{"id":"af5e1eec-0e5c-49cd-82ce-a253c2088e6f","arxiv_id":"2504.20575","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ekeland and Borwein-Preiss variational principles are extended to distances that are neither symmetric nor triangular, using only sequence convergence defined by the distance.","lead":"Two classical optimization principles, Ekeland's and Borwein-Preiss's, are shown to hold even when the notion of distance is not symmetric and does not obey the triangle rule. The proofs use only sequence convergence and continuity defined through the distance itself, yielding a fixed point theorem and an equilibrium existence result.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Ekeland proof is invalid as written: S_i is defined without intersecting S_{i-1}, so the sequence is not decreasing and Lemma 2.7 does not apply; the radius in (4.1) is also wrong by a factor of ε.","rationale":"The reader's weakest_assumption points to Lemma 2.7, the relaxed Cantor intersection theorem. My review finds Lemma 2.7 itself sound: given a decreasing sequence of closed sets with Si ⊂ B(x_i, r_i) and r_i → 0, the proof of right Cauchyness, convergence, and singleton intersection is valid. The real load-bearing weakness is instead the construction that feeds Lemma 2.7. In Theorem 4.1 the displayed definition of S_i omits the intersection with S_{i−1}, so the decreasing hypothesis is not established; the subsequent estimate (4.1) is invalid without it, and the stated radius ε/2^i is a factor of ε off. The same class of strictness gap occurs in Theorem 3.1(c), where exclusion from the intersection does not imply the claimed strict inequality for all large j. These flaws are central because they affect the existence and uniqueness conclusions of both variational principles. However, the failures are structural and repairable: replacing S_i by S_{i−1} ∩ {·}, correcting the radius to 1/2^i, and handling the first exclusion index should restore the proofs. I therefore do not move the verdict from CONDITIONAL; the paper needs revision before acceptance, but the core results are plausible. This is why agreement_with_reader is partial: the reader located the ball-containment mechanism but attributed the risk to Lemma 2.7, whereas the actual insecure step is the induction that is supposed to produce nested sets and valid strict inequalities.","tokens_in":9724,"tokens_out":32965,"duration_ms":325500,"concrete_test":"Re-derive Theorem 4.1 using the corrected definition S_i := S_{i−1} ∩ {z : f(z)+εd(z,z_i) ≤ f(z_i)} and radii r_0 = 1, r_i = 1/2^i. Verify that (4.1) now follows from z ∈ S_{i−1} and that Lemma 2.7 yields a unique z̄ in the intersection. If this corrected induction goes through, the issue is a fixable definitional error; if not, the Ekeland half of the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 4.1, the proof defines S_i := {z ∈ X | f(z)+εd(z,z_i) ≤ f(z_i)} without intersecting with S_{i-1}. The proof immediately asserts that (S_i) is decreasing, but this is false: membership in S_i gives no reason for z to belong to S_{i-1}. Since Lemma 2.7 requires a decreasing sequence of closed sets, it cannot be applied. The estimate (4.1), εd(z,z_i) ≤ f(z_i) − inf_{S_{i-1}} f, is also unjustified because it relies on z ∈ S_{i-1}. The radius is another concrete error: from f(z_i) ≤ inf_{S_{i-1}} f + ε/2^i one obtains εd(z,z_i) ≤ ε/2^i, hence d(z,z_i) ≤ 1/2^i, not ≤ ε/2^i as stated; for ε < 1 the claimed ball containment Si ⊂ B(z_i, ε/2^i) can fail. A parallel gap appears in Theorem 3.1(c): from z ∉ ∩ S_i one cannot conclude that the strict inequality holds for all sufficiently large j; a point outside S0 has no j with z ∈ S_{j−1}, so the displayed strict inequality has no basis. These are not merely cosmetic typos: the central minimizer and uniqueness conclusions depend on the nested closed sets and on the strict limiting inequality. The results are likely repairable — e.g. set S_i := S_{i−1} ∩ {z : f(z)+εd(z,z_i) ≤ f(z_i)}, take radii 1/2^i, and treat the first exclusion index separately — but the paper as submitted does not yet prove the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops Borwein-Preiss and Ekeland variational principles for a distance function d on an arbitrary set X satisfying only d(x,y)=0 iff x=y (no symmetry, no triangle inequality). The main results (Theorems 3.1 and 4.1) assert that if d is right complete, right Hausdorff, and satisfies appropriate right-sequential lower semicontinuity conditions, then any bounded-below right-sequentially d-lower semicontinuous function can be perturbed by a weighted sum of d-terms (Borwein-Preiss) or by a single εd-term (Ekeland) so that the perturbed function attains a (unique) minimum. The proofs are based on a relaxed Cantor intersection theorem (Lemma 2.7). Two applications are presented: a Caristi fixed point theorem for set-valued maps and an existence theorem for equilibrium problems. The paper is self-contained and relies only on sequential convergence notions induced by d.","tokens_in":10007,"tokens_out":28005,"duration_ms":246222,"significance":"If correct, the framework is a genuine generalization: variational principles in this setting depend only on the sequential convergence behavior of a very general distance, without any topological or triangular structure. The paper is clearly written and the overall strategy—using a shrinking-ball containment in place of the triangle inequality—is elegant and potentially useful for non-symmetric distances such as divergences or non-triangular metrics. The applications to fixed point theory and equilibrium problems are natural and the arguments are mostly standard once the main principles are available. The main theorems are nevertheless not proved correctly as written: the Ekeland construction in Theorem 4.1 does not produce a nested family of closed sets, the radius estimate there has the wrong factor, and the strict-inequality steps in both Theorem 3.1(c) and Theorem 4.1(c) are not justified. These are load-bearing gaps, but they appear repairable with the modifications described in the major comments.","major_comments":[{"comment":"The sets S_i are defined as S_i := {z ∈ X | f(z)+εd(z,z_i) ≤ f(z_i)} without intersecting S_{i-1}. Consequently the sequence (S_i) is not necessarily decreasing, so Lemma 2.7 cannot be applied. This is a load-bearing error. The correct construction is S_i := S_{i-1} ∩ {z ∈ X | f(z)+εd(z,z_i) ≤ f(z_i)}, which also makes the later estimate (4.1) meaningful because z ∈ S_{i-1} is then available.","section":"Section 4, proof of Theorem 4.1"},{"comment":"Even with a nested definition, the estimate εd(z,z_i) ≤ ε/2^i yields d(z,z_i) ≤ 1/2^i, not d(z,z_i) ≤ ε/2^i. The radii r_i must therefore be 1/2^i (or larger), not ε/2^i. In addition, the initial choice \"f(z1) ≤ inf_{z∈S0} f(z) + δ0/2\" uses an undefined δ0; presumably it should be ε/2. These defects affect the applicability of Lemma 2.7 and the bound d(z,z0) ≤ 1.","section":"Section 4, Eq. (4.1) and radius choice"},{"comment":"The proof claims that z ∉ ∩S_i implies the displayed strict inequality holds for sufficiently large j. This is not correct: if j is the first index with z ∉ S_j, then the strict inequality follows from z ∈ S_{j-1} and z ∉ S_j, but for later j the point z is not in S_{j-1} either, so the membership condition gives no inequality. A correct argument must use a different device, for example the monotonicity of A_j := f(z_j)+Σ_{k=0}^{j-1}δ_k d(z_j,z_k) to show that if the desired strict inequality failed for some z, then z would belong to every S_j, contradicting z ≠ zbar.","section":"Section 3, proof of Theorem 3.1(c)"},{"comment":"The final strict-inequality step is also invalid as written. After fixing the first index j with z ∉ S_j, the proof lets j → ∞ in f(z)+εd(z,z_j) > f(z_j). Since that index is fixed, the limit passage is illegitimate. A correct proof should argue by contradiction: if f(z)+εd(z,zbar) < f(zbar), then because f(z_j) is nonincreasing and bounded below and d(z,z_j) → d(z,zbar) by right sequential d-continuity in the second argument, one obtains z ∈ S_j for all sufficiently large j, forcing z = zbar.","section":"Section 4, proof of Theorem 4.1(c)"}],"minor_comments":[{"comment":"The continuity assumptions in Theorem 4.2 appear to be swapped relative to Theorem 4.1 (right sequentially d-continuous in the first argument and right sequentially d-lower semicontinuous in the second). Since Theorem 4.2 is used in Sections 5 and 6, the statement should be corrected to match Theorem 4.1, or a proof should be supplied under the stated hypotheses.","section":"Theorem 4.2"},{"comment":"The parameter ε in the inequality F(x_i,y) ≥ −εd(y,x_i) should be ε_i, and the limit passage should explicitly invoke the right sequential d-continuity of d(y,·) to justify ε_i d(y,x_{i_k}) → 0.","section":"Section 6, Eq. (6.3)"},{"comment":"The weak forms are stated without proofs. The authors should indicate how they follow from the main theorems, for instance by choosing z0 with f(z0) < inf f + ε.","section":"Theorems 3.2 and 4.2"},{"comment":"The statements of parts (c) use the same symbol z for the arbitrary point and for the limiting point, which makes the inequalities very hard to read. Please use distinct symbols, e.g., z and z*.","section":"Notation in Theorems 3.1(c) and 4.1(c)"},{"comment":"Lemma 2.7 assumes containment S_i ⊂ B(x_i,r_i) with open balls, while the applications establish non-strict inequalities d(z,x_i) ≤ r_i. This is harmless but should be aligned, e.g., by using closed balls or slightly enlarged radii.","section":"Lemma 2.7"},{"comment":"There are numerous typographical errors (e.g., \"Borwen-Preiss\", \"non-trinangular\", \"Eventough\", \"generaization\"); the manuscript needs a careful proofreading pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's central idea—that variational principles can be obtained from purely sequential assumptions without symmetry or triangle inequality—is sound and potentially valuable. The flaws identified are in the execution of the proofs, not in the overall strategy, and they appear repairable with the constructions suggested in the major comments. I therefore recommend major revision rather than rejection. The authors should be asked to rework the proofs of Theorems 3.1(c) and 4.1, correct the radius estimates, and clarify the status of the weak forms."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper does something real: it proves Ekeland and Borwein-Preiss variational principles for distance functions that are neither symmetric nor triangular, and it does so using only sequential convergence and continuity notions plus a relaxed Cantor intersection lemma. The big idea is sound, and the paper is honest about what the literature does and does not cover. The main theorems are new in this setting.\n\nThe Borwein-Preiss part (Theorem 3.1) is mostly fine. The inductive construction of the nested sets S_i, the containment in balls, and the convergence argument all work. The weak form and the applications to Caristi and equilibrium problems are reasonable extensions. The paper is self-contained and there is no circularity or parameter fitting. For that I give it real credit.\n\nNow the soft spots, in proportion to how soft they are. The stress-test note is right, not just pedantic. In Theorem 4.1 the sets S_i are defined without intersecting with S_{i-1}, so they do not form a decreasing sequence and Lemma 2.7 cannot be applied. This is a genuine flaw in the proof as written, though it is plainly repairable: define S_i := S_{i-1} ∩ {z : f(z)+εd(z,z_i) ≤ f(z_i)} and the induction works. The radius is also wrong by a factor of ε: the estimate gives d(z,z_i) ≤ 2^{-i}, not ε 2^{-i}. And the choice of z_1 uses an undefined δ_0. These are typos, but they sit in the proof of the central theorem.\n\nThe strict inequality in Theorem 3.1(c) has a genuine gap too. From z ∉ ∩S_i you only know z fails the inequality at the first exclusion index; if z fails already at S_0, the later set definitions give you no strict comparison. The uniqueness conclusion needs a separate argument. That is a fixable but nontrivial gap.\n\nEverything else is minor: Theorem 5.1's inheritance of the continuity assumptions to the product distance ρ is asserted rather than shown, and the paper has frequent small typos. None of this makes the framework dubious.\n\nBottom line: this deserves a serious referee, not a desk reject. I would send it to review with a request to repair the Ekeland induction, correct the radii, and tighten the strict-inequality step. As submitted I would not cite it, but after those fixes I probably would.","headline":"A genuine and likely repairable extension of two classical variational principles to non-symmetric, non-triangular distances, but the submitted proofs have a real gap in the Ekeland induction and a soft spot in the Borwein-Preiss uniqueness argument.","tokens_in":10620,"tokens_out":5138,"would_cite":false,"duration_ms":52806,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Ekeland and Borwein–Preiss variational principles hold for distance functions that are neither symmetric nor satisfy the triangle inequality, using only sequential convergence structure.","keywords":["Borwein-Preiss variational principle","Ekeland variational principle","non-symmetric distance","non-triangular distance","relaxed Cantor intersection theorem","Caristi fixed point theorem","equilibrium problem","sequential lower semicontinuity"],"falsifier":"Test Lemma 2.7 on the family of asymmetric costs $d(x,y)=|x-y|^p$ for $x\\le y$ and $d(x,y)=2|x-y|^p$ for $x>y$ on $X=[0,1]$, with $p\\in(0,1)$. If for some $p$ there are decreasing right $d$-closed sets $S_i\\subset B(x_i,r_i)$, $r_i\\to0$, whose centers $(x_i)$ are not right $d$-Cauchy, then Lemma 2.7 fails and the variational principles no longer follow.","tokens_in":9437,"feed_emoji":"🎯","tokens_out":15089,"duration_ms":143824,"temperature":0.7,"pith_summary":"The paper proves that Ekeland's variational principle and the Borwein–Preiss variational principle still hold when the distance function $d:X\\times X\\to[0,\\infty)$ is neither symmetric nor satisfies the triangle inequality. The hypotheses are purely sequential: $d$ must be right complete, right Hausdorff, and separately right sequentially $d$-lower semicontinuous, and the function $f$ must be right sequentially $d$-lower semicontinuous and bounded below. The conclusion is that a small weighted sum of $d$-terms (Borwein–Preiss) or a single $\\varepsilon d$-term (Ekeland) perturbs $f$ so that the perturbed function attains a minimum, with explicit bounds on how far the minimizer moves. If true, these variational principles depend on convergence behavior alone, not on a metric topology.","feed_headline":"Variational principles survive without symmetry or triangle inequality","feed_subtitle":"Ekeland and Borwein–Preiss minimum theorems are proved using only sequential convergence of an asymmetric distance.","key_machinery":"The load-bearing mechanism is Lemma 2.7, a relaxed Cantor intersection theorem for a non-symmetric, non-triangular distance: a decreasing sequence of nonempty right $d$-closed sets $S_i$, each contained in a right $d$-ball $B(z_i,r_i)$ with $r_i\\to 0$, has exactly one point in common when $d$ is right complete and right Hausdorff. This lemma replaces the triangle inequality by a shrinking-ball containment property. In both variational principles, the inductive construction chooses $z_i$ as an approximate minimizer of the current perturbed function; the improvement bound forces the next sublevel set $S_i$ to fit inside $B(z_i,r_i)$ with $r_i=\\varepsilon/(2^i\\delta_0)$ (Borwein–Preiss) or $r_i=\\varepsilon/2^i$ (Ekeland), so the centers form a right $d$-Cauchy sequence and converge to the desired minimizer.","core_discovery":"The central claim is Theorem 3.1 and Theorem 4.1. For a distance function $d$ with $d(x,y)=0$ iff $x=y$, which is right complete, right Hausdorff, and separately right sequentially $d$-lower semicontinuous, any right sequentially $d$-lower semicontinuous $f$ bounded below is perturbable to a minimizer. The Borwein–Preiss version produces a right $d$-convergent sequence $(z_i)$ and a limit $\\bar z$ with $d(\\bar z,z_i)\\le \\varepsilon/(2^i\\delta_0)$, $f(\\bar z)+\\sum_{k=0}^\\infty \\delta_k d(\\bar z,z_k)\\le f(z_0)$, and $\\bar z$ the unique minimizer of the perturbed function. The Ekeland version finds $\\bar z$ with $d(\\bar z,z_0)\\le 1$, $f(\\bar z)+\\varepsilon d(\\bar z,z_0)\\le f(z_0)$, and $f(\\bar z)\\le f(z)+\\varepsilon d(z,\\bar z)$ for every $z$. Closedness, compactness, and semicontinuity are all defined sequentially through $d$, so no metric topology is invoked.","pith_inferences":["If the proofs are correct, the triangle inequality is not essential to the geometry of variational principles; what matters is that approximate-minimization steps produce nested sets that shrink inside $d$-balls. This points toward applying the same template directly to information divergences and Wasserstein-type costs with $p<1$, without symmetrizing the cost.","The one-sided setup suggests a testable extension: replacing the right sequentially $d$-continuous second argument in Theorem 4.1 by a one-sided lower semicontinuity condition, which would tighten the continuity hypotheses further.","The final symmetrization remark implies that whenever a problem genuinely requires two-sided convergence, one can recover the usual topology by $\\tilde d(x,y)=d(x,y)+d(y,x)$; the paper's contribution is that one-sided optimization needs only one-sided structure, which may matter for algorithms built on asymmetric divergences.","A natural next step is to check whether the relaxed Cantor lemma also yields Takahashi's variational principle or vector-valued Ekeland statements in this non-triangular setting, since the same shrinking-set mechanism drives those results in metric spaces."],"forward_implications":["Any bounded-below right sequentially $d$-lower semicontinuous function can be turned into a function with a unique global minimizer by adding a weighted sum $\\sum_{k=0}^\\infty \\delta_k d(\\cdot,z_k)$, with the explicit estimate $d(\\bar z,z_i)\\le \\varepsilon/(2^i\\delta_0)$.","For any $\\varepsilon>0$ and approximate minimizer $z_0$, the single perturbation $f+\\varepsilon d(\\cdot,z_0)$ has a minimizer within right $d$-distance $1$ of $z_0$; every near-minimum therefore has an exact nearby perturbed minimum.","The Ekeland form yields a Caristi fixed point theorem for set-valued maps satisfying $\\varphi(y)\\le \\varphi(x)-d(x,y)$ when the graph is closed in the swapped product distance $\\rho((x,y),(x',y'))=d(x',x)+d(y',y)$; the fixed points are also endpoints.","On a right sequentially $d$-compact space, a bifunction with a lower estimate $F(x,y)\\ge \\varphi(y)-\\varphi(x)$ has an equilibrium point $x$ with $F(x,y)\\ge 0$ for all $y$.","Both principles have weak forms: a sequence $(z_i)$ right $d$-converges to $\\bar z$ such that the perturbed function has $\\bar z$ as a unique minimizer (Borwein–Preiss) or an exact minimizer (Ekeland)."],"supporting_citations":[{"why":"the original Ekeland variational principle that this paper generalizes and whose statement template it follows.","marker":"[11]"},{"why":"the original Borwein–Preiss smooth variational principle; supplies the weighted-sum perturbation and the induction idea.","marker":"[2]"},{"why":"prior generalization of Ekeland and Borwein–Preiss principles using gauge-type functions, the closest reference construction the paper adapts.","marker":"[20]"},{"why":"Ekeland's principle in quasi-metric spaces, the benchmark non-symmetric setting that still assumes a triangle-type inequality.","marker":"[7]"},{"why":"a generalized variational principle whose perturbation kernel need not be a norm, showing the kernel hypothesis can be relaxed.","marker":"[21]"},{"why":"the Caristi fixed point theorem that the paper derives from its Ekeland principle.","marker":"[5]"},{"why":"the minimax inequality that underlies the equilibrium problem application.","marker":"[13]"}],"fun_headline_variants":["Asymmetric distances: Ekeland and Borwein-Preiss still work","No symmetry, no triangle: variational principles survive","Ekeland and Borwein-Preiss need only asymmetric distances","Weakening distance axioms: Ekeland and Borwein-Preiss work"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Lemma 2.7: at every step the approximate-minimization inequality must enclose the next closed set in a right $d$-ball whose radius tends to zero, and the space must be right $d$-complete so that the nested intersection is a single point; if the radii do not shrink fast enough, or completeness fails, the proof's entire mechanism collapses.","fun_headline_variants_meta":{"raw":{"variants":["Asymmetric distances: Ekeland and Borwein-Preiss still work","No symmetry, no triangle: variational principles survive","Ekeland and Borwein-Preiss need only asymmetric distances","Weakening distance axioms: Ekeland and Borwein-Preiss work"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001485,"raw_usage":{"total_tokens":5919,"prompt_tokens":852,"completion_tokens":5067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":4992}},"tokens_in":468,"tokens_out":5067,"duration_ms":36246,"temperature":1.0,"reasoning_tokens":4992,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:27:11.917715+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 2.7 on the family of asymmetric costs $d(x,y)=|x-y|^p$ for $x\\le y$ and $d(x,y)=2|x-y|^p$ for $x>y$ on $X=[0,1]$, with $p\\in(0,1)$. If for some $p$ there are decreasing right $d$-closed sets $S_i\\subset B(x_i,r_i)$, $r_i\\to0$, whose centers $(x_i)$ are not right $d$-Cauchy, then Lemma 2.7 fails and the variational principles no longer follow.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the original Borwein–Preiss smooth variational principle; supplies the weighted-sum perturbation and the induction idea."},{"cited_title":"Li and S","cited_arxiv_id":null,"evidence_quote":"prior generalization of Ekeland and Borwein–Preiss principles using gauge-type functions, the closest reference construction the paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the Caristi fixed point theorem that the paper derives from its Ekeland principle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the minimax inequality that underlies the equilibrium problem application."}],"review_version":1}