{"id":"b1073b69-c888-408f-b1fc-a64faf6fb30a","arxiv_id":"2412.20168","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A general nonlinear conjugate gradient framework for set-valued optimization is proposed, with global convergence results for Dai-Yuan, Polak-Ribiere-Polyak, and Hestenes-Stiefel variants under general ordering cones.","lead":"Set optimization problems, where the objective returns a set of points instead of a single number, arise in decision-making under uncertainty. The authors extend nonlinear conjugate gradient methods to such problems, prove global convergence for general ordering cones, and test the methods numerically.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's Wolfe line-search existence assumes condition (16) (componentwise lower bound along the whole ray), which is not among the standing Assumptions 3.1–3.3 used by the convergence theorems; Algorithm 1's well-definedness therefore lacks support as written.","rationale":"The reader identified the same technical gap I consider most load-bearing: Theorem 3.1 requires condition (16) to guarantee a Wolfe step, and this condition is absent from the standing assumptions used in the convergence theorems. I agree this is the right place to focus: all global convergence results pass through the line search, so if Step 5 can be undefined the whole framework collapses. However, I do not fully endorse the claim that 'the line search may have no valid step' without additional qualification. In the actual algorithm, each iterate x_k lies in the bounded level set L, and the Armijo condition (14a) implies x_k + alpha d_k in L whenever it holds. Since an unbounded ray cannot be contained in a bounded set, (17) cannot hold for all alpha > 0 under Assumption 3.1 alone, so the existence proof may be repairable by replacing (16) with this boundedness argument. Thus the concern is a genuine proof gap rather than a demonstrated counterexample to the convergence theorems. The verdict remains CONDITIONAL: the paper should add condition (16) to the standing assumptions, or explicitly prove Theorem 3.1 under the level-set boundedness and x_k in L, so that the stated convergence results cover the algorithm as defined.","tokens_in":29749,"tokens_out":30041,"duration_ms":295980,"concrete_test":"Check whether Theorem 3.1's conclusion can be derived from Assumptions 3.1–3.3 together with the inductive invariant x_k in L (which holds because any alpha satisfying (14a) forces x_k + alpha d_k in L). Attempt to modify the proof by replacing (16) with the claim: if (17) held for all alpha > 0, then the entire ray {x_k + alpha d_k : alpha > 0} would lie in the bounded level set L, a contradiction. If this modified proof succeeds, then (16) is not load-bearing for the algorithm and only the theorem statement needs adjustment; if it fails, condition (16) must be added to Assumptions 3.1–3.3 or Algorithm 1 must be modified to include a bounded line search.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that all proposed nonlinear conjugate gradient schemes are globally convergent under the usual assumptions (Assumptions 3.1–3.3). But the convergence theorems (4.1, 5.1, 5.3) concern sequences generated by Algorithm 1, and Algorithm 1 is only well-defined if Step 5 has a solution. Section 4.1 explicitly states that Step 5 is well-defined because of Theorem 3.1. Yet Theorem 3.1 assumes condition (16): there exists A in R^m such that A ≺_K f_{a_j}(x + alpha d) for all j and all alpha > 0. This condition is not part of Assumptions 3.1–3.3 and is not restated in the convergence theorems. In the proof, (16) is used in Step 2 to conclude that the Armijo-type inequality (17) fails for large alpha, so that a finite maximal interval [0, alpha_bar] exists; then the curvature condition is forced by a mean-value argument. Without (16), the proof breaks at that point, and the paper gives no other argument that a Wolfe step exists. The gap is not purely cosmetic: for example, with F(x) = ((x, x^2)) and K = R_+^2, x0 = 1, the descent direction d = -1 satisfies Assumptions 3.1–3.3, but (16) fails because the first component tends to -infinity along the ray; the algorithm can still produce Wolfe steps, but the theorem as stated does not cover this case. Thus the hypotheses stated in the convergence theorems are insufficient to guarantee that the iterates are defined.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops nonlinear conjugate gradient methods for unconstrained set-valued optimization problems whose objective maps have finite image sets under the lower set less preorder. It introduces a notion of K-descent direction and standard/strong Wolfe conditions for set-valued functions, proves an existence theorem for Wolfe steps, derives a Zoutendijk-like inequality, and then states global convergence results for a general conjugate gradient scheme and for Dai-Yuan, Polak-Ribière-Polyak, and Hestenes-Stiefel variants. The numerical section compares the proposed methods with two existing conjugate gradient methods on five test instances, including one with a non-finitely generated ordering cone.","tokens_in":30156,"tokens_out":8984,"duration_ms":93840,"significance":"The intended contribution is meaningful: if the proofs are completed, the paper would extend nonlinear conjugate gradient methods to set optimization without requiring the ordering cone to be finitely generated and without imposing a regularity condition on the solution set. The proposed Zoutendijk-like condition and the treatment of a non-polyhedral cone in the numerical experiments are potentially useful advances over the vector-optimization CG literature, and the paper also engages with an open problem raised in that literature. However, as written, several load-bearing hypotheses are either unstated in the standing assumptions or not verified for the proposed algorithms, and one central proof is deferred to a prior work by reference. These gaps prevent the paper from currently supporting its main convergence claims.","major_comments":[{"comment":"The existence of a Wolfe step in Theorem 3.1 is proved only under the additional assumption (16), which is not part of Assumptions 3.1–3.3 and is not restated in Theorems 4.1, 5.1, or 5.3. Section 4.1 explicitly justifies Step 5 by Theorem 3.1, so the sequences whose convergence is asserted are not proved to exist without (16). The hypothesis (16) is not a consequence of the standing assumptions: for F(x)={(x,x^2)} with K=R_+^2, x0=1, and the descent direction d=-1, the first component tends to -infinity along the ray, so (16) fails even though Assumptions 3.1–3.3 can be satisfied. The authors need either to add (16) to the standing assumptions and to the convergence theorems, or to prove Wolfe-step existence under a weaker condition that is actually satisfied by the algorithm's iterates.","section":"Theorem 3.1 / Section 4.1"},{"comment":"Theorem 5.2, which is the vehicle for the PRP and HS convergence results, has no proof: the text states only 'The lines of the proof are similar to [30, Theorem 5.10]'. The setting here differs from [30] in essential ways (set-valued objective, variable partition sets P_k, variable cardinality omega_k, and a different step-length condition), so this deferral is not sufficient for a load-bearing result. Moreover, Theorem 5.3 assumes that d_k satisfies the sufficient descent condition (12) for beta_k=max{beta_PRP_k,0} or max{beta_HS_k,0}, but no lemma in the manuscript establishes this condition for these choices of beta_k; Proposition 4.1 gives a sufficient-descent condition of the form (28), and the PRP/HS formulas are not shown to fit it. Without such a verification, Theorem 5.3 does not establish convergence of the PRP and HS methods as defined in Algorithm 1.","section":"Theorems 5.2 and 5.3"},{"comment":"The numerical example 6.4 is degenerate: in the displayed definition of f_i(x), the bracketed expression equals (1,-1) for every i in [5], so all five component functions are identical and F(x) is always a singleton. This example therefore does not exercise the finite-cardinality set-valued structure, and the comparisons in Tables 4 and 5, as well as the discussion of the two ordering cones K1 and K2, are not informative for the proposed methods. The example should be corrected or replaced.","section":"Section 6, Example 6.4"}],"minor_comments":[{"comment":"There is a placeholder '(Theorem )' in the sentence preceding Algorithm 1; the theorem number should be filled in.","section":"Section 4.1"},{"comment":"In the strong-Wolfe branch, the expression 'F_k(x_k)+alpha_k d_k,d_k)' should read F_k(x_k+alpha_k d_k,d_k).","section":"Algorithm 1, Step 5"},{"comment":"The heading 'Hastenes-Stiefel' is a typo; the standard spelling is 'Hestenes-Stiefel', as used in the abstract.","section":"Section 5 heading"},{"comment":"The symbol alpha^diamond is introduced without definition, and the sentence 'that is, (17) is not true for all alpha in [0,alpha^diamond], where alpha^diamond > bar-alpha' is unclear; it should be rephrased.","section":"Theorem 3.1, proof of Step 2"},{"comment":"The phrase 'the relation (17) does not not hold for all alpha in [0,infty)' contains a double negative and should be corrected to 'does not hold for all alpha in [0,infty)'.","section":"Theorem 3.1, proof of Step 2"},{"comment":"The divergence condition (31) is an additional assumption, not a consequence of the standing assumptions; since none of the subsequent results verifies (31), the global-convergence claim for the general scheme should be stated as conditional on (31) whenever this theorem is cited.","section":"Theorem 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is appropriate in scope for a mathematical optimization journal, and the intended generalization is worth pursuing. However, the convergence statements are currently not supported as written because the Wolfe line search existence theorem requires an extra assumption that is not carried into the convergence theorems, and the PRP/HS convergence argument rests on an unproved sufficient-descent hypothesis and a proof deferred to a prior paper. The numerical section also contains a degenerate example. These are fixable in a revision, but they are not merely cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's genuine advance is the extension of nonlinear conjugate gradient methods (DY, PRP, HS) to set-valued optimization with non-finitely generated ordering cones, using a Gerstewitz scalarization and a compact generator of the dual cone. Prior work, including the authors' own, covered only FR and CD for finitely generated cones. The Zoutendijk-like condition in Theorem 3.2 is a clean extension, and the convergence proofs follow standard templates. The numerical experiments include a genuinely non-finitely generated ice-cream cone, which is a nice stress test.\n\nThe main soft spot is real. Theorem 3.1 assumes condition (16), a uniform lower bound on all component functions along the entire search ray, yet this condition is not among the standing Assumptions 3.1–3.3, nor is it restated in the convergence theorems. Section 4.1 invokes Theorem 3.1 to justify Step 5 of Algorithm 1, so the algorithm's well-definedness depends on an unstated hypothesis. The stress-test example—F(x) = {(x, x^2)} with K = R_+^2, x0 = 1, d = -1—does satisfy Assumptions 3.1–3.3 and gives a K-descent direction, but (16) fails because the first component goes to -infinity along the ray. The algorithm can still produce a Wolfe step there, so it is not a counterexample to the convergence claims, but it does show the proof cannot go through as written. This is patchable by adding a coercivity-type assumption, but it should be explicit.\n\nTwo smaller issues: Theorem 5.2's proof is deferred to a prior work with only \"lines of the proof are similar,\" which is a transparency gap in a new setting. And Example 6.4 is degenerate—all f_i are identical as written, so it does not test the multi-function structure. Minor, but worth cleaning up.\n\nThis deserves a serious referee. The core contribution is solid and the central gap is fixable. A referee should ask for a clear statement and discussion of condition (16) (or an alternative existence argument) in the convergence theorems, and a corrected Example 6.4. The paper is for researchers in set optimization and first-order methods; it is a useful contribution to an active subfield.","headline":"Real extension of CG to set-valued problems with general cones, but line search existence carries an unstated uniform lower-bound hypothesis that the convergence theorems don't assume.","tokens_in":30637,"tokens_out":7932,"would_cite":true,"duration_ms":77442,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J53","90C29","90C47"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the Dai-Yuan, Polak-Ribière-Polyak, and Hestenes-Stiefel conjugate gradient schemes extend to set-valued optimization and converge globally, with no requirement that the ordering cone be finitely generated.","keywords":["set optimization","nonlinear conjugate gradient methods","lower set less ordering","Wolfe conditions","global convergence","Gerstewitz scalarizing function","finite-cardinality set-valued mappings","non-finitely generated ordering cones"],"falsifier":"Run the paper's Algorithm 1 with the Dai-Yuan rule ($\\beta_k=\\eta\\beta_k^{\\mathrm{DY}}$, $0\\le\\eta<(1-\\sigma)/(1+\\sigma)$) on Example 6.5's problem with the non-finitely generated cone $K_2=\\{y\\in\\mathbb{R}^3: y_3\\ge\\sqrt{y_1^2+y_2^2}\\}$ from the paper's 100 initial points, recording the partial sums of $F_k^2(x_k,d_k)/\\|d_k\\|^2$ and the norms $\\|u_k\\|$; Theorem 3.2 predicts bounded partial sums and Theorem 5.1 predicts $\\liminf_{k\\to\\infty}\\|u_k\\|=0$, so any run where the partial sums diverge while every line search reports a valid Wolfe step, or where $\\|u_k\\|$ stays bounded away from zero, refutes the central claims. A second probe targets well-definedness: instrument the Wolfe line search to flag when no $\\alpha>0$ satisfies (14) on a $K$-descent direction, since the existence guarantee (Theorem 3.1) rests on condition (16) — any flagged instance would show exactly where the guarantee chain depends on that extra boundedness-below condition.","tokens_in":29534,"feed_emoji":"📉","tokens_out":30859,"duration_ms":279915,"temperature":0.7,"pith_summary":"This paper extends the classical nonlinear conjugate gradient (CG) machinery — the Dai-Yuan, Polak-Ribière-Polyak, and Hestenes-Stiefel update rules — to set-valued optimization, where the objective $F(x)=\\{f_1(x),\\dots,f_p(x)\\}$ returns one of finitely many continuously differentiable images compared by the lower set less order (writing $A\\preceq_\\ell B$ when $B\\subseteq A+K$ for a closed, convex, pointed, solid cone $K$). The paper's central claim is that by replacing the gradient with the minimizer $u(x)$ of a scalarized auxiliary problem built from the Gerstewitz function, one can define meaningful descent directions, sufficient-decrease and Wolfe line searches, and a Zoutendijk-like summation inequality for set-valued problems, and that all three proposed conjugate gradient schemes are then globally convergent: $\\liminf_{k\\to\\infty}\\|u_k\\|=0$, meaning the iterates asymptotically satisfy the necessary stationarity condition for local weak minimality. The authors stress two features: the ordering cone does not have to be finitely generated, so the results cover cones such as the second-order cone excluded by earlier vector-CG and set-CG theory, and no regularity assumption on the solution set is needed. Because set optimization is the standard model for robust vector optimization under finite uncertainty, a provably convergent first-order method gives a practical tool for that class, and the numerical experiments show the three rules performing competitively with the existing Fletcher-Reeves and conjugate-descent methods for set-valued objectives, including on a problem with a non-finitely generated cone.","feed_headline":"Proven convergent: conjugate-gradient methods for set-valued maps","feed_subtitle":"The three classical update rules work even when the ordering cone is not finitely generated.","key_machinery":"The load-bearing object is the scalarized subproblem $\\varphi(x)=\\min_{(a,d)\\in P_x\\times\\mathbb{R}^n}\\big[F_a(x,d)+\\tfrac12\\|d\\|^2\\big]$ with $F_a(x,d)=\\max_{j\\in[\\omega(x)]}\\psi_e\\big(\\nabla f_{a_j}(x)^\\top d\\big)$, where $\\psi_e$ is the Gerstewitz scalarizing function for the cone $K$; its minimizer $(a_k,u_k)$ supplies both the stopping test ($u_k=0$ characterizes stationary points) and the base descent direction. Strong convexity in $d$ yields the key inequality $-F_k(x_k,u_k)>\\tfrac12\\|u_k\\|^2$ at nonstationary iterates, which converts any lower bound on $|F_k|$ into a lower bound on $\\|u_k\\|^2$ and is what turns a Zoutendijk-style sum into a convergence statement. The second piece is the Wolfe-type line search (displays (14)–(15)) whose sufficient-decrease half bounds $F(x_{k+1})$ above, in the set order, by the linearized images $f_{a_j}(x_k)+\\rho\\alpha_k F_k(x_k,d_k)e$; Lipschitz continuity of the gradients and monotonicity of $G(A)=\\inf_{z\\in A}\\psi_e(z)$ with respect to $\\preceq_\\ell$ then produce the Zoutendijk-like inequality of Theorem 3.2, $\\sum_k F_k^2(x_k,d_k)/\\|d_k\\|^2<+\\infty$. The third piece is the regular restart in Step 4 of Algorithm 1, resetting $d_k=u_k$ when $|F_{k-1}(x_k,d_{k-1})|<F_k(x_k,d_{k-1})$, which guarantees $d_k$ is always a $K$-descent direction; access to non-finitely generated cones comes from representing the dual cone through a compact generator $C\\subset K^*$ of the form $\\{w\\in K^*:w^\\top e=1\\}$, used in the existence proof of Wolfe steps, while PRP and HS convergence is routed through the bounded-step Property (*) of [7] that controls $\\beta_k$ when consecutive steps are short.","core_discovery":"The paper's discovery, on its own terms, is that the entire convergent-descent template of smooth optimization survives when the objective is a finite-cardinality set-valued map $F(x)=\\{f_1(x),\\dots,f_p(x)\\}$ ordered by the lower set less relation, provided the gradient is replaced by the minimizer $(a_k,u_k)$ of the auxiliary scalarized problem $\\varphi(x)=\\min_{(a,d)\\in P_x\\times\\mathbb{R}^n}\\big[\\max_{j\\in[\\omega(x)]}\\psi_e(\\nabla f_{a_j}(x)^\\top d)+\\tfrac12\\|d\\|^2\\big]$, where $\\psi_e$ is the Gerstewitz scalarizing function for the cone $K$ and the inner maximum runs over the active indices of the minimal elements of $F(x)$. With this surrogate the paper defines $K$-descent directions, a sufficient descent condition, and standard and strong Wolfe conditions for set-valued functions (displays (14) and (15)); proves that a step satisfying them exists along any $K$-descent ray (Theorem 3.1) using a compact generator of the dual cone $K^*$, which is how non-finitely generated cones enter; and proves a Zoutendijk-like inequality $\\sum_k F_k^2(x_k,d_k)/\\|d_k\\|^2<+\\infty$ for iterates following Wolfe steps (Theorem 3.2). The convergence theorems — Theorem 4.1 for the general scheme with any $\\beta_k\\ge 0$ under a divergence condition, Theorem 5.1 for a bounded fraction of the Dai-Yuan parameter, and Theorems 5.2–5.3 for the Polak-Ribière-Polyak and Hestenes-Stiefel parameters via the bounded-step Property (*) of [7] — all conclude $\\liminf_{k\\to\\infty}\\|u_k\\|=0$: a subsequence of iterates reaches the necessary stationarity condition for local weak minimality, under the usual Lipschitz and bounded-level-set assumptions and with no restriction that the ordering cone be finitely generated.","pith_inferences":["The line search's well-definedness (Theorem 3.1) needs the extra boundedness condition (16) — all component functions bounded below in the cone order along every search ray — which is not among the standing assumptions (3.1)–(3.3) invoked by the convergence theorems. My reading: the practical guarantee is conditional on the search never descending to $-\\infty$ in the cone order along a ray; a smoo","The theory treats the subproblem $\\varphi$ at each iterate as an exact oracle and requires the minimal elements of $F(x_k)$ to be computed at every step; on large-$p$ or high-dimensional instances these inner computations, rather than the CG iteration itself, are likely to dominate the runtime. A testable extension is to replace the exact solve with a few accelerated subgradient steps and check nu","Convergence is to stationarity, and stationarity is only necessary for local weak minimality; the paper does not quantify how often the terminal iterates are actually minimal. In Example 6.1 it reports (by inspection of plots) that the generated points are weakly minimal, so an empirical study of the minimality rate on other instances would sharpen the practical reading of the theorems.","Because the Zoutendijk-like inequality and the Property (*) argument are stated in the same shape as the scalar and vector cases, the same proof skeleton plausibly transfers to other descent templates — for instance projected or proximal variants for constrained set optimization — which would make the paper's contribution a general recipe for set-valued first-order methods rather than three isolat"],"forward_implications":["The three classical update rules (Dai-Yuan, Polak-Ribière-Polyak, Hestenes-Stiefel) now carry global-convergence guarantees for set optimization under any closed, convex, pointed, solid ordering cone, including non-finitely generated cones such as the second-order cone of Example 6.5, a case the prior conjugate gradient theories for vector and set objectives explicitly left open.","The framework gives a reusable template: any future choice of $\\beta_k$ that is nonnegative and satisfies the bounded-step Property (*) inherits global convergence from Theorems 3.2 and 5.2, so extending the family of convergent set-CG methods reduces to verifying a bounded-step condition rather than re-proving the convergence argument.","For robust counterpart problems with finitely many scenarios ($F(x)=\\{f_1(x),\\dots,f_p(x)\\}$), the algorithms provide a provably convergent first-order method; the numerical comparison shows the three new rules are competitive with the existing Fletcher-Reeves and conjugate-descent baselines in iterations and runtime, with the PRP and HS rules ahead on some instances and DY ahead on the cone probl","Because the convergence target is stationarity, which is necessary for local weak minimality, the methods are suitable as a first-order engine that produces candidate limit points to be certified as weakly minimal afterward through the partition-set reduction of Lemma 2.4.","The regular restart in Step 4 — resetting $d_k=u_k$ whenever $|F_{k-1}(x_k,d_{k-1})|<F_k(x_k,d_{k-1})$ — is what guarantees every direction is a $K$-descent direction, and the authors identify as future work the search for a Wolfe-type line search that removes this restart, which would let the iterates keep full CG direction updates more often."],"supporting_citations":[{"why":"Provides the set-optimization problem class (finite-cardinality set-valued maps), the lower set less order, the partition-set reduction of Lemma 2.4, the stationary-point notion, and two of the numerical test instances.","marker":"[2]"},{"why":"The vector-optimization conjugate gradient framework being extended: Wolfe conditions for vector objectives, the Zoutendijk-style convergence template, the Property (*) approach, and the explicitly noted open problem of non-finitely generated cones.","marker":"[30]"},{"why":"Supplies the existing Fletcher-Reeves and conjugate-descent nonlinear CG methods for set-valued objectives that serve as the numerical baselines and whose finite-cone restriction the new methods remove.","marker":"[24]"},{"why":"Source of the bounded-step Property (*) and the associated proof strategy used to establish convergence of the PRP and HS variants in Theorem 5.2.","marker":"[7]"},{"why":"Monograph supplying the Gerstewitz scalarizing function properties (Lipschitz continuity, monotonicity, representability of $-K$ via $\\psi_e\\le 0$) used throughout the descent and Wolfe analysis.","marker":"[21]"},{"why":"The vector-function Armijo condition that the set-valued sufficient-decrease condition extends.","marker":"[3]"},{"why":"Supplies the domination property for compact sets (Proposition 2.1) used to pass from pointwise bounding of the active functions to the set-level inclusions in the sufficient-decrease argument.","marker":"[6]"},{"why":"The practical Wolfe line search implementation (initial step, backtracking parameters) adopted in the numerical experiments.","marker":"[29]"}],"fun_headline_variants":["Set-valued CG methods converge without finite generator","Conjugate gradients for set-valued maps: global convergence proof","Classical CG rules extended to set-valued optimization","Nonlinear CG for set-valued objectives, cones need not be finite","Set-valued Wolfe conditions enable CG convergence theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the component functions stay bounded below, in the cone's ordering sense, along every search ray the algorithm tries: this is condition (16) inside Theorem 3.1, it is what guarantees a usable step size exists, and it is stronger than the bounded-level-set assumptions that the convergence theorems themselves are stated under — if it fails, the algorithm may have no admissible step even though all the assumptions used in the convergence proofs hold.","fun_headline_variants_meta":{"raw":{"variants":["Set-valued CG methods converge without finite generator","Conjugate gradients for set-valued maps: global convergence proof","Classical CG rules extended to set-valued optimization","Nonlinear CG for set-valued objectives, cones need not be finite","Set-valued Wolfe conditions enable CG convergence theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000276,"raw_usage":{"total_tokens":1805,"prompt_tokens":1262,"completion_tokens":543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":878,"completion_tokens_details":{"reasoning_tokens":478}},"tokens_in":878,"tokens_out":543,"duration_ms":5814,"temperature":1.0,"reasoning_tokens":478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:29:28.308727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's Algorithm 1 with the Dai-Yuan rule ($\\beta_k=\\eta\\beta_k^{\\mathrm{DY}}$, $0\\le\\eta<(1-\\sigma)/(1+\\sigma)$) on Example 6.5's problem with the non-finitely generated cone $K_2=\\{y\\in\\mathbb{R}^3: y_3\\ge\\sqrt{y_1^2+y_2^2}\\}$ from the paper's 100 initial points, recording the partial sums of $F_k^2(x_k,d_k)/\\|d_k\\|^2$ and the norms $\\|u_k\\|$; Theorem 3.2 predicts bounded partial sums and Theorem 5.1 predicts $\\liminf_{k\\to\\infty}\\|u_k\\|=0$, so any run where the partial sums diverge while every line search reports a valid Wolfe step, or where $\\|u_k\\|$ stays bounded away from zero, refutes the central claims. A second probe targets well-definedness: instrument the Wolfe line search to flag when no $\\alpha>0$ satisfies (14) on a $K$-descent direction, since the existence guarantee (Theorem 3.1) rests on condition (16) — any flagged instance would show exactly where the guarantee chain depends on that extra boundedness-below condition.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the set-optimization problem class (finite-cardinality set-valued maps), the lower set less order, the partition-set reduction of Lemma 2.4, the stationary-point notion, and two of the numerical test instances."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The vector-optimization conjugate gradient framework being extended: Wolfe conditions for vector objectives, the Zoutendijk-style convergence template, the Property (*) approach, and the explicitly noted open problem of non-finitely generated cones."},{"cited_title":"Optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the existing Fletcher-Reeves and conjugate-descent nonlinear CG methods for set-valued objectives that serve as the numerical baselines and whose finite-cone restriction the new methods remove."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the bounded-step Property (*) and the associated proof strategy used to establish convergence of the PRP and HS variants in Theorem 5.2."},{"cited_title":"Sp ringer, Heidelberg (2015)","cited_arxiv_id":null,"evidence_quote":"Monograph supplying the Gerstewitz scalarizing function properties (Lipschitz continuity, monotonicity, representability of $-K$ via $\\psi_e\\le 0$) used throughout the descent and Wolfe analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The vector-function Armijo condition that the set-valued sufficient-decrease condition extends."},{"cited_title":"In: Recent Developments in Vector Optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the domination property for compact sets (Proposition 2.1) used to pass from pointwise bounding of the active functions to the set-level inclusions in the sufficient-decrease argument."},{"cited_title":"ACM Trans","cited_arxiv_id":null,"evidence_quote":"The practical Wolfe line search implementation (initial step, backtracking parameters) adopted in the numerical experiments."}],"review_version":1}