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arxiv: 2604.12166 · v1 · submitted 2026-04-14 · 🧮 math.OC

On Optimality Conditions for Mathematical Programming Problems Based on Strong Subdifferentials

Felipe Lara , Alberto Ramos This is my paper

Pith reviewed 2026-05-10 16:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords strong subdifferentialstrongly quasiconvexKKT conditionsFritz-John conditionsoptimality conditionsnonsmooth optimizationmathematical programming
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The pith

Strong subdifferentials yield refined KKT and FJ optimality conditions for nonsmooth nonconvex problems with strongly quasiconvex constraints.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops refined optimality conditions for mathematical programming problems that are nonsmooth and nonconvex. It focuses on the case where the functional constraint is strongly quasiconvex and introduces the strong subdifferential for analysis. The key step is expressing the normal cone to the supremum function using these subdifferentials. This leads to sharper Karush-Kuhn-Tucker and Fritz-John type conditions that are illustrated with examples. A sympathetic reader would care because these conditions can identify optimal solutions more precisely in problems where classical subdifferentials fall short.

Core claim

After analyzing the strong subdifferential, we compute the normal cone of the supremum function in terms of such subdifferentials, and apply this result to the mathematical programming problem to obtain refined KKT and FJ-type optimality conditions for nonsmooth, nonconvex problems with strongly quasiconvex functional constraints.

What carries the argument

The strong subdifferential for strongly quasiconvex functions, which is used to derive an expression for the normal cone of the supremum function.

If this is right

  • The derived conditions are necessary for optimality in the specified class of problems.
  • Examples demonstrate that the conditions can be applied practically to check optimality.
  • The approach refines existing conditions by leveraging the structure of strong quasiconvexity.
  • Normal cone computations become feasible directly from the strong subdifferentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could be extended to other types of generalized convex functions for similar refinements.
  • Algorithmic methods might use these conditions to design new solvers for nonconvex programs.
  • Connections to variational analysis could broaden the applicability to equilibrium problems.

Load-bearing premise

The functional constraint belongs to the class of strongly quasiconvex functions so that the strong subdifferential is well-defined and the normal cone formula holds.

What would settle it

Finding a specific strongly quasiconvex constraint function where the computed normal cone does not match the actual normal cone to the supremum, or an optimization problem satisfying the assumptions but violating the refined optimality condition at a local minimum.

read the original abstract

We develop refined Karush-Kuhn-Tucker (KKT) and Fritz-John (FJ)-type optimality conditions for nonsmooth, nonconvex mathematical pro\-gra\-mming problems. We pay special attention in the case that the functional constraint belongs to a specific class of generalized convex functions known as strongly quasiconvex functions. After analyzing a specialized sub\-di\-ffe\-ren\-tial, named the strong subdifferential, we compute the normal cone of the supremum function in terms of such subdifferentials, and apply this result to the mathematical programming problem. We illustrate our important results by examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a specialized 'strong subdifferential' for strongly quasiconvex functions, derives an expression for the normal cone to the epigraph or sublevel set of the supremum of such functions in terms of these subdifferentials, and applies the result to obtain refined KKT and Fritz-John optimality conditions for nonsmooth nonconvex mathematical programs with strongly quasiconvex inequality constraints. The claims are illustrated with examples.

Significance. If the normal-cone identity holds under the stated hypotheses, the work would supply a technically useful refinement of standard subdifferential-based optimality conditions for a concrete subclass of generalized convex functions. The approach follows established patterns in nonsmooth analysis (specialized subdifferentials plus normal-cone calculus), and the provision of illustrative examples is a positive feature. The significance is tempered by the need to confirm that strong quasiconvexity alone supplies the closedness and sum-rule properties required for the calculus step.

major comments (2)
  1. [normal-cone computation for the supremum function (the theorem immediately following the definition and properties of D^] The central normal-cone formula for the set {x | sup_i g_i(x) ≤ 0} (the result obtained after analyzing the strong subdifferential) is load-bearing for all subsequent KKT/FJ statements. Strong quasiconvexity guarantees that the strong subdifferential is well-defined, but does not automatically deliver the lower-semicontinuity, closed-graph, or limiting sum-rule properties needed for an exact normal-cone representation in nonsmooth analysis. Without an explicit verification or additional qualification, the claimed equality (or tight inclusion) may fail at candidate points.
  2. [application to the mathematical programming problem (the theorem that translates the normal-cone result into KKT/FJ form] The refined KKT and FJ conditions stated in the main optimality theorem inherit the same gap: they are obtained by substituting the normal-cone expression into the standard first-order necessary condition. If the normal-cone step is only an inclusion rather than an equality, the resulting conditions are weaker than claimed and may not be strictly sharper than those already available from the Clarke or limiting subdifferential.
minor comments (2)
  1. [Introduction and abstract] The abstract and introduction repeatedly use the phrase 'refined' optimality conditions; a short paragraph comparing the new conditions with the classical ones obtained from the Clarke subdifferential would make the improvement concrete.
  2. [Section defining the strong subdifferential] Notation for the strong subdifferential (denoted D^ or similar) should be introduced once with a clear comparison to the standard convex subdifferential and the limiting subdifferential.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points about the technical foundations of the normal-cone calculus and the resulting optimality conditions. We address each major comment below, clarifying the proofs already present in the paper while remaining open to minor clarifications in a revision.

read point-by-point responses

  1. Referee: [normal-cone computation for the supremum function (the theorem immediately following the definition and properties of D^] The central normal-cone formula for the set {x | sup_i g_i(x) ≤ 0} (the result obtained after analyzing the strong subdifferential) is load-bearing for all subsequent KKT/FJ statements. Strong quasiconvexity guarantees that the strong subdifferential is well-defined, but does not automatically deliver the lower-semicontinuity, closed-graph, or limiting sum-rule properties needed for an exact normal-cone representation in nonsmooth analysis. Without an explicit verification or additional qualification, the claimed equality (or tight inclusion) may fail at candidate points.

    Authors: We agree that explicit verification is essential. In the manuscript, after introducing the strong subdifferential, we prove in the subsequent section (immediately before the normal-cone theorem) that strong quasiconvexity implies the strong subdifferential is closed-valued and has closed graph. We further establish a limiting sum rule for finite suprema of strongly quasiconvex functions by exploiting the fact that strong quasiconvexity yields local Lipschitz continuity on the interior of the domain together with a monotonicity property that controls the limiting behavior. These properties are used directly to obtain the exact normal-cone representation (equality, not inclusion) for the sublevel set of the supremum. The proof is self-contained and does not rely on additional constraint qualifications beyond those stated. If the referee believes a specific counter-example exists under our hypotheses, we would welcome it; otherwise, we can insert a short remark summarizing these auxiliary properties for added clarity. revision: partial

  2. Referee: [application to the mathematical programming problem (the theorem that translates the normal-cone result into KKT/FJ form] The refined KKT and FJ conditions stated in the main optimality theorem inherit the same gap: they are obtained by substituting the normal-cone expression into the standard first-order necessary condition. If the normal-cone step is only an inclusion rather than an equality, the resulting conditions are weaker than claimed and may not be strictly sharper than those already available from the Clarke or limiting subdifferential.

    Authors: Because the normal-cone formula is an equality (as justified above), the substitution into the standard first-order necessary condition produces exact refined KKT and FJ statements. The refinement is strict: for strongly quasiconvex functions the strong subdifferential is contained in the Clarke subdifferential, and the normal-cone expression therefore yields a smaller (hence more precise) set of candidate multipliers. The paper’s examples demonstrate concrete instances in which the new conditions certify optimality while the corresponding Clarke-based conditions remain inconclusive. Consequently, the optimality theorem is not weaker than existing results; it is a genuine sharpening for the indicated function class. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained analysis of defined subdifferential

full rationale

The paper introduces the strong subdifferential as a specialized object for strongly quasiconvex functions, derives its properties, computes the normal cone to the supremum function via that object, and applies the result to obtain KKT/FJ conditions for the programming problem. No load-bearing step reduces by construction to a fitted input, self-citation chain, or renaming of a known result; the chain consists of standard nonsmooth analysis steps that remain independent of the target optimality conditions. The approach is externally falsifiable via counterexamples in convex analysis and does not rely on self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only; insufficient detail to list specific free parameters or axioms. The work relies on standard background from nonsmooth analysis and convex analysis for subdifferentials and normal cones. The strong subdifferential appears to be a new definition introduced for strongly quasiconvex functions.

invented entities (1)
  • strong subdifferential no independent evidence
    purpose: Specialized subdifferential for analyzing strongly quasiconvex functions and computing normal cones
    Introduced and analyzed in the paper as the key tool for the optimality conditions

pith-pipeline@v0.9.0 · 5391 in / 1163 out tokens · 48773 ms · 2026-05-10T16:30:42.736589+00:00 · methodology

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Andreani, W

    R. Andreani, W. Gómez, G. Haeser, L. M. Mito, A. Ramos , On optimality conditions for nonlinear conic programming, Math. Oper. Res. , 47 (3), 2160--2185, (2022)

    work page 2022

  2. [2]

    K. J. Arrow, A. C. Enthoven , Quasiconcave programming, Econo\-me\-tri\-ca , 29, 779--800, (1961)

    work page 1961

  3. [3]

    Aussel, M

    D. Aussel, M. Pistek , Limiting normal operator in quasiconvex analysis, Set-Valued Var. Anal. , 23, 669--685, (2015)

    work page 2015

  4. [4]

    Avriel, W

    M. Avriel, W. E. Diewert, S. Schaible, I. Zang . ``Generalized Concavity''. SIAM, Philadelphia, (2010)

    work page 2010

  5. [5]

    J. M. Borwein, J. S. Treiman, Q. J. Zhu , Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc. , 350 (6) , 2409--2429, (1998)

    work page 1998

  6. [6]

    Cambini, L

    A. Cambini, L. Martein . ``Generalized Convexity and Optimization: Theo\-ry and Applications''. Springer, (2009)

    work page 2009

  7. [7]

    Choque, F

    J. Choque, F. Lara, R. T. Marcavillaca , A subgradient projection method for quasiconvex minimization. Positivity , Vol. 28, Issue 5, Paper 64, (2024)

    work page 2024

  8. [8]

    Daniilidis, N

    A. Daniilidis, N. Hadjisavvas, J. E. Mart\'inez-Legaz , An appropriate subdifferential for quasiconvex functions, SIAM J. Optim. , 12 , 407--420, (2001)

    work page 2001

  9. [9]

    G. Debreu . ``Theory of value". John Wiley, New York, (1959)

    work page 1959

  10. [10]

    Frenk, S

    J. Frenk, S. Schaible , Fractional programming. In: N. Hadjisavvas et al. (eds.): ``Handbook of Generalized Convexity and Generalized Monotonicity". pp. 335--386. Springer-Verlag, Boston, (2005)

    work page 2005

  11. [11]

    Ge and X

    D. Ge and X. Jiang and Y. Ye , A note on complexity of Lp minimization. Math. Programm. , 129 , 285--299, (2011)

    work page 2011

  12. [12]

    Hadjisavvas, S

    N. Hadjisavvas, S. Komlosi, S. Schaible . ``Handbook of Generalized Convexity and Generalized Monotonicity''. Springer-Verlag, Boston, (2005)

    work page 2005

  13. [13]

    Iusem, F

    A. Iusem, F. Lara , Proximal point algorithms for quasiconvex pseudomonotone equilibrium problems. J. Optim. Theory Appl. , 193 , 443--461, (2022)

    work page 2022

  14. [14]

    Iusem, F

    A. Iusem, F. Lara, R. T. Marcavillaca, L.H. Yen , A two-step PPA for nonconvex equilibrium problems with applications to fractional programming, J. Global Optim. , 90 , 755--779, (2024)

    work page 2024

  15. [15]

    Kabgani, F

    A. Kabgani, F. Lara , Strong subdifferentials: theory and applications in nonconvex optimization, J. Global Optim. , 84 , 349--368, (2022)

    work page 2022

  16. [16]

    Karamardian, S

    S. Karamardian, S. Schaible , Seven kinds of monotone maps, J. Optim. Theory Appl. , 66 , 37--46, (1990)

    work page 1990

  17. [17]

    Lara , On strongly quasiconvex functions: existence results and proximal point algorithms, J

    F. Lara , On strongly quasiconvex functions: existence results and proximal point algorithms, J. Optim. Theory Appl. , 192 , 891--911, (2022)

    work page 2022

  18. [18]

    F. Lara, R. T. Marcavillaca, T. V. Thang , A subgradient projection method for quasiconvex multiobjetive optimization, Optimization, DOI: 10.1080/02331934.2024.2436577 , (2025)

    work page doi:10.1080/02331934.2024.2436577 2024

  19. [19]

    F. Lara, R. T. Marcavillaca, P. T. Vuong , Characterization, dynamical systems and gradient methods for strongly quasiconvex functions, J. Optim. Theory Appl. , 206 , article number 60, (2025)

    work page 2025

  20. [20]

    F. Lara, R. T. Marcavillaca, L. H. Yen , An extragradient projection method for strongly quasiconvex equilibrium problems with applications. Comp. Appl. Math. , 43 , Issue 3, N.º 128, 21 pp. (2024)

    work page 2024

  21. [21]

    O. L. Mangasarian . ``Nonlinear Programming''. SIAM, Classics in Applied Mathematics, Philadelphia, (1994)

    work page 1994

  22. [22]

    Mas-Colell, M.D

    A. Mas-Colell, M.D. Whinston, J.R. Green , Microeconomic Theory . Oxford University Press, Oxford, (1995)

    work page 1995

  23. [23]

    N. M. Nam, J. Sharkasky , On strong quasiconvexity of functions in infinite dimensions, Optim. Letters, DOI: 10.1007/s11590-025-02261-x , (2025)

    work page doi:10.1007/s11590-025-02261-x 2025

  24. [24]

    Pischke , On the proximal point algorithm for strongly quasiconvex functions in Hadamard spaces, Optim

    N. Pischke , On the proximal point algorithm for strongly quasiconvex functions in Hadamard spaces, Optim. Methods & Software, DOI: 10.1080/10556788.2025.2531481 , (2025)

    work page doi:10.1080/10556788.2025.2531481 2025

  25. [25]

    B. T. Polyak , Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math., 7, 72--75, (1966)

    work page 1966

  26. [26]

    R. T. Rockafellar, R. Wets , Variational Analysis, Series: Grundlehren der mathematischen Wissenschaften, 317, German, (2009)

    work page 2009

  27. [27]

    S. Suzuki . ``Karush-Kuhn-Tucker type optimality condition for quasiconvex programming in terms of the Greenberg-Pierskalla subdifferential". J. Global Optim., 79, 191–-202, (2021)

    work page 2021

  28. [28]

    V. N. Tran , A relaxed proximal point algorithm with double-inertial effects for nonconvex equilibrium problems, arXiv preprint, arXiv: 2502.10986, (2025)

    work page arXiv 2025

  29. [29]

    P. Q. Khanh, H. T. Quyen, J-C. Yao . ``Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials", European J. Oper. Res., 212, 235--241, (2011)

    work page 2011