Pith. sign in

REVIEW 3 major objections 4 minor 106 references

Directional first order approach for a class of bilevel programs

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read For a class of bilevel problems with nonconvex lower level, the lower-level constraint is locally equivalent to a directional first-order condition, yielding directional KKT necessary conditions at local minimizers.

desk verdict Directional reformulation is a good idea and Theorem 3.1 is clean, but the central sufficient condition for its key hypothesis is false; the paper currently overstates its reach. read the letter →

arxiv 2602.22573 v2 pith:5YF3KDQN submitted 2026-02-26 math.OC

Kuang Bai , Wei Yao , Jane J. Ye , Jin Zhang This is my paper
classification math.OC MSC 90C3090C4649J52
keywords bilevelprogrammingnonconvexlower-levelprogramdirectionalfirst-orderapproachKKTconditionsmetricsubregularitynormal-conegraphreformulationprincipal-agentproblemsset-constrainedoptimization
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 targets bilevel programs whose lower-level feasible set does not depend on the upper variable, but where the lower-level objective may be nonconvex. The classical first-order approach replaces the lower-level solution constraint by a stationarity condition, and is only known to be sound under lower-level convexity; value-function reformulations avoid that convexity but produce nonsmooth programs with degenerate constraints. The paper shows that, under two directional assumptions — a single-valued directional localization of the first-order stationary map and directional inner semicontinuity of the true solution map — the constraint y∈S(x) is exactly equivalent to the regular-normal-cone stationarity condition (y,−∇_y f(x,y))∈gph N̂_Y on a directional neighborhood, a cone-like wedge around a chosen direction. On that basis it derives a directional KKT necessary condition for the resulting single-level set-constrained reformulation, provided a nonzero critical direction exists. A modified principal-agent example shows a case where the classical first-order approach fails but the directional version still yields a valid necessary condition.

What carries the argument

The load-bearing objects are two set-valued maps: the true lower-level solution map S(x)=argmin_{y∈Y} f(x,y) and the first-order stationary map S_FO(x)={y∈Y | 0∈∇_y f(x,y)+N̂_Y(y)}. The graph of the regular normal cone N̂_Y, written gph N̂_Y, converts the generalized equation into the set constraint (y,−∇_y f(x,y))∈gph N̂_Y. The proof mechanism is the single-valued directional localization: if S_FO(x)∩(ȳ+εB) is a singleton for every x in a directional neighborhood and S has nonempty values there, then the two maps agree, making the first-order condition an exact surrogate for the lower-level constraint. The necessary conditions then come from directional variational analysis applied to the s

What would settle it

Exhibit a lower-level program satisfying the paper's localization and inner-semicontinuity assumptions in which S_FO(x) contains a point that is not a true lower-level minimizer arbitrarily close to (x̄,ȳ) along the chosen direction; computing S and S_FO for that instance settles Theorem 3.1. Alternatively, a local minimizer satisfying every hypothesis of Theorem 4.3 but admitting no directional KKT multipliers would falsify the main necessary condition.

Watch

Extended reading notes

Core claim

The central claim is that, for bilevel programs with lower-level feasible set independent of the upper variable, the constraint y∈S(x) coincides with the first-order stationarity constraint (y,−∇_y f(x,y))∈gph N̂_Y over a directional neighborhood of a candidate local minimizer, provided the stationary map S_FO(x)={y∈Y : 0∈∇_y f(x,y)+N̂_Y(y)} has a single-valued directional localization near (x̄,ȳ) in the direction u and the solution map S is directionally inner semicontinuous there. This local equivalence (Theorem 3.1) makes the bilevel program locally equivalent to a set-constrained program restricted to a directional neighborhood. Then, under directional metric subregularity of the constra

Load-bearing premise

The argument would collapse if, near the point in question, the lower-level problem had more than one competing stationary point along the chosen direction, or if a minimizer had no nonzero direction along which the upper objective is unchanged to first order; the paper acknowledges the second limitation explicitly.

Editorial extensions

If this is right

  • Nonconvex lower-level programs satisfying the directional localization and inner-semicontinuity conditions can be reformulated as single-level set-constrained programs, bypassing value functions and their nonsmooth constraints.
  • Setting the direction u=0 recovers the classical first-order/KKT approach, so the directional theory contains the convex lower-level case as a special case.
  • The directional KKT condition uses a directional limiting normal cone that is generally smaller than the classical one, so the necessary conditions are sharper than the standard M-stationary condition.
  • Box constraints Y=∏[a_i,b_i], common in principal-agent models, admit explicit localization tests (second-order sufficient condition or vanishing critical cone), making the reformulation checkable in applications.
  • The modified principal-agent example shows a local minimizer at which the classical first-order approach finds no minimizer on the stationarity surface, while the directional reformulation yields a local solution and directional KKT multipliers.

Reading between the lines

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

  • If the directional equivalence can be shown to hold on a union of directional neighborhoods covering all relevant solutions, the local theorem could be assembled into a global single-level reformulation for nonconvex bilevel programs — a step the paper does not take.
  • When S(x) consists of finitely many solution branches, the directional inner-semicontinuity condition suggests a computational strategy: enumerate the stationary branches, pick the branch containing the candidate solution, and choose a direction u that separates it from competing branches; each branch then gives a tractable directional SCOP.
  • The main theorem is silent when no nonzero critical direction exists; extending the analysis to second-order or higher-order critical directions could cover additional local minimizers in principal-agent-type examples.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies a class of bilevel programs in which the lower-level feasible set Y is independent of the upper-level variable. It proposes replacing the lower-level solution constraint y∈S(x) by the first-order stationarity condition (y,−∇_y f(x,y))∈gph \hat N_Y, but only on a directional neighborhood. Theorem 3.1 gives an equivalence between y∈S(x) and the first-order condition over such a neighborhood, assuming (i) a single-valued directional localization of the stationary map S_FO and (ii) directional inner semicontinuity of the solution map S. Theorem 3.2 provides sufficient conditions for (i), including for polyhedral convex Y a second-order sufficient optimality condition. Section 4 derives directional KKT conditions for the resulting set-constrained problem using directional metric subregularity and results from Ouyang–Ye–Zhang; Theorem 4.3 is the main necessary-optimality statement. Example 4.1 is a modified Mirrlees example intended to show that the directional first-order approach can succeed where the classical approach fails. Section 5 explicitly acknowledges that the approach requires existence of a nonzero critical direction.

Significance. If correct, the directional reformulation would provide a genuinely useful way to handle nonconvex lower-level problems without introducing value functions, and it is motivated by principal–agent problems with box constraints. The paper's central equivalence framework (Theorem 3.1) is logically sound under the stated assumptions, and the use of directional variational analysis in Section 4 is systematic and appropriately builds on prior work. The authors also deserve credit for explicitly flagging the critical-direction limitation in Section 5. However, the paper's main sufficient-condition toolkit contains a false theorem: Theorem 3.2(iv-a) is contradicted by a simple polyhedral box example, and its corollary Theorem 3.3(a) is therefore also false. Since the principal-agent motivation leans on this box-constraint case, the current version substantially overstates what is established.

major comments (3)
  1. [Section 3.1, Theorem 3.2(iv-a)] The claimed implication is false. Take Y=[0,1]^2, (x̄,ȳ)=(0,0), and f(x,y)=1/2(y_1^2+4y_1y_2+y_2^2)-x_1y_1-x_2y_2. Then ∇_y f(0,0)=0, so the critical cone K=R_+^2 and for every nonzero w∈K, w^T∇²_yy f w = w_1^2+4w_1w_2+w_2^2>0, so the SOSC in (iv-a) holds. Yet for x=(ε,ε) with 0<ε<1, the inclusion 0∈∇_y f(x,y)+N_Y(y) has three distinct solutions near ȳ: (ε,0), (0,ε), and (ε/3,ε/3). Thus S_FO has no single-valued localization at (0,0), contradicting the theorem. The proof's step that condition (6) of [10, Theorem 2E.6] is implied by SOSC on K is incorrect: (6) quantifies over span(K) and includes an orthogonality condition to lin(K), while SOSC on K only controls directions in K, not in span(K)\K. In the example span(K)=R^2 and the Hessian is indefinite on R^2, so (6) fails despite SOSC on K holding.
  2. [Section 3.1, Theorem 3.3(a)] The box-constrained specialization in Theorem 3.3(a) inherits the error in Theorem 3.2(iv-a) and is false for the same counterexample. Since Section 1.1 motivates principal–agent problems precisely through box constraints, this is not a peripheral issue. The authors should either replace (iv-a) and 3.3(a) with a correct sufficient condition—for instance a strong-regularity condition in the spirit of Theorem 3.4, or strong monotonicity on Y−Y as in Theorem 3.2(iii)—or state and prove the additional hypothesis (e.g. control on span(K)) under which the localization actually holds.
  3. [Section 4, Example 4.1] The only concrete demonstration of the new approach rests on several unproved assertions: “one can verify that all bilevel feasible points in a small neighborhood U0 of (¯x,¯y) satisfy y≥y0”; the claim that S(¯x)={−0.957,+0.957}; and the inequality ∇_x f(¯x,0.957)u < ∇_x f(¯x,−0.957)u, where no direction u is specified. Because this example is used to show that the directional KKT conditions are non-vacuous and to compute ν̄, these claims need either detailed verification or a precise reference to the Mirrlees analysis. As written, the example is more an illustration than a proof.
minor comments (4)
  1. [Section 3.1, proof of Theorem 3.2(iv-b)] The sentence “can be implied by, hence is equivalent to” is logically unclear. Since the theorem only needs sufficiency, the authors should state exactly whether (7) is equivalent to condition (5) or merely implies it, and adjust the surrounding text accordingly.
  2. [Abstract and Introduction] The abstract mentions “M-stationary condition,” but the paper does not define this term; the theorems state conditions with limiting normal cones. Please align the terminology with the body or define M-stationarity.
  3. [Throughout] There are several typos and formatting issues: “W ords” in the keywords, “Linear Indepence” in Example 4.1, “directional neighborhood of of the origin” in Definition 2.4, and “For anx,S FO(x)” in the introduction. These should be cleaned up.
  4. [Example 4.1] The computed value ν̄ = 1/2 e^{(y0+1)^2} ≈ 23.1 is asserted without derivation. Please show the substitution into the KKT system so the reader can verify the directional stationarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the reformulation equivalence is proved from stated localization/semicontinuity hypotheses; same-author citations are to published theorems, not the paper's own conclusion.

full rationale

Verdict: no significant circularity (score 2 only because the manuscript contains several same-author citations; none of them force the conclusion). The central derivation chain is: Theorem 3.1 claims that under (i) single-valued directional localization of the stationary map S_FO and (ii) S(x)∩(ȳ+ε_yB)≠∅ on the directional neighborhood, the constraints y∈S(x) and (y,−∇_yf(x,y))∈gph bN_Y coincide there. The proof is an in-paper deduction: every solution is stationary, so S(x)∩ball ⊆ S_FO(x)∩ball; single-valuedness plus nonemptiness gives the equality. Condition (i) is a hypothesis about S_FO alone, not a restatement of the equivalence, so the theorem is not self-definitional, and no quantity is fitted to make the reformulation hold. Verifiable sufficient conditions for (i) are argued in place, citing standard external theorems (Dontchev–Rockafellar [10, Thms 2E.6/2E.8/2F.7] and [9]) whose assumptions do not contain the target result. Section 4 imports the directional KKT multiplier existence from [30, Prop 4.1] (Ouyang–Ye–Zhang, overlapping authorship) via Proposition 4.2; this is a published, peer-reviewed proposition with stated assumptions not including the bilevel conclusion, so under the review rules it is independent support, not a self-citation chain. [1] (Bai–Ye) supplies definitions (directional inner semicontinuity) and is likewise background, with the sufficient conditions proved in the paper (Proposition 3.1). The paper explicitly concedes in Section 5 that 'the application of our approach is limited by the assumption of existence of a nonzero critical direction'; that is an applicability limitation, not a disguised conclusion, and the critical direction (u,v) is an assumed object with ∇F(x̄,ȳ)(u,v)=0, not an output of the theory. Finally, the skeptic's counterexample to Theorem 3.2(iv-a) (SOSC on the critical cone claimed to imply a Lipschitz single-valued localization) attacks the validity of an external-theorem application; if correct it falsifies a sufficient condition used to verify Theorem 3.1(i), but a false implication is a correctness risk, not constructional circularity, and is therefore not scored here. Honest non-finding: the derivation reduces to stated hypotheses and standard published theorems, not to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on problem-class assumptions (smoothness, nonempty solution sets) and on structural assumptions (single-valued localization, directional inner semicontinuity, existence of a critical direction, directional metric subregularity) that are stated in the theorems. No numerical parameters are fitted; the direction u and v are part of the analysis, not free parameters.

assumptions (7)
  • domain assumption S(x) (lower-level solution set) is nonempty for every x
    Stated in Section 1 after problem (BP); ensures the bilevel program is well-defined and the solution map has values.
  • domain assumption F, f, G are sufficiently smooth
    Stated in Section 1; needed for gradients, Jacobians, and variational analysis calculus (normal cones, directional derivatives).
  • ad hoc to paper The first-order stationary map S_FO has a single-valued localization over the directional neighborhood (Theorem 3.1(i))
    Central structural assumption: the stationary condition 0∈∇_y f+bN_Y must select a unique branch near (¯x,¯y) in direction u. Without it, the first-order condition can include spurious stationary points not solving the lower level.
  • domain assumption The solution map S intersects a fixed neighborhood of ¯y on the directional neighborhood (Theorem 3.1(ii)) or is directionally inner semicontinuous
    Ensures the true solution branch passes through (¯x,¯y) along direction u; needed to identify S(x) with S_FO(x) on the relevant set. This is essentially a calmness/branch-continuity condition on the lower-level solutions.
  • ad hoc to paper There exists a nonzero critical direction (u,v)∈L(¯x,¯y) with ∇F(¯x,¯y)(u,v)=0 (Theorem 4.3)
    Needed to apply the directional KKT theory. The authors concede in Section 5 that the application of the approach is limited by existence of such a direction.
  • domain assumption Directional metric subregularity / NNAMCQ / affine+polyhedral structure of the constraint system (Section 4, Proposition 2.1)
    Constraint qualification required to convert directional local optimality into KKT-type multiplier conditions; various sufficient conditions are listed in Proposition 2.1.
  • domain assumption Inf-compactness (Proposition 3.1) when used to establish directional inner semi-continuity
    Used in Propositions 3.1-3.2 and Theorems 4.6-4.7 to guarantee the solution map has convergent sequences; Example 3.1 shows failure without it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Directional first order approach for a class of bilevel programs." pith.science (2026). https://pith.science/paper/5YF3KDQN

@misc{pith2026260222573,
  author       = {Pith},
  title        = {Pith review of: Directional first order approach for a class of bilevel programs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YF3KDQN}},
  note         = {Machine review of arXiv:2602.22573}
}
read the original abstract

In this paper, we study a class of bilevel optimization program, where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order reformulation of a bilevel program requires the convexity of the lower level program while reformulations involving the value function result in difficult optimization problems. In this paper we propose a directional first order approach which does not require convexity of the lower level program. First, we propose some conditions under which the lower level program can be equivalently characterized by its first order condition over a directional neighborhood around the local optimal condition. Next we give some conditions under which the classical first order optimality condition in the form of M-stationary condition still holds as a necessary optimality condition for the first order reformulation of the bilevel program even when the lower level program is nonconvex.

Figures

Figures reproduced from arXiv: 2602.22573 by the authors.

Figure 1
Figure 1. Mirrlees example In [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

106 extracted references · 4 linked inside Pith

  1. [1]

    J. W. Alexander , A proof and extension of the Jordan-Brouwer separation theorem , Trans. Amer. Math. Soc., 23(1992), pp. 333-349

    1992

  2. [2]

    Bai and J

    K. Bai and J. J. Ye , Directional necessary optimality conditions for bilevel programs , Math. Oper. Res., 47(2022), pp. 1169-1191

    2022

  3. [3]

    Bai and J

    K. Bai and J. J. Ye , Directional subdifferential of the value function , Commun. Optim. Theory, 35(2023), pp. 1-36

    2023

  4. [4]

    Bai and J

    K. Bai and J. J. Ye , Directional derivative of the value function for parametric set-constrained optimization problems , preprint, arXiv:2311.03604

    arXiv

  5. [5]

    K. Bai, J. J. Ye and J. Zhang , Directional quasi-/pseudo-normality as sufficient conditions for metric subregularity , SIAM J. Optim., 29(2019), pp. 2625-2649

    2019

  6. [6]

    B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer , Non-Linear Parametric Optimization , Birkh \"a user, Basel, 1982

    1982

  7. [7]

    E. M. Bednarczuk, L. I. Minchenko and K. E. Rutkowski , On Lipschitz continuity of a class of set-valued mappings , to appear in Optimization

  8. [8]

    Benko, Matúš, Michal Červinka, and Tim Hoheisel , Sufficient conditions for metric subregularity of constraint systems with applications to disjunctive and ortho-disjunctive programs , Set-Valued and Variational Analysis 2(2021), pp. 1-35

    2021

Show all 106 references
  1. [9]

    Bector, S

    C.R. Bector, S. Chandra, and J.Dutta , Principles of Optimization Theory . Alpha Science International, Harrow, UK, 2005

    2005

  2. [10]

    Benko, H

    M. Benko, H. Gfrerer and J. V. Outrata , Calculus for directional limiting normal cones and subdifferentials , Set-Valued Var. Anal., 27(2019), pp. 713-745

    2019

  3. [11]

    Benko, H

    M. Benko, H. Gfrerer, J. J. Ye, J. Zhang and J. C. Zhou , Second-order optimality conditions for general nonconvex optimization problems and variational analysis of disjunctive systems , SIAM J. Optim., 33(2023), pp. 2625-2653

    2023

  4. [12]

    D. P. Bertsekas , Nonlinear Programming, 2nd ed. , Athena Scientific, Belmont, MA, 1999

    1999

  5. [13]

    Bertsekas, A.E

    D.P. Bertsekas, A.E. Ozdaglar , Pseudonormality and a Lagrange multiplier theory for constrained optimization , J. Optim. Theory Appl., 114(2002), pp 287-343

    2002

  6. [14]

    Bertsekas, and A.Nedi c , A.E

    D.P. Bertsekas, and A.Nedi c , A.E. Ozdaglar , Convex Analysis and Optimization , Belmont, MA, Athena Scientific, 2003

    2003

  7. [15]

    J. F. Bonnans and A. Shapiro , Perturbation analysis of optimization problems , Springer, New York, 2000

    2000

  8. [16]

    J. M. Borwein and D. Preiss , A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions , Trans. AMS., 303(1987), pp 517-527

    1987

  9. [17]

    Bracken and J

    J. Bracken and J. McGill , Mathematical programs with optimization problems in the constraints , Oper. Res., 21(1973), pp. 37-44

    1973

  10. [18]

    F. H. Clarke , Optimization and nonsmooth analysis , Classics Appl. Math. 5, SIAM, Philadelphia, PA, 1990

    1990

  11. [19]

    F. H. Clarke and M. Darrough , Optimal Incentive schemes: existence and characterization , Econ. Lett., 5 (1980), pp. 305-310

    1980

  12. [20]

    J. R. Conlon , Two new conditions supporting the first‐order approach to multisignal principal–agent problems , Econometrica, 77(2009), pp. 249-278

    2009

  13. [21]

    J. P. C\^ot\'e, P. Marcotte and G. Savard , A bilevel modeling approach to pricing and fare optimisatioin in airline industry . Jounal of managing and pricing management, 2(2003), pp. 23-36

    2003

  14. [22]

    Dempe , Foundations of bilevel programming , Kluwer Academic, Dordrecht, 2002

    S. Dempe , Foundations of bilevel programming , Kluwer Academic, Dordrecht, 2002

    2002

  15. [23]

    Dempe , A necessary and sufficient optimality condition for bilevlel programming problems

    S. Dempe , A necessary and sufficient optimality condition for bilevlel programming problems . Optimization, 25(1992), pp. 341-354

    1992

  16. [24]

    Dempe and J

    S. Dempe and J. Dutta , Is bilevel programming a special case of mathematical program with equlibrium constraints? Math. Program., 131(2012), pp. 37-48

    2012

  17. [25]

    Dempe, J

    S. Dempe, J. Dutta and B. S. Mordukhovich , New necessary optimality conditions in optimistic bilevel programming , Optimization, 56(2007), pp. 577-604

    2007

  18. [26]

    Dempe and A

    S. Dempe and A. B. Zemkoho , The generalized Magasarian-Fromovitz constraint qualification and optimality conditions for bilevel programs , J. Optim. Theory Appl., 148(2011), pp. 46-68

    2011

  19. [27]

    A. L. Dontchev and R. T. Rockafellar , Characterizations of strong regularity for variational inequalities over polyhedral convex sets , SIAM J. Optim., 6(1996), pp. 1087-1105

    1996

  20. [28]

    A. L. Dontchev and R. T. Rockafellar , Implicit functions and solution mappings , Springer, New York, 2009

    2009

  21. [29]

    Fernández-Blanco, J

    R. Fernández-Blanco, J. M. Arroyo and N. Alguacil , On the solution of revenue-and network-constrained day-ahead market clearing under marginal pricing—Part I: An exact bilevel programming approach. , IEEE T. Power Syst., 1(2016), pp. 208-219

    2016

  22. [30]

    L. Gao, J. J. Ye, H. A. Yin, S. Z. Zeng and J. Zhang , Value function based difference-of-convex algorithm for bilevel hyperparameter selection problems , ICML (2022), pp. 7164-7182

    2022

  23. [31]

    Gauvin and F

    J. Gauvin and F. Dubeau , Differential properties of the marginal function in mathematical programming . Math. Programming Stud., 19(1982), pp. 101-119

    1982

  24. [32]

    Gfrerer , On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs , SIAM J

    H. Gfrerer , On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs , SIAM J. Optim., 23(2013), pp. 632-665

    2013

  25. [33]

    Gfrerer , Second-order optimality conditions for scalar and vector optimization problems in Banach spaces , SIAM J

    H. Gfrerer , Second-order optimality conditions for scalar and vector optimization problems in Banach spaces , SIAM J. Control Optim., 45 (2006), pp. 972-997

    2006

  26. [34]

    Gfrerer , On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs

    H. Gfrerer , On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs . Set-Valued Var. Anal., 21(2013), pp. 151-176

    2013

  27. [35]

    Gfrerer , On metric pseudo-(sub) regularity of multifunctions and optimality conditions for degenerated mathematical programs , Set-Valued Var

    H. Gfrerer , On metric pseudo-(sub) regularity of multifunctions and optimality conditions for degenerated mathematical programs , Set-Valued Var. Anal., 22(2014), pp. 79-115

    2014

  28. [36]

    Gfrerer , Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints , SIAM J

    H. Gfrerer , Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints , SIAM J. Optim., 24 (2014), pp. 898-931

    2014

  29. [37]

    Gfrerer and D

    H. Gfrerer and D. Klatte , Lipschitz and H\"older stability of optimization problems and generalized equations , Math. Program., 158(2016), pp. 35-75

    2016

  30. [38]

    Gfrerer and B

    H. Gfrerer and B. Mordukhovich , Robinson stability of parametric constraint systems via variational analysis , SIAM J. Optim. 27(2017), pp. 438-465

    2017

  31. [39]

    Gfrerer and J.J

    H. Gfrerer and J.J. Ye , New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis , SIAM J. Optim., 27 (2017), pp. 842-865

    2017

  32. [40]

    Gfrerer and J

    H. Gfrerer and J. J. Ye , New sharp necessary optimality conditions for mathematical programs with equilibrium constraints , Set-Valued Var. Anal., 28(2020), pp. 395-426

    2020

  33. [41]

    Gfrerer, J

    H. Gfrerer, J. J. Ye and J. Zhou , Second-order optimality condition for nonconvex set-constrained optimization problems , preprint arXiv (1911.04076)

    1911 arXiv

  34. [42]

    Ginchev and B

    I. Ginchev and B. S. Mordukhovich , On directionally dependent subdifferentials , C.R. Bulg. Acad. Sci., 64(2011), pp. 497-508

    2011

  35. [43]

    B. M. Glover and B. D. Craven , A Fritz John optimality condition using the approximate subdifferential , J. Optim. Theory Appl., 82 (1994), pp. 253–265

    1994

  36. [44]

    Guillemin and A

    V. Guillemin and A. Pollack , Differential topology , AMS Chelsea Publishing, 2010

    2010

  37. [45]

    L. Guo, J. J. Ye and J. Zhang , Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity , SIAM J. Optim., 23(2013), pp. 2295-2319

    2013

  38. [46]

    L.Guo, J. J. Ye, and J. Zhang , Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity , SIAM Journal on Optimization 23.4 (2013): 2295-2319

    2013

  39. [47]

    Guo, G-H

    L. Guo, G-H. Lin, J. J. Ye, and J. Zhang , Sensitivity analysis of the value function for parametric mathematical programs with equlibrium constraints , SIAM J. Optim., 24(2014), pp. 1206-1237

    2014

  40. [48]

    Hatcher , Algebraic topology , Cambridge University Press, 2002

    A. Hatcher , Algebraic topology , Cambridge University Press, 2002

    2002

  41. [49]

    Henrion, J.V

    R. Henrion, J.V. Outrata and T. Surowiec , On regular coderivatives in parametric equilibria with non-unique multipliers , Math. Program., 136(2012), pp. 111-131

    2012

  42. [50]

    Henrion and T

    R. Henrion and T. Surowiec , On calmness conditions in convex bilevel programming , Appl. Anal., 90 (2010), pp. 951-970

    2010

  43. [51]

    M. R. Hestenes , Optimization Theory: The Finite Dimensional Case , Wiley, New York, 1975

    1975

  44. [52]

    V. T. Hieu and A. Takeda , Computing local minimizers in polynomial optimization under genericity conditions , 92(2025), pp. 909-932

    2025

  45. [53]

    Janin , Directional derivative of the marginal function in nonlinear programming , Math

    R. Janin , Directional derivative of the marginal function in nonlinear programming , Math. Program. Study, 21(1984), pp. 110--126

    1984

  46. [54]

    John , Extremum problems with inequalities as subsidiary conditions , Traces and emergence of nonlinear programming (pp

    F. John , Extremum problems with inequalities as subsidiary conditions , Traces and emergence of nonlinear programming (pp. 197-215). Springer Basel, 2014

    2014

  47. [55]

    Kalashnikov, S

    V. Kalashnikov, S. Dempe, G. Pérez-Valdés, N. Kalashnykova and J. Camacho-Vallejo , Bilevel programming and applications , Math. Probl. Eng., 2015

    2015

  48. [56]

    Ko cvara and J

    M. Ko cvara and J. V. Outrata , Effective reformulations of the truss topology design problem. Opt. Eng., 7(2006), pp. 201-219

    2006

  49. [57]

    R. Z. Ke, W. Yao, J. J. Ye and J. Zhang , Generic property of the partial calmness condition for bilevel programming problems , SIAM J. Optim., 32(2022), pp. 604-634

    2022

  50. [58]

    Kirkegaard , Moral hazard and the spanning condition without the first-order approach , Games

    R. Kirkegaard , Moral hazard and the spanning condition without the first-order approach , Games. Econ. Behav., 102 (2017), pp. 373–387

    2017

  51. [59]

    Kunapuli, K

    G. Kunapuli, K. P. Bennet, J. Hu and J. S. Pang , Classification model selection via bilevel programming , Optim. methods Softw., 23(2008), pp. 475-489

    2008

  52. [60]

    Lachhwani and A

    K. Lachhwani and A. Dwivedi , Bi-level and multi-Level programming problems: taxonomy of literature review and research issues , Arch. Compt. Method E., 4(2018), pp. 847-877

    2018

  53. [61]

    R. Liu, L. Ma, X. Yuan, S. Zeng and J. Zhang , Bilevel Integrative Optimization for ill-posed inverse problems , preprint, arXiv:1907.03083, 2019

    1907 arXiv

  54. [62]

    Lang , Fundamentals of differential geometry , Springer-Verlag, New York, 1999

    S. Lang , Fundamentals of differential geometry , Springer-Verlag, New York, 1999

    1999

  55. [63]

    P. Long, B. Wang and X. Yang , Calculus of directional subdifferentials and coderivatives in Banach spaces , Positivity, 21(2017), pp. 223-254

    2017

  56. [64]

    D. G. Luenberger , Optimization by vector space methods , Wiley and Sons, New York, 1969

    1969

  57. [65]

    Z. Q. Luo, J. S. Pang and D. Ralph , Mathematical programs with equilibrium constraints , Cambridge University Press, 1996

    1996

  58. [66]

    Matou e k , Lectures on discrete geometry , Springer-Verlag, New York, 2002

    J. Matou e k , Lectures on discrete geometry , Springer-Verlag, New York, 2002

    2002

  59. [67]

    O. L. Mangasarian, S. Fromovitz , The Fritz John necessary optimality conditions in the presence of equality and inequality constraints , J. Math. Anal. Appl., 17(1967), pp. 37-47

    1967

  60. [68]

    Minchenko and S

    L. Minchenko and S. Stakhovski , Parametric nonlinear programming problems under the relaxed constant rank condition , SIAM J. Optim., 21(2011), pp. 314-332

    2011

  61. [69]

    Mirrlees , The optimal structure of incentives and authority within an organization , Bell j

    J. Mirrlees , The optimal structure of incentives and authority within an organization , Bell j. econ., 7(1976), pp. 105-131

    1976

  62. [70]

    Mirrlees , The theory of moral hazard and unobservable behaviour: Part I , Rev

    J. Mirrlees , The theory of moral hazard and unobservable behaviour: Part I , Rev. Econom. Stud., 66(1999), pp. 3–22

    1999

  63. [71]

    Mirrlees , The theory of moral hazard and unobservable behaviour-Part I

    J. Mirrlees , The theory of moral hazard and unobservable behaviour-Part I . Rev. Econom. Stud. 66(1999), pp. 3-22

    1999

  64. [72]

    B. S. Mordukhovich , Sensitivity analysis in nonsmooth analysis , in Theoretical Aspects of Industrial Design, edited by D.A.Field and V.Komkov. SIAM Proc. Appl. Math. 58(1992), 32-46, Philadelphia, Pennsylvania

    1992

  65. [73]

    B. S. Mordukhovich , Variational analysis and generalized differentiation I. Basic theory , Springer-Verlag, Berlin, 2006

    2006

  66. [74]

    B. S. Mordukhovich, N. M. Nam and N. D. Yen , Subgradients of marginal functions in parametric mathematical programming , Math. Program., 116(2009), pp. 369-396

    2009

  67. [75]

    J. V. Outrata , Necessary optimality conditions for Stackelberg problems , J. Optim. Theory Appl., 76(1993), pp. 305-320

    1993

  68. [76]

    Nie , Optimality conditions and finite convergence of Lasserre’s hierarchy , Math

    J. Nie , Optimality conditions and finite convergence of Lasserre’s hierarchy , Math. Program., 146(2014), pp. 97-121

    2014

  69. [77]

    Nie , The hierarchy of local minimums in polynomial optimization , Math

    J. Nie , The hierarchy of local minimums in polynomial optimization , Math. Program., 151(2015), pp. 555-583

    2015

  70. [78]

    J. V. Outrata, M. Ko\^cvara and J. Zowe , Nonsmooth approach to optimization problems with equlibrium constraints: theory, applications and numerical results , Kluwer, Dordrecht, The Netherlands, 1998

    1998

  71. [79]

    Ouyang, J

    W. Ouyang, J. J. Ye and B. Zhang , New second-order optimality conditions for directional optimality of a general set-constrained optimization problem , SIAM J. Optim., 35(2024), pp. 1274-1299

    2024

  72. [80]

    Renner and K

    P. Renner and K. Schmedders , A polynomial optimization approach to principal–agent problems , Econometrica, 83(2015), pp. 729-769

    2015

  73. [81]

    S. M. Robinson , Some continuity properties of polyhedral multifunctions , Mathematical Programming at Oberwolfach, Math. Program. Stud., 14, Springer Berlin, 1981, pp. 206–214

    1981

  74. [82]

    S. M. Robinson , Stability theory for systems of inequalities, II : Differentiable nonlinear systems , SIAM J. Numeri. Anal., 13(1976), pp. 497-513

    1976

  75. [83]

    R. T. Rockafellar , Convex analysis , Princeton University Press, 1970

    1970

  76. [84]

    R. T. Rockafellar and R. J-B. Wets , Variational analysis , Springer, Berlin, 1998

    1998

  77. [85]

    W. P. Rogerson , The first-order approach to principal-agent problems , Econometrica, 53(1985), pp. 1357-1367

    1985

  78. [86]

    Scheel and S

    H. Scheel and S. Scholtes , Mathematical programs with complementarity constraints: stationarity, optimaltiy and sensitivity . Math. Oper. Res., 25(2000), pp. 1-22

    2000

  79. [87]

    H. V. Stackelberg , The theory of the market economy , Oxford University Press, Oxford, England, 1954 (English translation of Marktform und gleichgewicht, Springer-Verlag, Berlin, 1934)

    1954

  80. [88]

    von Stackelberg , Marktform and Gleichgewicht , Springer-Verlag, Berlin, 1934, engl

    H. von Stackelberg , Marktform and Gleichgewicht , Springer-Verlag, Berlin, 1934, engl. transl.: The theory of the market economy , Oxford University Press, Oxford, England, 1954

    1934

  81. [89]

    J. J. Ye and J. Zhang , Enhanced Karush-Kuhn-Tucker Condition and Weaker Constraint Qualification . Math. Program., 139(2013), pp. 353-381

    2013

  82. [90]

    Xu and J

    M. Xu and J. J. Ye , Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs , J. Global Optim., to appear, arXiv(1912.12372)

    1912 arXiv

  83. [91]

    J. J. Ye , Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints , SIAM J. Optim., 10(2000), pp. 943-962

    2000

  84. [92]

    J. J. Ye , Constraint qualifications and optimality conditions in bilevel optimization, Bilevel Optimization: Advances and Next Challenges , ch. 8, Springer Optimization and its Applications, 2020

    2020

  85. [93]

    J. J. Ye , Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints , SIAM J. Math. Anal. Appl.,307(2005), pp. 350-369

    2005

  86. [94]

    J. J. Ye and X. Y. Ye , Necessary optimality conditions for optimization problems with variational inequality constraints , Math. Oper. Res., 22(1997), pp. 977-997

    1997

  87. [95]

    J. J. Ye and J. Zhou , Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems , Math. Program., 171(2018), pp. 361--395

    2018

  88. [96]

    J. J. Ye, X. M. Yuan, S. Z. Zeng and J. Zhang , Difference of convex algorithms for bilevel programs with applications in hyperparameter selection , Math. Program., 198(2023), pp. 1583-1616

    2023

  89. [97]

    J. J. Ye and J. C. Zhou , Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems , Math. Program., 171(2018), pp. 361–395

    2018

  90. [98]

    J. J. Ye and D. L. Zhu , Optimality conditions for bilevel programming problems , Optimization, 33(1995), pp. 9-27

    1995

  91. [99]

    J. J. Ye and D. L. Zhu , A note on optimality conditions for bilevel programming problems , Optimization, 39(1997), pp. 361-366

    1997

  92. [100]

    J. J. Ye and D. L. Zhu , New necessary optimality conditions for bilevel programs by combining MPEC and value function approaches , SIAM J. Optim., 20(2010), pp. 1885-1905

    2010

  93. [101]

    J. J. Ye, D. L. Zhu and Q. J. Zhu , Exact penalization and necessary optimality conditions for generalized bilevel programming problems , SIAM J. Optim., 2(1997), pp. 481-507

    1997

  94. [102]

    K. Bai, J. J. Ye and J. Zhang , Directional quasi/pseudo-normality as sufficient conditions for metric subregularity , preprint

  95. [103]

    J. M. Borwein and Q. J. Zhu , Viscosity solutins and viscosity subderivatives in smooth Banach spaces with applications to metric regularity , SIAM J. Control Optim., 34(1996), pp. 1568--1591

    1996

  96. [104]

    Borwein and Q

    J.M. Borwein and Q. J. Zhu , Techniques of Variational Analysis , Springer, Vancouver, 2004

    2004

  97. [105]

    N. T. Hang, B. S. Mordukhovich and M. E. Sarabi , Second-order variational analysis in second-order cone programming , Math. Program., 2018, pp. 1--42

    2018

  98. [106]

    B. S. Mordukhovich, Y. H. Shao and Q. J. Zhu , Viscosity coderivatives and their limiting behavior in smooth Banach spaces , Positivity, 4(2000), pp. 1--39

    2000

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.