Pith. sign in

REVIEW 3 major objections 3 minor 36 references

Active-Set Identification in Noisy and Stochastic Optimization

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read The paper proves that active-set identification in constrained optimization still works when function and derivative values are noisy, provided the point is close to a local minimizer and the noise is small enough.

desk verdict Extends active-set identification to noisy function/derivative values in a genuinely useful way, but the proof of the main LP-LPEC theorem contains a false inequality that needs fixing before the central guarantee is credible. read the letter →

arxiv 2509.00888 v1 pith:VHSND6VF submitted 2025-08-31 math.OC cs.NAmath.NA

Frank E. Curtis , Daniel P. Robinson , Lara Zebiane This is my paper · ORCID
classification math.OCcs.NAmath.NA MSC 90C3090C55
keywords active-setidentificationnoisyoptimizationstochasticconstraintqualificationsLP-LPECquadraticprogrammingsubproblemKKTconditionsconstrainedneuralnetworktraining
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 asks whether a classical optimization tool—identifying which inequality constraints are exactly satisfied at a solution—still works when function and derivative values carry errors. It proves two ways to do it: one estimates Lagrange multipliers by solving a linear program with equilibrium constraints, and the other computes a primal step from a quadratic program. Under standard constraint qualifications and a second-order condition, both recover the exact active set once the iterate is close enough to a local minimizer and the noise is small enough. This matters because correct identification turns an inequality-constrained problem into an equality-constrained one, which simplifies multiplier estimation, search-direction computation, and sensitivity analysis in noisy and stochastic settings.

What carries the argument

The two load-bearing mechanisms are the noisy KKT-residual functions (25), which convert noisy Lagrange multiplier candidates into a scalar residual whose two-sided comparison to the exact residual is controlled by Lemma 4.2, and the perturbed KKT linear system (36) for the QP method, whose solution is controlled by a standard conditioning bound for linear systems (Lemma 2.1). Lemma 4.2 guarantees that, when the deterministic error bounds are small, the noisy residual lies close to the exact KKT residual, which lets the LP-LPEC threshold in (32) separate active from inactive constraints. Lemma 4.4 uses the conditioning bound to show that, under LICQ and strict complementarity, the QP step fr

What would settle it

With exact function and derivative values, run both subproblem strategies (30) and (33) on a dense grid of points within 10^-6 of a local minimizer satisfying the paper's constraint qualifications and parameter choices. The theorems assert that every estimate equals the true active set; a single mismatch at an arbitrarily close point would refute the central claim. Then repeat with increasing fixed noise levels to map where the 'sufficiently small noise' threshold sits.

Watch

Extended reading notes

Core claim

The central claim is that active-set identification, long proved for exact problem data, survives when the objective and constraints are evaluated with errors. The paper isolates two subproblem templates. The first, LP-LPEC, solves a linear program with equilibrium constraints to estimate Lagrange multipliers; Theorem 4.1 states that under MFCQ and the second-order condition, if x is close enough to a local minimizer x* and the six error bounds are small enough, its active-set estimate equals A(x*). The second solves a quadratic program for a primal step; Theorem 4.2 states that under LICQ and strict complementarity, the estimate formed from the linearized constraint values at the step likew

Load-bearing premise

The proofs require that, in a whole neighborhood of the minimizer, the approximate function and derivative values stay within fixed error bounds (the six bounds in (23)); if those bounds are exceeded, the residual inequality that separates the active set from the inactive set can break down.

Editorial extensions

If this is right

  • When the LP-LPEC conditions hold, solving the noisy LP (30) at a sufficiently close point recovers A(x*) exactly, so the estimate can be trusted inside deterministic-noise algorithms.
  • When LICQ and strict complementarity hold, the QP-based estimate from (33) also recovers A(x*) exactly with fixed, sufficiently small noise, without requiring noise to shrink as x approaches x*.
  • For a convergent stochastic algorithm, both strategies identify the active set correctly at infinitely many iterations with probability one, if each iterate has probability p of meeting the uniform noise bounds.
  • Correct active-set identification reduces the local problem to equality constraints, making it possible to estimate Lagrange multipliers and restrict search directions to a smaller, better-conditioned manifold.
  • The numerical demonstrations show the strategies working on a constrained neural-network training problem, where enforcing monotonicity constraints improved out-of-sample accuracy in the tested setting.

Reading between the lines

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

  • For stochastic mini-batch estimates, the uniform deterministic bound (23) rarely holds with a fixed epsilon; the practical consequence is that the LP-LPEC method needs noise to shrink near x*, whereas the QP method can tolerate fixed small noise once iterates enter a neighborhood. The paper's stochastic corollaries reveal this by conditioning on the bound holding with probability p at each iterate
  • If active-set identification succeeds, an inequality-constrained noisy problem becomes locally equality-constrained, so existing equality-constrained stochastic SQP convergence and multiplier-estimation guarantees should transfer; the paper hints at this line in its conclusion.
  • A testable engineering rule suggested by the proofs is to choose the QP penalty parameter above the multiplier bound and the threshold parameters so the right-hand side of (32) stays below the smallest inactive-constraint gap; the numerical section shows that parameter choices matter.
  • A practical diagnostic emerges: monitor whether the uniform noise-bound event in (23) holds at each iterate, and treat the active-set estimate as unreliable when it fails, even if the objective appears converged.
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 / 3 minor

Summary. The paper extends classical active-set identification results for constrained optimization (in the tradition of Oberlin and Wright) to settings where objective and constraint function and derivative evaluations are corrupted by deterministic or stochastic noise. Two identification strategies are analyzed: an LP-LPEC approach based on noisy multiplier estimation (Section 4.1) and a QP-based approach based on a noisy primal step (Section 4.2). The main deterministic results, Theorems 4.1 and 4.2, assert that, under suitable constraint qualifications and second-order conditions, and for sufficiently small uniform noise, the noisy active-set estimates equal the true active set at a nearby local minimizer. Sections 4.3 states corollaries for use inside a stochastic algorithm, and Section 5 presents illustrative numerical experiments on two-dimensional problems and a constrained neural-network training problem.

Significance. If the theorems were fully established, this would be a useful and natural extension of the Oberlin-Wright framework: it would give explicit, uniform noise tolerances under which LP-LPEC and QP active-set estimates are exact, and the QP-based Theorem 4.2 is a new variant beyond the LP/trust-region version in [29]. The paper is also careful to attribute the exact-setting foundations to [29] and to use standard perturbation bounds in Lemma 4.4. However, the current proof of Theorem 4.1 contains a false inequality in exactly the step that establishes active inclusion, and the stochastic theorems in Section 4.3 are very close to assuming their own conclusions. The deterministic QP result and the overall architecture are plausible, but the central LP-LPEC claim is unproven as written.

major comments (3)
  1. [§4.1, proof of Theorem 4.1(b)] The active-inclusion proof contains the chain, for i in A(x*): |tilde{c}_i(x)| <= ... <= epsilon_c + (hat(beta) bar(rho))^sigma <= epsilon_c + (hat(beta)(tilde{bar(rho)}+epsilon_rho))^sigma <= (beta tilde{bar(rho)})^sigma. The last inequality is algebraically impossible in general: hat(beta) > beta, sigma in (0,1), and tilde{bar(rho)}+epsilon_rho >= tilde{bar(rho)} imply (hat(beta)(tilde{bar(rho)}+epsilon_rho))^sigma > (beta tilde{bar(rho)})^sigma whenever tilde{bar(rho)}>0, so adding epsilon_c>0 cannot make the left side smaller. This is not a cosmetic slip: the condition i in eALP(yx,zx;x) requires tilde{c}_i(x) >= -(beta tilde{bar(rho)})^sigma, and the displayed chain is precisely what is supposed to prove that. Consequently Theorem 4.1(b) is unproven, and Theorem 4.3 inherits the gap. A repair may be possible by choosing hat(beta)<beta and using bar(rho) approx tilde{bar(rho)} near x
  2. [§4.1, Theorem 4.1 statement and proof] The theorem states 'for arbitrary M >= 1 let epsilon_rho and epsilon_bar(rho) be defined as in Lemma 4.2', but the proof immediately replaces M by the constant M defined in (10), and subproblem (30) uses that same constant. The value of M is load-bearing: the bound z <= M1 in (30) must be large enough to contain the true multiplier set D(x*), and the lower bound M^{-1}tilde{rho} - epsilon_rho in part (a) is only meaningful for such an M. If an arbitrary small M, e.g., M=1, is used while the true multipliers have larger norm, subproblem (30) can exclude all valid multiplier estimates and the conclusion fails. The statement should fix M to be the quantity in (10), or at least require M >= max{||c(x*)||_infty, max_{D(x*)} ||(y*,z*)||_infty} + 1, with epsilon_rho and epsilon_bar(rho) defined for that M.
  3. [§4.3, Theorems 4.3–4.4] The stochastic theorems are near-tautological. Theorem 4.3 assumes, with probability at least p at each iterate, that (23) holds, that bar(rho)(X_k,Y_k,Z_k)>0, and that the error bound is sufficiently small as required by Theorem 4.1. But (23) is a deterministic uniform bound, and the 'sufficiently small' condition is exactly the hypothesis of the deterministic identification theorem. Thus the theorem essentially says: if the deterministic identification conditions hold at each iterate with probability at least p, then identification holds with probability at least p from some iteration onward. This is a conditional corollary, not a stochastic convergence guarantee. For mini-batch stochastic optimization, where (23) fails with positive probability at typical iterate points, the theorem does not provide a verifiable sufficient condition. The paper should either derive bounds on the probab
minor comments (3)
  1. [§3, proof of Theorem 3.3, Eq. (20)] In the display after Eq. (20), '[r_x]_i = 0' appears for i not in bA, but r is the equality-constraint slack vector of dimension p. This should presumably be '[t_x]_i = 0', the slack for the inequality constraints. Please correct.
  2. [§4.3, Theorem 4.4] The list of thresholds in the assumption reads '(epsilon_x, epsilon_e, epsilon_c, epsilon_nabla f, epsilon_nabla c, epsilon_nabla c, nu)'; the sixth entry should be epsilon_nabla e.
  3. [§5.1] The numerical noise model adds independent uniform noise on every evaluation, while the theory assumes deterministic uniform bounds (23) valid at all x. The experiment is therefore a heuristic illustration, not a verification of the theorem's hypotheses. Please state this explicitly so readers do not infer that the experiment tests the stochastic theorems.

Circularity Check

1 steps flagged · score 4.0 of 10

Deterministic noisy identification theorems are non-circular; stochastic corollaries (Thms 4.3-4.4) are near-tautological because their probabilistic assumptions already contain Theorem 4.1/4.2's sufficient conditions.

  1. self definitional [Section 4.3, Theorem 4.3 (and analogously Theorem 4.4)]
    "suppose that P[(23) holds with x = Xk ∧ ¯ρ(Xk, Yk, Zk) > 0 where (Yk, Zk) is computed to solve (30) with x = Xk ∧ ˜ϵx is as small as required in Theorem 4.1 | Fk ∧ ∥Xk − x∗∥ ≤ˆϵ] ≥ p ... it holds with probability at least p that ... eALP(Yk, Zk; Xk) = A∗."

    The hypothesis literally conditions on '˜ϵx is as small as required in Theorem 4.1,' and Theorem 4.1 defines ˜ϵx as the threshold below which the noisy LP-LPEC estimate equals A(x∗). Thus the event whose probability is assumed at every iterate is essentially the event whose probability the theorem concludes: the stochastic guarantee reduces to the deterministic conclusion of Theorem 4.1, exported as a probabilistic conditional. Theorem 4.4 does the same for the QP method by assuming the thresholds '(ϵx, ϵe, ϵc, ϵ∇f, ϵ∇c, ϵ∇c, ν) as defined in Theorem 4.2'. The theorem is valid as a conditional statement, but it is a near-restatement of its own hypothesis rather than an independent stochastic identification guarantee.

full rationale

The deterministic core of the paper is not circular. Theorem 4.1 extends the external Oberlin-Wright exact result [29] to noisy settings using explicit uniform error bounds (23); the proof does not fit any parameter to the active set, and the conclusion eALP(yx,zx;x)=A(x*) is not among the assumptions. The same holds for Theorem 4.2, which uses Lemma 2.1 and Lemma 4.4 to perturb the exact QP identification theorem; the input is the uniform noise bound, not the active set. The only circular/tautological element is the stochastic wrapper in Theorems 4.3-4.4. These theorems assume that with probability p the exact sufficient conditions of the corresponding deterministic theorem hold at each iterate, and then conclude that with probability p the estimate is correct. That is a conditional corollary whose hypothesis contains the conclusion's sufficient condition, so it is a near-restatement and inflates the apparent contribution. The paper itself partially flags this difficulty in §4.1 and §4.3 by noting that the noise may need to shrink near x*. The omitted proof for ϵρ in Lemma 4.2 is a missing detail but not circularity. The skeptic's reported false inequality in the proof of Theorem 4.1(b) is a correctness concern, not a circularity, so it does not affect the circularity score. Self-citations in the references are background and not load-bearing; [29] is independent external work. Overall, the central deterministic identification results have independent content, so the score is moderate rather than high.

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

The central claim rests on standard mathematical tools, stated smoothness and constraint-qualification hypotheses, and the uniform noise-bound model. No new entities are introduced. The free parameters are algorithmic hyperparameters that the theory leaves unspecified and the experiments choose by hand or heuristic.

free parameters (5)
  • β = 0.7071 in 2D experiments via heuristic 1/max{||∇f(x)||∞, ||c(x)||∞, ||∇c(x)||∞} at a random grid point; 1e-12 in NN expe
    User-defined constant in the active-set estimate (12)/(32); the 2D value is scaled to the problem data, a data-dependent choice.
  • σ = 0.7
    User-defined exponent in (12)/(32); fixed by hand in all experiments.
  • M = 1e8 in experiments
    Upper bound on Lagrange multipliers in subproblem (10)/(30); set to a large practical constant instead of the theoretical value from (10).
  • θ = 5
    Regularization weight in QP subproblems (14)/(33); chosen by hand.
  • ν = 100 (2D), 1e2 (NN)
    Exact-penalty parameter in (33); theory requires ν sufficiently large but does not provide a constructive bound.
assumptions (5)
  • standard math Farkas' theorem and the perturbed linear-system bound (Lemma 2.1, from [6]/[30])
    Used in §2 for the QP feasibility discussion and in §4.2 (Lemma 4.4) to bound the distance between noisy and exact solutions.
  • standard math Oberlin-Wright results: Theorem 3.1, Theorem 3.2, [29, Theorem 3.4, Lemma 3.5, Lemma 3.6, Theorem 3.7]
    Imported black boxes for exact active-set identification; the noisy theorems in §4 rely on these bounds and on the exact LP-LPEC results.
  • domain assumption Problem functions f, e, c are continuously differentiable
    Stated in §2; needed for continuity, Taylor-type arguments, and perturbation continuity in Lemmas 4.2-4.4.
  • domain assumption Constraint qualifications: MFCQ plus second-order condition (4) for the LP approach; LICQ plus strict complementarity and strict local minimizer for the QP approach
    Explicitly assumed in Theorems 4.1 and 4.2 and in their proofs; these conditions guarantee bounded multipliers and nondegenerate active-set structure.
  • domain assumption Deterministic uniform noise bounds (23)
    Defines the entire noise model; all error bounds ερ and ε¯ρ in Lemma 4.2 are derived from these uniform bounds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Active-Set Identification in Noisy and Stochastic Optimization." pith.science (2026). https://pith.science/paper/VHSND6VF

@misc{pith2026250900888,
  author       = {Pith},
  title        = {Pith review of: Active-Set Identification in Noisy and Stochastic Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHSND6VF}},
  note         = {Machine review of arXiv:2509.00888}
}
read the original abstract

Identifying active constraints from a point near an optimal solution is important both theoretically and practically in constrained continuous optimization, as it can help identify optimal Lagrange multipliers and essentially reduces an inequality-constrained problem to an equality-constrained one. Traditional active-set identification guarantees have been proved under assumptions of smoothness and constraint qualifications, and assume exact function and derivative values. This work extends these results to settings when both objective and constraint function and derivative values have deterministic or stochastic noise. Two strategies are proposed that, under mild conditions, are proved to identify the active set of a local minimizer correctly when a point is close enough to the local minimizer and the noise is sufficiently small. Guarantees are also stated for the use of active-set identification strategies within a stochastic algorithm. We demonstrate our findings with two simple illustrative examples and a more realistic constrained neural-network training task.

Figures

Figures reproduced from arXiv: 2509.00888 by the authors.

Figure 1
Figure 1. Heatmaps for the problem to minimize f1 subject to c(x) ≤ 0 for noise levels ϵ = 0 (deterministic), 10−2 , and 10−1 , respectively. The top row corresponds to the LP-LPEC approach based on computing Lagrange multiplier estimates; the bottom row corresponds to the QP approach based on computing a primal step. The value of the heatmap at each point indicates the fraction of times that the active set A(x∗) was identifi… view at source ↗
Figure 2
Figure 2. Heatmaps for the problem to minimize f2 subject to c(x) ≤ 0. The details are the same as stated in the caption for [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Over the iterates of an algorithm to solve (40), the number of correctly identified elements of the [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Over the iterates of an algorithm to solve (40), the number of correctly identified elements of the [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 33 canonical work pages

  1. [29]

    Christina Oberlin and Stephen J. Wright. Active Set Identification in Nonlinear Programming. SIAM Journal on Optimization , 17(2):577, 2006

    work page 2006

  2. [1]

    Berahas, Richard H

    Albert S. Berahas, Richard H. Byrd, and Jorge Nocedal. Derivative-free optimization of noisy functions via quasi-Newton methods. SIAM Journal on Optimization , 29(2):965–993, 2019

    work page 2019

  3. [2]

    Berahas, Frank E

    Albert S. Berahas, Frank E. Curtis, Michael J. O’Neill, and Daniel P. Robinson. A stochastic sequential quadratic optimization algorithm for nonlinear equality constrained optimization with rank-deficient jacobians. Mathematics of Operations Research, 2023

    work page 2023

  4. [3]

    Berahas, Frank E

    Albert S. Berahas, Frank E. Curtis, Daniel P. Robinson, and Baoyu Zhou. Sequential quadratic op- timization for nonlinear equality constrained stochastic optimization. SIAM Journal on Optimization , 31(2):1352–1379, 2021

    work page 2021

  5. [4]

    Bertsekas

    Dimitri P. Bertsekas. Convex Optimization Theory . Athena Scientific, Belmont, Mass, 2009

    work page 2009

  6. [5]

    Curtis, and Jorge Nocedal

    L´ eon Bottou, Frank E. Curtis, and Jorge Nocedal. Optimization methods for large-scale machine learning. SIAM Review, 60(2):223–311, 2018

    work page 2018

  7. [6]

    Burden and John Douglas Faires

    Richard L. Burden and John Douglas Faires. Numerical Analysis . Thomson Brooks/Cole, Belmont, CA, 8th edition, 2006

    work page 2006

  8. [7]

    James V. Burke. On the identification of active constraints II: The nonconvex case. SIAM Journal on Numerical Analysis, 27(4):1081–1102, 1990

    work page 1990

Show all 36 references
  1. [8]

    Burke and Jorge J

    James V. Burke and Jorge J. Mor´ e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25(5):1197–1211, 1988. 25

    1988

  2. [9]

    Byrd, Nicholas I

    Richard H. Byrd, Nicholas I. M. Gould, Jorge Nocedal, and Richard A. Waltz. On the convergence of successive linear-quadratic programming algorithms. SIAM Journal on Optimization , 16(2):471–489, 2005

    2005

  3. [10]

    Byrd, Nicholas I.M

    Richard H. Byrd, Nicholas I.M. Gould, Jorge Nocedal, and Richard A. Waltz. An algorithm for non- linear optimization using linear programming and equality constrained subproblems. Mathematical Programming, 100(1):27–48, 2003

    2003

  4. [11]

    Curtis, Shima Dezfulian, and Andreas W¨ achter

    Frank E. Curtis, Shima Dezfulian, and Andreas W¨ achter. An interior-point algorithm for continuous nonlinearly constrained optimization with noisy function and derivative evaluations. arXiv preprint arXiv:2502.11302, 2025

    2025 arXiv

  5. [12]

    Curtis, Xin Jiang, and Qi Wang

    Frank E. Curtis, Xin Jiang, and Qi Wang. Almost-sure convergence of iterates and multipliers in stochastic sequential quadratic optimization. Journal of Optimization Theory and Applications , 2024

    2024

  6. [13]

    Curtis, Michael J

    Frank E. Curtis, Michael J. O’Neill, and Daniel P. Robinson. Worst-case complexity of an SQP method for nonlinear equality constrained stochastic optimization. Mathematical Programming, 205:431–483, 2024

    2024

  7. [14]

    Curtis, Daniel P

    Frank E. Curtis, Daniel P. Robinson, and Baoyu Zhou. Sequential quadratic optimization for stochas- tic optimization with deterministic nonlinear inequality and equality constraints. SIAM Journal on Optimization, 34(4):3592–3622, 2024

    2024

  8. [15]

    Curtis, Daniel P

    Frank E. Curtis, Daniel P. Robinson, and Baoyu Zhou. A stochastic inexact sequential quadratic optimization algorithm for nonlinear equality-constrained optimization. INFORMS Journal on Opti- mization, 2024

    2024

  9. [16]

    Curtis and Katya Scheinberg

    Frank E. Curtis and Katya Scheinberg. Optimization methods for supervised machine learning: From linear models to deep learning. In INFORMS Tutorials in Operations Research, chapter 5, pages 89–114. Institute for Operations Research and the Management Sciences (INFORMS), 2017

    2017

  10. [17]

    On the convergence of interior-point methods for bound- constrained nonlinear optimization problems with noise

    Shima Dezfulian and Andreas W¨ achter. On the convergence of interior-point methods for bound- constrained nonlinear optimization problems with noise. arXiv preprint arXiv:2405.11400 , 2024

    2024 arXiv

  11. [18]

    On the accurate identification of active constraints

    Francisco Facchinei, Andreas Fischer, and Christian Kanzow. On the accurate identification of active constraints. SIAM Journal on Optimization , 9(1):14–32, 1998

    1998

  12. [19]

    Mahoney, and Mladen Kolar

    Yuchen Fang, Sen Na, Michael W. Mahoney, and Mladen Kolar. Fully stochastic trust-region sequential quadratic programming for equality-constrained optimization problems.SIAM Journal on Optimization, 34(2):2007–2037, 2024

    2007

  13. [20]

    Hager and M

    William W. Hager and M. Seetharama Gowda. Stability in the presence of degeneracy and error estimation. Mathematical Programming, 85(1):181–192, May 1999

    1999

  14. [21]

    Han and Olvi L

    S.-P. Han and Olvi L. Mangasarian. Exact penalty functions in nonlinear programming. Mathematical Programming, 17(1):251–269, December 1979

    1979

  15. [22]

    Warren Hare and Adrian S. Lewis. Identifying active constraints via partial smoothness and prox- regularity. Journal of Convex Analysis , 11:251–266, 2004

    2004

  16. [23]

    Heart Disease

    Andras Janosi, William Steinbrunn, Matthias Pfisterer, and Robert Detrano. Heart Disease. UCI Machine Learning Repository, 1989. DOI: https://doi.org/10.24432/C52P4X

    1989 doi

  17. [24]

    Adrian S. Lewis. Active sets, nonsmoothness, and sensitivity. SIAM Journal on Optimization, 13(3):702– 725, 2002. 26

    2002

  18. [25]

    Mangasarian and Stan Fromovitz

    Olvi L. Mangasarian and Stan Fromovitz. The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints. Journal of Mathematical Analysis and Applications , 17(1):37–47, 1967

    1967

  19. [26]

    An adaptive stochastic sequential quadratic programming with differentiable exact augmented lagrangians

    Sen Na, Mihai Anitescu, and Mladen Kolar. An adaptive stochastic sequential quadratic programming with differentiable exact augmented lagrangians. Mathematical Programming, pages 1–71, jun 2022

    2022

  20. [27]

    Inequality constrained stochastic nonlinear optimization via active-set sequential quadratic programming

    Sen Na, Mihai Anitescu, and Mladen Kolar. Inequality constrained stochastic nonlinear optimization via active-set sequential quadratic programming. Mathematical Programming, pages 1–75, 2023

    2023

  21. [28]

    Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer New York, 2006

    2006

  22. [30]

    Ortega and Werner Rheinboldt

    James M. Ortega and Werner Rheinboldt. Numerical Analysis: A Second Course . Elsevier Science, Burlington, 2014

    2014

  23. [31]

    Constrained optimization in the presence of noise

    Figen Oztoprak, Richard Byrd, and Jorge Nocedal. Constrained optimization in the presence of noise. SIAM Journal on Optimization , 33(3):2118–2136, 2023

    2023

  24. [32]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970

    1970

  25. [33]

    A trust region method for noisy unconstrained optimization

    Shigeng Sun and Jorge Nocedal. A trust region method for noisy unconstrained optimization. Mathe- matical Programming, pages 1–28, 2023

    2023

  26. [34]

    Stephen J. Wright. Identifiable surfaces in constrained optimization. SIAM Journal on Control and Optimization, 31(4):1063–1079, 1993

    1993

  27. [35]

    Stephen J. Wright. Modifying SQP for degenerate problems. SIAM Journal on Optimization, 13(2):470– 497, 2002

    2002

  28. [36]

    Analysis of the BFGS method with errors

    Yuchen Xie, Richard H Byrd, and Jorge Nocedal. Analysis of the BFGS method with errors. SIAM Journal on Optimization , 30(1):182–209, 2020. 27

    2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.