Pith. sign in

REVIEW 47 references

A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A BFGS quasi-Newton method is claimed to converge to robust weakly efficient solutions of finite-scenario uncertain multiobjective problems under a regularity condition on the limit point.

arxiv 2505.13967 v1 pith:4DJ5UKGF submitted 2025-05-20 math.OC

K. Gupta , D. Ghosh , C. Tammer , X. Zhao , J. C. Yao This is my paper · ORCID
classification math.OC
keywords optimizationproblemsquasi-newtonproposedschemevectormethodclass
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 wants to solve optimization problems with several objectives, where each objective's formula depends on an uncertain scenario selected from a finite list. The standard trick is to turn the uncertain problem into a robust counterpart, treating the set of scenario-outcomes as a set to be minimized under the 'upper set less' ordering. Earlier work used a Newton method that demands strong convexity; this paper swaps in BFGS curvature approximations to handle nonconvex objectives. At each iteration, the algorithm identifies the maximal scenario-outcomes (the Pareto-like frontier of the current objective set), enumerates a finite collection of 'partition' index tuples, and for each tuple solves a convex minimax problem to get a descent direction. It then takes an Armijo step. The authors prove, under a regularity condition on the limit point, that accumulation points are stationary, and they state conditions for local superlinear convergence. However, the write-up has serious issues: the quadratic model uses a single n-by-n matrix B for an m-dimensional vector objective, which does not type-check when m > 1; the abstract claims convergence to weakly efficient points while the theorems only establish stationarity; the superlinear proof assumes the whole sequence converges and treats the minimax direction as if it minimized a weighted sum, without proof. As a result, the claimed result, as stated, is not supported.
Extended reading notes

Core claim

The abstract states: 'Under commonly used assumptions, any limit point of a sequence generated by the proposed quasi-Newton technique is found to be a robust weakly efficient point of the problem.' The theorems actually prove a weaker statement: Theorem 4.4 concludes that accumulation points that are regular are stationary points of (SOP), not that they are weakly efficient. If the strongest statement were true, the method would solve finite-scenario uncertain multiobjective problems without convexity.

Load-bearing premise

The accumulation point xbar is a regular point of FU (Definition 3.5): Max(FU(xbar),K) equals WMax(FU(xbar),K) and the number of maximal elements omega(x) is constant near xbar. This stability of the partition set is loaded into Theorem 4.4 and Lemma 4.1 to guarantee that the selected index tuple beta_k eventually stabilizes; without it, the iterate-wise vector optimization problem can change discontinuously and the descent/stationarity arguments fail. The assumption is not implied by twice differentiability or by the finite uncertainty set.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The method inherits the finite-scenario, twice-differentiability, and proper-cone framework from the problem statement. It then adds two heavy extras: BFGS updates are assumed to preserve K-positive definiteness without convexity, and limit points must be regular. The superlinear result adds uniform bounds and continuity on B and the Hessians. These are not derived from the data and none are verified by the numerical section. The algorithm also requires a hand-chosen initial Hessian approximation and scalarization element.

free parameters (2)
  • Initial Hessian approximation B(x0, ξ_i) = identity by default in experiments (not stated explicitly)
    Chosen by hand for each scenario to start the BFGS recursion; convergence proofs assume uniform bounds aI <= B <= bI but do not specify how to choose B0 to satisfy them.
  • Scalarization element e = (1,...,1)^T in numerical tests
    Fixed element in int(K) used to define the Gerstewitz function; the choice affects the measure of descent but not the theoretical claims.
assumptions (5)
  • domain assumption Uncertainty set U is finite with p scenarios; each F(·,ξ_i) is twice continuously differentiable on R^n.
    Stated in Section 2 assumptions (i)-(iii); finite cardinality is needed to form the partition set P_x and the finite max in phi_x.
  • domain assumption K is a proper cone (closed, convex, pointed, solid) with fixed e in int(K); ordering is induced by K.
    Section 2; standard in vector and set optimization.
  • ad hoc to paper The BFGS update preserves K-positive definiteness of B at every iterate, i.e., the curvature condition s_k^T y_i^k > 0 holds.
    Section 4 around Eq. (4.1); for nonconvex F this is not guaranteed, and the paper provides no safeguarding strategy (damping/restart), despite claiming no convexity.
  • ad hoc to paper Accumulation points are regular points of F_U (Definition 3.5).
    Used in Theorem 4.4 and Lemma 4.1 to stabilize omega(x) and P_x; not implied by the problem data.
  • ad hoc to paper For superlinear convergence, B and nabla^2 F satisfy uniform bounds and uniform continuity conditions (Theorem 4.5 (i)-(iii)).
    These are strong extra hypotheses; the proof of Theorem 4.5 additionally assumes the whole sequence converges to the accumulation point, which is not established.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality." pith.science (2026). https://pith.science/paper/4DJ5UKGF

@misc{pith2026250513967,
  author       = {Pith},
  title        = {Pith review of: A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DJ5UKGF}},
  note         = {Machine review of arXiv:2505.13967}
}
read the original abstract

In this article, we derive an iterative scheme through a quasi-Newton technique to capture robust weakly efficient points of uncertain multiobjective optimization problems under the upper set less relation. It is assumed that the set of uncertainty scenarios of the problems being analyzed is of finite cardinality. We also assume that corresponding to each given uncertain scenario from the uncertainty set, the objective function of the problem is twice continuously differentiable. In the proposed iterative scheme, at any iterate, by applying the \emph{partition set} concept from set-valued optimization, we formulate an iterate-wise class of vector optimization problems to determine a descent direction. To evaluate this descent direction at the current iterate, we employ one iteration of the quasi-Newton scheme for vector optimization on the formulated class of vector optimization problems. As this class of vector optimization problems differs iterate-wise, the proposed quasi-Newton scheme is not a straight extension of the quasi-Newton method for vector optimization problems. Under commonly used assumptions, any limit point of a sequence generated by the proposed quasi-Newton technique is found to be a robust weakly efficient point of the problem. We analyze the well-definedness and global convergence of the proposed iterative scheme based on a regularity assumption on stationary points. Under the uniform continuity of the Hessian approximation function, we demonstrate a local superlinear convergence of the method. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 2505.13967 by the authors.

Figure 1
Figure 1. ‘e value of FU at each iteration generated by Algorithm 1 for the initial point x0 = −2.8372 for Example 5.1 In [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Output of Algorithm 1 for Example 5.2 ‘e performance of Algorithm 1 and comparing it with the Newton method [17] for Exam￾ple 5.2 is shown in [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Output of Algorithm 1 for Example 5.3 Example 5.4. [17] Consider the UMOP with the tri-objective function F : R 2 ×U → R 3 de￾fined as F(x,ξ ) := 1 2   kx−l1 −ξ k 2 kx−l2 −ξ k 2 kx−l3 −ξ k 2  , where l1 := [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Output of Algorithm 1 for Example 5.4: ‘e value of xk at each iter￾ation for three different initial points: {(−2,−2) ⊤,(−2,10) ⊤, and (10,−2) ⊤}, as generated by Algorithm 1 in 5.4 quasi-Newton method since the function F is strongly convex for each ξ , and the Newton…
Figure 5
Figure 5. Figure 5: Robust weakly efficient points generated by Algorithm 1 for a few test problems from [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality

    I/n.sc/t.sc/r.sc/o.sc/d.sc/u.sc/c.sc/t.sc/i.sc/o.sc/n.sc Uncertain multiobjective optimization problems (UMOPs) encompass a category of opti- mization problems where multiple uncertain objectives of a conflicting nature must be ad- dressed. In many practical optimization problems, the objectives are inherently uncertain due to incomplete information, imper...

    work page Pith review arXiv 2007

  2. [2]

    /T_he notation ‖·‖ denotes either the Euclidean norm for vectors or the spectral norm for mat rices, depending on the context

    N/o.sc/t.sc/a.sc/t.sc/i.sc/o.sc/n.sc /a.sc/n.sc/d.sc P/r.sc/e.sc/l.sc/i.sc/m.sc/i.sc/n.sc/a.sc/r.sc/i.sc/e.sc/s.sc /T_he set of all nonempty subsets of Rm is represented as P(Rm). /T_he notation ‖·‖ denotes either the Euclidean norm for vectors or the spectral norm for mat rices, depending on the context. /T_he cardinality of a finite set A is denoted by |...

    work page

  3. [3]

    First, we start this section by mentioning some important index-related s et-valued mapping sets

    O/p.sc/t.sc/i.sc/m.sc/a.sc/l.sc/i.sc/t.sc/y.sc C/o.sc/n.sc/d.sc/i.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section, we report optimality conditions for robust weakly effic ient points of P(U ). First, we start this section by mentioning some important index-related s et-valued mapping sets. /T_he following notions are introduced in [23]. Definition 3.1. Consider ...

    work page

  4. [4]

    For each function F(·,ξ i) : Rn → Rm, we use the following quadratic approximation around xk: q(x,ξ i) := F(xk,ξ i) +∇ F(xk,ξ i)⊤(x − xk) + 1 2 (x − xk)⊤B(xk,ξ i)(x − xk)

    /Q_u.sc/a.sc/s.sc/i.sc/hyphen.scN/e.sc/w.sc/t.sc/o.sc/n.sc M/e.sc/t.sc/h.sc/o.sc/d.sc /a.sc/n.sc/d.sc I/t.sc/s.sc C/o.sc/n.sc/v.sc/e.sc/r.sc/g.sc/e.sc/n.sc/c.sc/e.sc A/n.sc/a.sc/l.sc/y.sc/s.sc/i.sc/s.sc For a given initial point x0, for each i ∈ [p], we construct a sequence {B(xk,ξ i)} of Hessian approximations of F(·,ξ i) beginning with an initial matrix...

    work page

  5. [5]

    /T_he testing of Algorithm1 is conducted in MATLAB R2023b

    N/u.sc/m.sc/e.sc/r.sc/i.sc/c.sc/a.sc/l.sc I/l.sc/l.sc/u.sc/s.sc/t.sc/r.sc/a.sc/t.sc/i.sc/o.sc/n.sc In this section, we show the performance of the proposed quasi-New ton Algorithm 1 on several numerical examples. /T_he testing of Algorithm1 is conducted in MATLAB R2023b. /T_he so/f_tware is installed on an IOS system with an 8-core CPU and 8 GB RAM. For t...

    work page 1958

  6. [6]

    In Figure 5, we have shown the generated robust weakly efficient point correspo nding to a randomly chosen initial point for test problems taken from Table5. /T_he black-colored points in each figure collectively present the value of FU (x0) at the initial point x0, blue-colored points present FU (xk) for intermediate xk’s, and the red-colored points collect...

    work page 1986

  7. [7]

    We adopted a set-valued optimization perspective to reformulate the problem as a determ inistic one

    C/o.sc/n.sc/c.sc/l.sc/u.sc/s.sc/i.sc/o.sc/n.sc In this paper, we have introduced a quasi-Newton method to determ ine weakly robust effi- cient solutions for UVOPs with an uncertainty set of finite cardinality. We adopted a set-valued optimization perspective to reformulate the problem as a determ inistic one. /T_he deterministic set optimization problem was ...

    work page 2022

  8. [8]

    W. Zhou, F. Wang, A PRP–based residual method for large-sc ale monotone nonlinear equations, Appl. Math Comput. 261 (2015), 1–7

    work page 2015

Show all 47 references
  1. [9]

    Ben-Tal, A

    A. Ben-Tal, A. Nemirovski, L.E. Ghaoui, Robust Optimizat ion, Princeton University Press, Princeton, 2009

    2009

  2. [10]

    Ehrgo/t_t, J

    M. Ehrgo/t_t, J. Ide, A. Sch ¨obel, Minmax robustness for multiobjective optimization pro blems, Eur. J. Oper. Res. 239 (2014), 17–31

    2014

  3. [11]

    Kuroiwa, G.M

    D. Kuroiwa, G.M. Lee, On robust multiobjective optimiza tion, Vietnam J. Math. 40 (2012), 305–317

    2012

  4. [12]

    Ehrgo/t_t, Multicriteria Optimization, Springer, Berlin, 2005

    M. Ehrgo/t_t, Multicriteria Optimization, Springer, Berlin, 2005

    2005

  5. [13]

    J. Ide, A. Sch ¨obel, Robustness for uncertain multiobjective optimization: a survey and analysis of different concepts, Spectr. 38 (2016), 235–271

    2016

  6. [14]

    J. Ide, E. K ¨obis, Concepts of efficiency for uncertain multiobjective optimization problems based on set order relations, Math. Oper. Res. 80 (2014), 99–127

    2014

  7. [15]

    Eichfelder, J

    G. Eichfelder, J. Jahn, Vector optimization problems and t heir solution concepts, In: Recent Developments in Vector Optimization, vol. 1, pp. 1–27, Springer, Berlin, 201 1. 28 K. GUPTA, D. GHOSH, C. TAMMER, X. ZHAO, J. C. YAO

  8. [16]

    E. K ¨obis, On robust optimization – a unified approach to robustness using a nonlinear scalarizing functional and relations to set optimization, Universit ¨ats-und Landesbibliothek Sachsen-Anhalt, 2014

    2014

  9. [17]

    Wei, C.R

    H.Z. Wei, C.R. Chen, S.J. Li, A unified characterization of multiobjective robustness via separation, J. Optim. /T_heory Appl. 179 (2018), 86–102

    2018

  10. [18]

    F. Wang, S. Liu, Y. Chai, Robust counterparts and robust effic ient solutions in vector optimization under uncertainty, Oper. Res. Le/t_t. 43 (2015), 293–298

    2015

  11. [19]

    C. Liu, Z. Gong, K.L. Teo, J. Sun, L. Cacce/t_ta, Robust multio bjective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Anal.: H ybrid Syst. 25 (2017), 1–20

    2017

  12. [20]

    Ghosh, N

    D. Ghosh, N. Kishor, Generalized ordered weighted aggre gation robustness: a new robust counterpart to solve uncertain multiobjective optimization problems, Eng. Optim. (2024), 1–32. h/t_tps://doi.org/10.1080/0305215X.2024.2418341

    2024

  13. [21]

    Kishor, D

    N. Kishor, D. Ghosh, X. Zhao, Generalized ordered weight ed aggregation robustness to solve uncertain single objective optimization problems, arXiv preprint arXi v:2410.03222 (2024)

    2024 arXiv

  14. [22]

    Kumar, M.A.T

    S. Kumar, M.A.T. Ansary, N.K. Mahato, D. Ghosh, Y. Shehu, Newton’s method for uncertain multiobjective optimization problems under finite uncertainty sets, J. Nonlinear Var. Anal. 7 (2023), 785–809

    2023

  15. [23]

    Kumar, M.A.T

    S. Kumar, M.A.T. Ansary, N.K. Mahato, D. Ghosh, Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set, Appl. Anal. ( 2024), 1–22. h/t_tps://doi.org/10.1080/00036811.2024.2426219

    2024

  16. [24]

    Ghosh, N

    D. Ghosh, N. Kishor, X. Zhao, A Newton method for uncertai n multiobjective optimization problems with finite uncertainty set, J. Nonlinear Var. Anal. 9 (2024), 81–110

    2024

  17. [25]

    J. Ide, E. K ¨obis, D. Kuroiwa, A. Sch ¨obel, C. Tammer, /T_he relationship between multi-objective ro bustness concepts and set-valued optimization, Fixed Point /T_heory Appl. 83 (2014), 1–20

    2014

  18. [26]

    Kuroiwa, Existence theorems of set optimization with set-valued maps, J Inform Optim Sci

    D. Kuroiwa, Existence theorems of set optimization with set-valued maps, J Inform Optim Sci. 24 (2003), 73–84

    2003

  19. [27]

    Kuroiwa, Natural Criteria of Set-Valued Optimization, Manuscript Shimane University, Japan, 1068 (1999), 164–170

    D. Kuroiwa, Natural Criteria of Set-Valued Optimization, Manuscript Shimane University, Japan, 1068 (1999), 164–170

    1999

  20. [28]

    Kuroiwa, Some duality theorems of set-valued optimi zation with natural criteria, In: T

    D. Kuroiwa, Some duality theorems of set-valued optimi zation with natural criteria, In: T. Tanaka, (ed.) Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp. 221–228, World Scientific, Singapore, 1999

    1999

  21. [29]

    Lovison, Singular continuation: Generating piecewise linea r approximations to Pareto sets via global analysis, SIAM J

    A. Lovison, Singular continuation: Generating piecewise linea r approximations to Pareto sets via global analysis, SIAM J. Optim. 21 (2011), 463–490

    2011

  22. [30]

    Bouza, E

    G. Bouza, E. /Q_uintana, C. Tammer, A steepest descent method for set optimization problems with set-valued mappings of finite cardinality, J. Optim. /T_heory Appl. 190 (2021), 711–743

    2021

  23. [31]

    Dennis Jr., J.J

    J.E. Dennis Jr., J.J. Mor ´e, /Q_uasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977), 46–89

    1977

  24. [32]

    G ¨opfert, H

    A. G ¨opfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Method s in Partially Ordered Spaces, Springer, New York, 2003

    2003

  25. [33]

    Jahn, T.X.D

    J. Jahn, T.X.D. Ha, New order relations in set optimizatio n, J. Optim. /T_heory Appl. 148 (2011), 209–236

    2011

  26. [34]

    Drummond, F.M.P

    L.M.G. Drummond, F.M.P. Ruppa, B.F. Svaiter, A quadrati cally convergent Newton method for vector opti- mization, Optim. 63 (2014), 661–677

    2014

  27. [35]

    A.A. Khan, C. Tammer, C. Z ˘alinescu, Set-valued Optimization, Springer, Berlin, 2014

    2014

  28. [36]

    Wright, Numerical Optimization, Springer, New York , 2006

    S.J. Wright, Numerical Optimization, Springer, New York , 2006

    2006

  29. [37]

    Fliege, L.M.G

    J. Fliege, L.M.G. Drummond, B.F. Svaiter, Newton’s metho d for multiobjective optimization, SIAM J. Optim. 20 (2009), 602–626

    2009

  30. [38]

    Ghosh, A

    D. Ghosh, A. Anshika, J.-C. Yao, X. Zhao, /Q_uasi-Newton met hod for set optimization problems with set- valued mapping given by finitely many vector-valued functions, ar Xiv preprint, arXiv:2501.04711 (2024)

    2024 arXiv

  31. [39]

    Povalej, /Q_uasi-Newton’s method for multiobjective optimi zation, J

    ˇZ. Povalej, /Q_uasi-Newton’s method for multiobjective optimi zation, J. Comput. Appl. Math. 255 (2014), 765–777

    2014

  32. [40]

    Huband, P

    S. Huband, P. Hingston, L. Barone, L. While, A review multiobjective test problem and a scalable test problem toolkit, IEEE Trans. Evol. Comput. 10 (2006), 477–506

    2006

  33. [41]

    Y. Jin, M. Olhofer, B. Sendhoff, Dynamic weighted aggregati on for evolutionary multiobjective optimization: Why does it work and how?, Proceedings of the Genetic and Evolutio nary Computation Conference, pp. 1042–1049, 2001. A QUASI-NEWTON METHOD FOR UNCERTAIN MULTIOBJECTIVE O...

    2001

  34. [42]

    Fliege, L.G

    J. Fliege, L.G. Drummond, B.F. Svaiter, Newton’s method f or multiobjective optimization, SIAM J. Optim. 20 (2009), 602–626

    2009

  35. [43]

    Zitzler, K

    E. Zitzler, K. Deb, L. /T_hiele, Comparison of multiobject ive evolutionary algorithms: Empirical results, Evol. Comput. 8 (2000), 173–195

    2000

  36. [44]

    Broyden, A new double–rank minimisation algorithm, Pr eliminary report, Am

    C.G. Broyden, A new double–rank minimisation algorithm, Pr eliminary report, Am. Math. Soc. Notices 16 (1969), 670

    1969

  37. [45]

    Fletcher, A new approach to variable metric algorithm s, Comput

    R. Fletcher, A new approach to variable metric algorithm s, Comput. J. 13 (1970), 317–322

    1970

  38. [46]

    Goldfarb, A family of variable–metric methods deriv ed by variational means, Math

    D. Goldfarb, A family of variable–metric methods deriv ed by variational means, Math. Comput. 24 (1970), 23–26

    1970

  39. [47]

    Shanno, Conditioning of quasi-Newton methods for functi on minimization, Math

    D.F. Shanno, Conditioning of quasi-Newton methods for functi on minimization, Math. Comput. 24 (1970), 647–656

    1970

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.