\\dot X_t)$, whose monotone decrease transfers to distance to the approximate optimum, and by Lemma 2.1, which transfers that to distance to the true optimum. For the discrete half, the same Lyapunov structure is discretized: implicit Euler gives Algorithm 1, and a hybrid Euler step with an extra sequence $y_k$ gives Algorithm 2.","core_discovery":"The paper's central claim is Theorem 3.1: under Assumption 3.1, the solution of the continuous-time system (3.4) converges to the unique optimal solution $x^*$ of the original problem, with the explicit rate $\\|X_t - x^*\\|_2 = O(e^{-\\beta t/2})$ when the objective is smooth. The mechanism is a two-level approximation: the log-barrier $\\Phi(x)=f(x)-\\frac{1}{c}\\sum_i\\log(s-g_i(x))$ has a unique minimizer $\\hat{x}^*$, Lemma 2.1 bounds the gap between $\\hat{x}^*$ and $x^*$ by $m/c + s\\|\\lambda^*\\|_1$, and Lemma 3.1 shows the flow converges to $\\hat{x}^*$ with the same exponential factor. Making $c(t)=e^{\\beta t}$ and $s(t)=e^{-\\beta t}$ balances the two errors, so the trajectory homes in on the true optimum at rate $e^{-\\beta t/2}$. The paper further derives discrete algorithms by Euler discretization: Algorithm 1 matches the continuous rate $O(1/\\sqrt{A_k})$, while Algorithm 2, a hybrid implicit-explicit scheme with an additional mirror-descent sequence, is claimed to achieve $O(1/k)$.","pith_inferences":["A natural extension is to make the barrier schedule adaptive, choosing $c$ from the current duality gap $\\|\\lambda^*\\|_1$ and using Lemma 2.1 as a stopping criterion; the paper's proof structure suggests the same Lyapunov machinery would carry over if $\\dot c/c$ stays comparable to $\\dot\\beta$.","The proof of Theorem 4.2 uses step sizes $\\eta_k \\le 1/(kL_\\Phi)$, while Algorithm 2 is stated with a fixed $\\eta$; whether the $O(1/k)$ rate holds under the fixed-step version is an open testable question.","Because the slack variable starts large and decays, the method can be started from infeasible points; this suggests a direct comparison with interior-point path-following, where the barrier parameter plays the same role and the path $\\hat{x}^*(c,s)$ is the central path."],"forward_implications":["With $\\beta_t = 2p\\log t$, the continuous system converges polynomially at rate $O(1/t^p)$, and with $\\beta_t = 2t^p$ it converges as $O(e^{-t^p})$, giving a tunable family of rates from the same Lyapunov argument.","Algorithm 1 inherits the continuous rate: with $A_k = (\\delta k)^{2p}$ it achieves $O(1/(\\delta k)^p)$, and with $A_k = e^{2\\delta k p}$ it achieves $O(e^{-\\delta k p})$.","Algorithm 2 achieves $O(1/k)$ convergence to $x^*$ for the smoothed problem, and the numerical experiments show it reaches a fixed accuracy with fewer iterations than the plain gradient scheme (Algorithm 3).","When all constraint functions are identically zero, the framework reduces to unconstrained accelerated methods, recovering the standard setting as a special case."],"supporting_citations":[{"why":"Supplies the log-barrier construction and the perturbation-sensitivity inequality used in Lemma 2.1.","marker":"[3]"},{"why":"Supplies the Bregman-Lagrangian variational framework from which the continuous-time system (3.4) is derived.","marker":"[23]"},{"why":"Supplies the Lyapunov analysis and the discretization perspective used to design the discrete-time algorithms.","marker":"[26]"},{"why":"Provides the dual-method treatment of constrained convex problems whose continuous-time exponential convergence the paper extends to inequality constraints.","marker":"[33]"},{"why":"Provides the continuous-time ODE modeling of acceleration that motivates the dynamical-system analysis.","marker":"[20]"},{"why":"Supplies Hamilton's principle used to derive the Euler-Lagrange equation for the Bregman Lagrangian.","marker":"[41]"},{"why":"Supplies the Lyapunov stability method used in Lemma 3.1 and Theorem 3.1.","marker":"[42]"},{"why":"Supplies the Euler discretization used to derive Algorithms 1, 2, and 3.","marker":"[43]"},{"why":"Supplies the quasi-monotone subgradient method to which Algorithm 3 reduces when the constraints vanish.","marker":"[44]"}],"fun_headline_variants":["Bregman flow exponentially hits constrained optima","Exponential convergence for constrained convex via barrier","Log-barrier dynamics race to optimum at e^{-βt}","Discrete O(1/k) algorithms from Bregman Lagrangian","Constrained convex solved by exponential Bregman flow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on being able to schedule the barrier parameter and slack as $c(t)=e^{\\beta t}$ and $s(t)=e^{-\\beta t}$ while the trajectory remains inside the open set $\\hat K$; if that scheduling cannot be maintained, the exponential tracking bound collapses.","fun_headline_variants_meta":{"raw":{"variants":["Bregman flow exponentially hits constrained optima","Exponential convergence for constrained convex via barrier","Log-barrier dynamics race to optimum at e^{-βt}","Discrete O(1/k) algorithms from Bregman Lagrangian","Constrained convex solved by exponential Bregman flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1567,"prompt_tokens":918,"completion_tokens":649,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":534,"tokens_out":649,"duration_ms":6325,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:51:32.222066+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically integrate (3.4) for the quadratic example (5.1) with $\\beta_t = 2t$ and measure $\\log\\|X_t - x^*\\|_2$; Theorem 3.1 predicts a slope approaching $-1$, meaning $\\|X_t - x^*\\|_2 \\sim e^{-t}$. A slope that flattens, or any iteration at which $s - g_i(X_t)$ reaches zero and the log-barrier gradient blows up, would falsify the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the log-barrier construction and the perturbation-sensitivity inequality used in Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bregman-Lagrangian variational framework from which the continuous-time system (3.4) is derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov analysis and the discretization perspective used to design the discrete-time algorithms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the dual-method treatment of constrained convex problems whose continuous-time exponential convergence the paper extends to inequality constraints."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the continuous-time ODE modeling of acceleration that motivates the dynamical-system analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Hamilton's principle used to derive the Euler-Lagrange equation for the Bregman Lagrangian."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov stability method used in Lemma 3.1 and Theorem 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Euler discretization used to derive Algorithms 1, 2, and 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-monotone subgradient method to which Algorithm 3 reduces when the constraints vanish."}],"review_version":1}