{"id":"9c98d379-e2de-477e-9e34-bbf3bb7d2671","arxiv_id":"2506.00638","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A boundary point is an epsilon-optimum of a reverse convex program exactly when every eps-prime subgradient of the constraint belongs to the union of eps-subdifferentials of scaled objectives.","lead":"This paper proves a necessary and sufficient condition for when a point on the boundary of a reverse convex constraint set is an approximate (epsilon) global optimum, using eps-subdifferentials. It extends prior exact-optimality characterizations to approximate solutions and covers equality constraints as a special case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 is correct as stated, but it characterizes only boundary points with h(bar x)=0 and a strict essential inequality; the abstract's claim to characterize all epsilon-optima overreaches.","rationale":"I checked the proof of Theorem 3 line by line and found it internally consistent. Lemma 1's two implications, Theorem 2's epsilon-efficiency criterion, and the scalarization step via Theorem 1 all cohere. The strict essential inequality is genuinely needed only for necessity (to force both scalarization weights positive), and h(bar x)=0 is genuinely needed for sufficiency. The reader's weakest assumption points to the real load-bearing issue: the abstract and introduction claim a characterization of approximate global optimal solutions, while the theorem establishes a criterion only for boundary solutions of a restricted 'nontrivial' type. The example f(x)=x^2, h(x)=x-1, epsilon=0.5 shows an interior epsilon-optimum that is not an unconstrained epsilon-minimizer and is not addressed by any result in the paper. This is not an internal inconsistency of the theorem, but it is a substantive mismatch between the paper's advertised contribution and its proved content. Consequently, the verdict should remain CONDITIONAL: the main theorem can stand, but the scope claims must be revised or the missing interior case must be treated.","tokens_in":12063,"tokens_out":23681,"duration_ms":215022,"concrete_test":"For the one-dimensional program min f(x)=x^2 subject to h(x)=x-1>=0 with epsilon=0.5, compute the exact epsilon-argmin set [1,sqrt(1.5)] and take bar x=1.1. Verify that bar x is an epsilon-optimum, that h(bar x)>0, and that no theorem in the paper (Theorem 3, Corollary 2, or Corollary 3) applies to certify it. If the abstract is revised to claim only a boundary characterization of nontrivial epsilon-optima, the concern is resolved; otherwise the paper should supply an additional treatment of interior epsilon-optima.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The advertised central result is that the paper characterizes approximate global optimal solutions (epsilon-optima) of reverse convex programs. What Theorem 3 actually proves is an if-and-only-if criterion for points satisfying h(bar x)=0 and inf_X f < f(bar x)-epsilon. The boundary condition is used in the sufficiency direction through Lemma 1, and the strict essential inequality is used in the necessity proof to rule out the degenerate scalarization cases lambda1=0 and lambda2=0 (Section 4, proof of Theorem 3). For epsilon>0, epsilon-argmin sets of reverse convex programs can contain interior feasible points. Consider f(x)=x^2, h(x)=x-1, epsilon=0.5: the feasible set is x>=1, inf=1, and the epsilon-argmin is [1,sqrt(1.5)]. The point bar x=1.1 is an epsilon-optimum with h(bar x)>0, and it is not an unconstrained epsilon-minimizer because inf_X f=0 is not >= f(1.1)-0.5=0.71. Neither Theorem 3 nor Corollary 2 gives any criterion for this point: Theorem 3 requires h=0, Corollary 2 characterizes only the h=0 subset of the epsilon-argmin, and the essential inequality in Theorem 3, while satisfied at 1.1, cannot compensate for the missing boundary condition. Thus the abstract's universal characterization claim is not supported by the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops ε-optimality conditions for reverse convex programs, i.e., problems of the form min f(x) subject to h(x) ≥ 0, where f and h are proper convex extended-valued functions on a real topological vector space. The method converts (ROP) into the unconstrained bicriteria difference program (f, 0) − (0, h) and applies the first author's earlier theory of ε-efficiency in difference vector optimization, together with the paper's Theorem 2, which extends the weak-efficiency criterion of [5] to the Pareto (efficient) ε-efficiency case. Theorem 3, the central result, states that for a feasible point \\bar{x} with h(\\bar{x}) = 0 and a strict essential inequality inf_X f < f(\\bar{x}) − ε, the point is an ε-minimizer of (ROP) if and only if every ε′-subgradient of h at \\bar{x} lies in the union over α > 0 of the (αε + ε′)-subdifferentials of αf. Setting ε = 0 recovers Hiriart-Urruty's exact characterization. Theorem 4 extends the criterion to additional convex constraints under Moreau–Rockafellar or Attouch–Brésis qualification conditions, and Corollaries 1–3 treat the nonlinear equality-constrained case. I examined the proof of Theorem 3 in detail and found it internally consistent.","tokens_in":12362,"tokens_out":52458,"duration_ms":429509,"significance":"If the results stand, this is a genuine contribution: to my knowledge it is the first approximate-optimality characterization for reverse convex programs, recovering and extending the exact-characterization literature [10, 14, 15]. The extended-value framework and the explicit use of Moreau–Rockafellar and Attouch–Brésis conditions are real improvements in scope, and the ε = 0 case is correctly reduced to Hiriart-Urruty's condition (Remark 1(d)). The proof of Theorem 3 is checkable: the scalarization steps, the case analysis excluding the degenerate multipliers λ₁ = 0 and λ₂ = 0, and the role of the strict essential inequality are all sound. The paper is also explicit about several limitations (Remarks 1(a)–(d) and 3(b)). The main weakness is that the abstract and Section 4 claim a characterization of all approximate global optimal solutions, whereas Theorem 3 only characterizes the boundary slice h(\\bar{x}) = 0 of the ε-argmin; for ε > 0, interior ε-optima exist and are left uncharacterized. This is a fixable framing issue rather than an error in the theorems.","major_comments":[{"comment":"The advertised scope exceeds what is proved. The abstract promises a characterization of approximate global optimal solutions of reverse programs, and Section 4 states that Theorem 3 completely characterizes the (nontrivial) ε-optimal solutions, but Theorem 3 requires h(\\bar{x}) = 0, which is essential in the sufficiency direction through the converse of Lemma 1; that converse is false for h(\\bar{x}) > 0 (example: f(x) = x², h(x) = x, ε = 1, \\bar{x} = 2 lies in E^e_{(1,0)}(f, −h) but is not in the ε-argmin of f over h ≥ 0). For ε > 0 the ε-argmin of a reverse program can contain interior points: with f(x) = x², h(x) = x − 1, ε = 0.5, the ε-argmin over h ≥ 0 is [1, √1.5], and \\bar{x} = 1.1 satisfies inf_X f = 0 < f(1.1) − 0.5 = 0.71 and is not an unconstrained ε-minimizer, yet neither Theorem 3 nor Corollary 2 gives any condition for it (Corollary 2 explicitly covers only the boundary slice ε-argmin_{h=0} f = ε-argmin_{h≥0} f ∩ {h = 0}). Remark 1(a) notes that h(\\bar{x}) = 0 is used only for sufficiency, but the paper does not draw the consequence: for ε > 0 the characterization covers only part of the ε-argmin (for ε = 0 the boundary restriction is harmless by Lemma 3). Please narrow the abstract and the Section 4 claim to the boundary part of the ε-argmin and add a remark presenting an interior ε-optimum not covered by the theorem.","section":"Abstract; Section 4 (Theorem 3, Lemma 1)"},{"comment":"The proofs of both directions of Theorem 3 invoke Theorem 1, quoted from [7], to pass from vector ε-subdifferential membership such as (0, x*) ∈ ∂^w_{(ε, ε′)}(f, 0)(\\bar{x}) or z* ∈ ∂^p_{ε+ε′}(f, 0)(\\bar{x}) to scalar ε-subdifferential conditions with a multiplier λ ∈ R²₊ \\ {0}. Theorem 3 is stated for an arbitrary real topological vector space X, with no local-convexity or lower-semicontinuity assumptions, but scalarization theorems of this type are usually proved via separation of convex sets, and the hypotheses under which [7, Theorem 1] holds are not reproduced in the paper. Please state the exact hypotheses of [7, Theorem 1], and if local convexity of X or additional regularity of f is required there, add the corresponding assumption to Theorem 3 and to Corollaries 1 and 3, whose proofs inherit the same step. This is a verification request rather than a claim of a definite error: I found no counterexample to Theorem 3, but the generality of its statement currently exceeds what can be checked from the manuscript and its cited sources.","section":"Section 4, proof of Theorem 3"}],"minor_comments":[{"comment":"The symbol ⊓ is used for set intersections ('S ⊓ dom F', 'Y₊ ⊓ −Y₊') without being defined; please define it or use ∩ throughout for readability.","section":"Section 2"},{"comment":"The paper calls h 'nonconcave' in the introduction, while the hypotheses and applications require h to be convex (with (H′) typically holding for convex l.s.c. functions, as noted in Remark 1(c)); please use 'convex' consistently to avoid ambiguity.","section":"Sections 1 and 4"},{"comment":"In the λ₁ = 0 case the clause '∂_{ε′}h(\\bar{x}) ⊇ ∂h(\\bar{x}) ≠ ∅ (by (H′))' combines two facts: (H′) applied with ε′ = 0 gives ∂h(\\bar{x}) ≠ ∅, and the inclusion ∂h(\\bar{x}) ⊆ ∂_{ε′}h(\\bar{x}) holds because ε′ ≥ 0; please spell these out separately.","section":"Section 4, proof of Theorem 3"},{"comment":"The weak-efficiency half of Theorem 2 is taken from [5, Theorem 2] and carries the necessity direction of Theorem 3; please state the exact hypotheses of [5, Theorem 2] (space assumptions, pointedness of the ordering cone, convexity of F) so the reader can verify that G = (0, h) under (H′) satisfies them.","section":"Section 3, Theorem 2"},{"comment":"The reverse implication of Corollary 1 is handled with 'by the same arguments as for Theorem 4'; since it relies on the qualification-free inclusion in (11) (Remark 2), a few explicit lines would make the proof easier to verify.","section":"Section 5.2, proof of Corollary 1"},{"comment":"Several displays are garbled in the submitted version (e.g., 'X − →R ∪ {+∞}', 'Min f (x)( h(x) ≥ 0', '∀ϵ′ ≥R2+ 0'); clean typesetting is needed before publication.","section":"Whole paper"},{"comment":"Consider adding to Remark 1(a) the sharpness example f(x) = x², h(x) = x, ε = 1, \\bar{x} = 2, which shows that the converse of Lemma 1 genuinely requires h(\\bar{x}) = 0.","section":"Remark 1(a)"},{"comment":"The proper-efficiency sets E^p_ε are defined but never used in the sequel; a sentence indicating they are included for completeness would help the reader.","section":"Section 2"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the gap between the advertised scope (abstract, Section 4) and the boundary-only statement of Theorem 3; if the authors narrow the claims and add the suggested examples, the paper is, in my view, publishable. I could not consult [5] and [7] directly; please ask the authors to reproduce the exact hypotheses of [5, Theorem 2] and [7, Theorem 1], since the proofs of Theorem 3 rest on them. The paper draws substantially on the first author's earlier work ([5]–[7]); the genuinely new content appears to be the efficient (σ = e) half of Theorem 2, the application to reverse convex programs, and the qualification-condition-based extensions, and the introduction should say so explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: Theorem 3 is correct, and the epsilon-extension is genuinely new, but the abstract promises more than the theorem delivers. It characterizes only epsilon-optima on the boundary h(bar x)=0 that satisfy a strict essential inequality, not all approximate global optima. The math underneath is solid; the presentation needs an honest scope fix.\n\nWhat is new and good: This is the first epsilon-optimality characterization for reverse convex programs. Prior exact conditions by Hiriart-Urruty, Strekalovsky, and Tuy are recovered when epsilon=0. The proof strategy is a natural bicriteria reduction, and the equality-constraint corollaries (Corollaries 1-3) appear to be new. I checked the proof of Theorem 3 carefully: the scalarization steps are consistent, and the essential inequality is exactly what rules out degenerate multiplier cases. The epsilon=0 case reduces properly to Hiriart-Urruty. The paper also carefully handles extended-value functions with Moreau-Rockafellar and Attouch-Brezis qualification conditions, which is a real plus.\n\nSoft spots: The abstract and introduction say the paper characterizes approximate global optimal solutions without qualification. Theorem 3 requires h(bar x)=0. For epsilon>0, epsilon-argmins of reverse convex programs can contain interior feasible points; the paper's own Corollary 2 only characterizes the boundary intersection, not the full epsilon-argmin. The stress-test example (f(x)=x^2, h(x)=x-1, epsilon=0.5) is correct: bar x=1.1 is an interior epsilon-optimum, and neither Theorem 3 nor Corollary 2 gives a criterion for it. This is a scope problem, not a proof error. Also, the weak-efficiency half of Theorem 2 is imported from El Maghri [5], so the main engine is prior work plus scalarization; that is not circular, but readers should know it. Minor: Section 4 calls h \"nonconcave\" when the assumptions make it convex.\n\nWho this is for: Researchers working on global optimality conditions for nonconvex programs, especially those who care about epsilon-optimality because numerical global solvers produce approximate solutions. The paper is a legitimate theoretical contribution and deserves a serious referee. My recommendation: send to peer review, but require the authors to revise the abstract and introduction to state explicitly that the characterization covers boundary epsilon-optima, or add a separate treatment of interior approximate optima.","headline":"The epsilon-optimality theorem is correct and genuinely new, but the abstract overstates its scope: it only characterizes boundary points with h(x)=0, not all approximate global optima.","tokens_in":12876,"tokens_out":6868,"would_cite":true,"duration_ms":56726,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C26","90C29","49J52"],"pacs":[],"model":"deepseek-v4-flash","headline":"A point with h(x̄)=0 is an ε-optimal solution of a reverse convex program exactly when every ε′-subgradient of h at x̄ lies in an (αε+ε′)-subgradient of some positive scaling of f.","keywords":["Reverse convex optimization","ε-Optimality","DC vector optimization","ε-Efficiency","ε-Subdifferentials","approximate global optimality","nonlinear equality constraint"],"falsifier":"Compute with $f(x)=x^2+x$, $h(x)=x$, and $\\varepsilon=0.2$ on $X=\\mathbb{R}$. At the boundary point $\\bar{x}=0$ the essential inequality holds ($\\inf_X f=-0.25 < f(0)-0.2=-0.2$), yet $x=0.1$ is feasible with $f(0.1)=0.11 \\le \\inf_{x\\ge0}(x^2+x)+0.2=0.2$, so $x=0.1$ is an $\\varepsilon$-optimal solution despite $h(0.1)>0$. This shows that no condition confined to the boundary $h(\\bar{x})=0$ can characterize all approximate optima of a reverse convex program; any proposed full characterization must account for interior $\\varepsilon$-optima, and this example tests whether it does.","tokens_in":11846,"feed_emoji":"🎯","tokens_out":19982,"duration_ms":212518,"temperature":0.7,"pith_summary":"Reverse convex programs minimize a convex function over a nonconvex feasible region of the form $h(x) \\ge 0$, where $h$ is itself convex. The paper establishes the first characterization of approximate ($\\varepsilon$-optimal) global solutions for this class of problems. At any boundary point $\\bar{x}$ with $h(\\bar{x})=0$, the point is an $\\varepsilon$-optimum precisely when every $\\varepsilon'$-subgradient of $h$ at $\\bar{x}$ is also an $(\\alpha\\varepsilon+\\varepsilon')$-subgradient of some scaled objective $\\alpha f$, for some $\\alpha>0$. The proof converts the reverse program into an unconstrained bicriteria difference-of-convex (DC) problem and applies an $\\varepsilon$-efficiency criterion for such problems; at $\\varepsilon=0$ the condition reduces to the known exact global-optimality criterion. The same route yields new conditions for reverse programs with additional convex constraints and for nonlinear equality constraints.","feed_headline":"Subdifferential inclusion settles ε-optimality","feed_subtitle":"For reverse convex programs, a boundary point is ε-optimal exactly when ε′-subgradients of h belong to scaled ε-subgradients of f.","key_machinery":"The load-bearing object is the $\\varepsilon$-subdifferential of an extended convex function, $\\partial_\\varepsilon \\varphi(\\bar{x}) = \\{x^* : \\varphi(x) \\ge \\varphi(\\bar{x}) + \\langle x^*, x-\\bar{x}\\rangle - \\varepsilon \\text{ for all } x\\}$. The paper embeds the reverse program in the bicriteria DC program $(f,0)-(0,h)=(f,-h)$ and shows, in Lemma 1, that $\\varepsilon$-optimality of the scalar problem is equivalent to $\\varepsilon$-efficiency of this vector problem, with the boundary condition used for the reverse direction. The decisive transfer theorem states that for difference vector programs, $\\varepsilon$-efficiency of $\\bar{x}$ is equivalent to the inclusion of every strong $\\varepsilon'$-subdifferential of the subtracted map in the appropriate $\\varepsilon'$-subdifferential of the first map, for all $\\varepsilon'\\ge0$; applying this to $F=(f,0)$ and $G=(0,h)$ and using scalarization of convex vector maps produces the scalar inclusion of Theorem 3. The standing hypothesis that all needed $\\varepsilon$-subdifferentials be nonempty is automatically satisfied for proper convex lower semicontinuous functions when $\\varepsilon>0$, and for $\\varepsilon=0$ under a standard closedness qualification.","core_discovery":"The paper's central result is Theorem 3: for convex $f$, for $h$ satisfying the hypothesis that every $\\varepsilon$-subdifferential of $h$ is nonempty, and for a boundary point $\\bar{x}$ with $h(\\bar{x})=0$ and $\\inf_X f < f(\\bar{x})-\\varepsilon$, the membership $\\bar{x} \\in \\varepsilon\\text{-argmin}_{h(x)\\ge 0} f(x)$ is equivalent to the inclusion $\\partial_{\\varepsilon'}h(\\bar{x}) \\subseteq \\bigcup_{\\alpha>0} \\partial_{\\alpha\\varepsilon+\\varepsilon'}(\\alpha f)(\\bar{x})$ holding for every $\\varepsilon' \\ge 0$. Here $\\partial_\\delta \\varphi$ is the set of linear functionals whose affine minorants stay within tolerance $\\delta$ of $\\varphi$. The equivalence is proved by rewriting the reverse program as the unconstrained bicriteria DC problem $(f,0)-(0,h)=(f,-h)$, applying a general $\\varepsilon$-efficiency criterion for difference vector optimization, and then scalarizing the convex vector map. The boundary condition $h(\\bar{x})=0$ is needed for the sufficiency direction, and the strict inequality is used in the necessity direction; together they isolate the nontrivial case the paper calls essential.","pith_inferences":["For $\\varepsilon>0$, the boundary hypothesis is a real restriction: approximate optima of reverse convex programs can lie strictly inside the feasible region. A complete characterization of all $\\varepsilon$-optimal points would need an extra tolerance on $h$ or a projection step that sends interior candidates to the boundary.","The union over $\\alpha>0$ can be read as a tolerance-transfer rule: the slack $\\varepsilon'$ allowed in the constraint is absorbed as an extra $\\varepsilon'$ of tolerance in the objective, rescaled by the same $\\alpha$ that weights objective versus constraint in the bicriteria formulation. This suggests a numerical test that scans $\\alpha$ and checks subgradient membership rather than solving the ","The equality-constraint criterion resembles an approximate multiplier rule in which $\\beta$ plays the role of a multiplier for the equality; comparing it with standard multiplier-rule conditions could show whether the scale parameter $\\alpha$ carries independent information.","The construction treats $h$ as a single reverse constraint; extending the same bicriteria idea to several simultaneous reverse constraints $h_i(x)\\ge 0$ would require a higher-dimensional ordering cone and would likely yield a vectorized version of the condition."],"forward_implications":["At $\\varepsilon=0$ with finite convex $f$ and $h$, Theorem 3 reduces to the previously known exact global-optimality criterion, so the new condition is a direct extension of the exact theory.","With additional convex inequality constraints $G(x)\\le 0$, the same proof gives an $\\varepsilon$-optimality criterion in terms of subdifferentials of $\\alpha f + \\mu\\cdot G$, provided a Slater-type or closedness qualification holds.","For the nonlinear equality constraint $h(x)=0$, the derived criterion uses subdifferentials of $\\alpha f + \\beta h$ with $\\alpha>0$, $\\beta\\ge0$; the paper notes this is new even for exact solutions.","Because the route runs through the bicriteria representation, future refinements of $\\varepsilon$-efficiency conditions for difference vector optimization will automatically supply refinements for reverse convex programs.","Under the paper's boundary-reduction lemma, when $f$ and $h$ are finite convex and the objective is strictly better at the unconstrained infimum than on the feasible set, equality-constrained and reverse-constrained $\\varepsilon$-optima coincide on the boundary $h=0$."],"supporting_citations":[{"why":"Supplies the ε-efficiency criterion for difference vector optimization that becomes the paper's Theorem 2, the engine of the main proof.","marker":"[5]"},{"why":"Provides the scalarization theorem for ε-subdifferentials of convex vector maps used to pass from vector to scalar inclusions in Theorem 3.","marker":"[7]"},{"why":"Establishes the exact-solution characterization that Theorem 3 recovers at ε=0 and extends.","marker":"[10]"},{"why":"Provides the boundary-reduction lemma (Lemma 3) and reverse-convex background used for the equality-constraint applications.","marker":"[16]"},{"why":"Contains the ε-subdifferential sum and composition rules used in deriving Theorem 4 and the equality-constraint corollary.","marker":"[20]"},{"why":"Supplies the closedness qualification that secures nonempty ε-subdifferentials for ε=0 in extended-valued settings.","marker":"[1]"},{"why":"Gives the ε-subdifferential composition rule for vector maps invoked alongside the sum rule in the proof of Theorem 4.","marker":"[6]"}],"fun_headline_variants":["Subdifferential inclusion resolves reverse convex ε-optimality","New test for ε-optimality in reverse convex programs","Reverse convex ε-optima: subdifferential inclusion criterion","Boundary ε-optimality tied to subdifferential inclusion","Characterizing ε-optimality in reverse convex via subdifferentials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The characterization applies only to points on the constraint boundary $h(\\bar{x})=0$ with the strict inequality $\\inf_X f < f(\\bar{x})-\\varepsilon$; for $\\varepsilon>0$, approximate optima of reverse convex programs can occur strictly inside the feasible region, and the theorem has nothing to say about those points.","fun_headline_variants_meta":{"raw":{"variants":["Subdifferential inclusion resolves reverse convex ε-optimality","New test for ε-optimality in reverse convex programs","Reverse convex ε-optima: subdifferential inclusion criterion","Boundary ε-optimality tied to subdifferential inclusion","Characterizing ε-optimality in reverse convex via subdifferentials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000447,"raw_usage":{"total_tokens":2277,"prompt_tokens":982,"completion_tokens":1295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":1209}},"tokens_in":598,"tokens_out":1295,"duration_ms":13490,"temperature":1.0,"reasoning_tokens":1209,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:02:41.728016+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute with $f(x)=x^2+x$, $h(x)=x$, and $\\varepsilon=0.2$ on $X=\\mathbb{R}$. At the boundary point $\\bar{x}=0$ the essential inequality holds ($\\inf_X f=-0.25 < f(0)-0.2=-0.2$), yet $x=0.1$ is feasible with $f(0.1)=0.11 \\le \\inf_{x\\ge0}(x^2+x)+0.2=0.2$, so $x=0.1$ is an $\\varepsilon$-optimal solution despite $h(0.1)>0$. This shows that no condition confined to the boundary $h(\\bar{x})=0$ can characterize all approximate optima of a reverse convex program; any proposed full characterization must account for interior $\\varepsilon$-optima, and this example tests whether it does.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ε-efficiency criterion for difference vector optimization that becomes the paper's Theorem 2, the engine of the main proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the scalarization theorem for ε-subdifferentials of convex vector maps used to pass from vector to scalar inclusions in Theorem 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the exact-solution characterization that Theorem 3 recovers at ε=0 and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the boundary-reduction lemma (Lemma 3) and reverse-convex background used for the equality-constraint applications."},{"cited_title":"World Scientific, Singapore (2002)","cited_arxiv_id":null,"evidence_quote":"Contains the ε-subdifferential sum and composition rules used in deriving Theorem 4 and the equality-constraint corollary."},{"cited_title":"In: Barroso, J","cited_arxiv_id":null,"evidence_quote":"Supplies the closedness qualification that secures nonempty ε-subdifferentials for ε=0 in extended-valued settings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the ε-subdifferential composition rule for vector maps invoked alongside the sum rule in the proof of Theorem 4."}],"review_version":1}