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Variational principles using a non-symmetric non-triangular distance

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.20575 v1 pith:T33QKNGF submitted 2025-04-29 math.FA math.OC

Natthaya Boonyam , Parin Chaipunya , Poom Kumam This is my paper · ORCID
classification math.FAmath.OC
keywords Borwein-PreissvariationalprincipleEkelandnon-symmetricdistancenon-triangularrelaxedCantorintersectiontheoremCaristifixedpointequilibriumproblemsequentiallowersemicontinuity
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
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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

4 major / 6 minor

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.

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 (4)
  1. [Section 4, proof of Theorem 4.1] 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.
  2. [Section 4, Eq. (4.1) and radius choice] 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.
  3. [Section 3, proof of Theorem 3.1(c)] 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.
  4. [Section 4, proof of Theorem 4.1(c)] 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.
minor comments (6)
  1. [Theorem 4.2] 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.
  2. [Section 6, Eq. (6.3)] 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.
  3. [Theorems 3.2 and 4.2] 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.
  4. [Notation in Theorems 3.1(c) and 4.1(c)] 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*.
  5. [Lemma 2.7] 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.
  6. [Throughout] There are numerous typographical errors (e.g., "Borwen-Preiss", "non-trinangular", "Eventough", "generaization"); the manuscript needs a careful proofreading pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variational principles are derived from explicit completeness and semicontinuity assumptions, with no fitted input or load-bearing self-citation.

full rationale

The derivation chain is self-contained and assumption-based. Theorems 3.1 and 4.1 are proved from explicit hypotheses (right completeness, right Hausdorff property, right sequential d-lower semicontinuity, and appropriate continuity of d), with the argument controlled by Lemma 2.7, a relaxed Cantor intersection theorem proved directly from those hypotheses. The constructed sublevel sets S_i are defined by the perturbation inequalities, and the ball containments used to apply Lemma 2.7 are obtained from approximate-minimization increments, not from the desired conclusion. No parameter is fitted to data, no target result is assumed in the proof, and the conclusion that the perturbed function has a minimizer is not used as an input. The only self-citation, reference [6], appears in the introduction as contextual discussion of prior generalized-distance Ekeland principles and is not load-bearing in any proof. Concerns that the Ekeland proof may contain a technical gap (e.g., non-decreasing sets or incorrect radius) are matters of correctness, not circularity, and do not change the circularity score. Overall, no circular step can be exhibited by quoting the paper; the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The paper's contribution is purely structural: it replaces metric axioms (symmetry, triangle inequality) with sequential convergence assumptions. The only implicit additions are standard choice and the explicitly stated completeness/continuity hypotheses.

assumptions (5)
  • standard math Axiom of dependent choice is used to select approximate minimizers z_i in the inductive constructions (Theorems 3.1 and 4.1).
    The proofs choose z_i at each step satisfying an approximate infimum inequality; such a choice principle is standard but unstated.
  • domain assumption d is a distance function with d(x,y)=0 iff x=y (Definition 2.1), and each theorem imposes explicit right-completeness, right-Hausdorff, and sequential lower semicontinuity conditions.
    These assumptions are stated in each theorem; they are the framework and not derived.
  • standard math Lemma 2.6: sublevel sets of right sequentially d-lower semicontinuous functions are right sequentially d-closed.
    Proved in the paper by a direct liminf argument; no background result.
  • domain assumption Lemma 2.7: a decreasing sequence of right d-closed sets, each contained in a shrinking right d-ball, has exactly one common point under right completeness and right Hausdorff.
    The relaxed Cantor intersection theorem is the central technical tool; its proof is given but is not reducible to the classical Cantor theorem because of the missing triangle inequality.
  • domain assumption In Theorem 5.1, the product distance ρ((x,y),(x',y'))=d(x',x)+d(y',y) is asserted to satisfy the hypotheses of Theorem 4.2.
    The authors state that this follows from the assumptions on d but do not provide a detailed verification; the inheritance of right sequential continuity and completeness is plausible but not spelled out.

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Pith. "Pith review of Variational principles using a non-symmetric non-triangular distance." pith.science (2026). https://pith.science/paper/T33QKNGF

@misc{pith2026250420575,
  author       = {Pith},
  title        = {Pith review of: Variational principles using a non-symmetric non-triangular distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T33QKNGF}},
  note         = {Machine review of arXiv:2504.20575}
}
read the original abstract

We consider Borwein-Preiss and Ekeland variational principles using distance functions that neither is symmetric nor enjoy the triangular inequality. All the given results rely exclusively on the convergence and continuity behaviors induced synthetically by the distance function itself without any topological implications. At the end of the paper, we also present two applications; the Caristi fixed point theorem and an existence theorem for equilibrium problems.

Discussion (0). Continue with ORCID to comment.

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