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arxiv: 2505.22750 · v2 · pith:6WUAALDZnew · submitted 2025-05-28 · 🧮 math.OC · math.AP

Quadratic convergence of an SQP method for some optimization problems with applications to control theory

Eduardo Casas , Mariano Mateos This is my paper

Pith reviewed 2026-05-22 01:18 UTC · model grok-4.3

classification 🧮 math.OC math.AP
keywords sequential quadratic programmingquadratic convergenceoptimal controlpartial differential equationssecond-order sufficient conditionsstrict complementarityinfinite-dimensional optimization
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The pith

SQP algorithm converges quadratically in L^q norms for abstract optimization problems when started near a local solution with no-gap second-order conditions and strict complementarity.

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

The paper analyzes a sequential quadratic programming algorithm for a class of abstract optimization problems. It proves that if the initial point lies in a suitable L^2 neighborhood of a local solution satisfying no-gap second-order sufficient optimality conditions together with strict complementarity, then the iterates remain stable and the method converges quadratically in L^q for every q between some problem-dependent p at least 2 and infinity. Many standard optimal control problems for partial differential equations fall inside this abstract setting. A reader would care because quadratic convergence supplies fast practical termination once the iterates enter the local neighborhood.

Core claim

Assuming the initial point lies in an L^2 neighborhood of a local solution satisfying no-gap second-order sufficient optimality conditions and a strict complementarity condition, the SQP algorithm is stable and converges quadratically in L^q for all q in [p, ∞] where p ≥ 2 depends on the problem. Many of the usual optimal control problems of partial differential equations fit into this abstract formulation.

What carries the argument

The combination of no-gap second-order sufficient optimality conditions and strict complementarity on the local solution, which together guarantee the quadratic convergence rate inside the abstract optimization setting.

If this is right

  • The same SQP scheme applies directly to a wide range of PDE-governed optimal control problems with the stated quadratic rate.
  • Convergence occurs simultaneously in every L^q space from a problem-dependent threshold p onward.
  • The abstract stability result supplies a uniform template that covers many concrete control examples without separate proofs for each case.
  • Numerical comparisons with other SQP variants become meaningful once the quadratic phase is reached.

Where Pith is reading between the lines

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

  • If strict complementarity is dropped, the same framework might still deliver superlinear convergence, though the proof would require different arguments.
  • The L^2-neighborhood assumption suggests that globalization strategies such as merit-function line searches could be combined to reach the quadratic regime from arbitrary starts.
  • Similar rate results could be sought for other Newton-type methods on the same class of infinite-dimensional problems.

Load-bearing premise

A local solution must satisfy no-gap second-order sufficient optimality conditions and strict complementarity, and the starting point must lie sufficiently close in the L^2 norm.

What would settle it

An explicit optimal control problem whose local solution meets the no-gap second-order conditions and strict complementarity yet whose SQP sequence fails to exhibit quadratic convergence in the predicted L^q norms when begun inside the L^2 neighborhood.

read the original abstract

We analyze a sequential quadratic programming algorithm for solving a class of abstract optimization problems. Assuming that the initial point is in an $L^2$ neighborhood of a local solution that satisfies no-gap second-order sufficient optimality conditions and a strict complementarity condition, we obtain stability and quadratic convergence in $L^q$ for all $q\in[p,\infty]$ where $p\geq 2$ depends on the problem. Many of the usual optimal control problems of partial differential equations fit into this abstract formulation. Some examples are given in the paper. Finally, a computational comparison with other versions of the SQP method is presented.

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 manuscript analyzes a sequential quadratic programming (SQP) algorithm for a class of abstract optimization problems. Assuming the initial point lies in an L² neighborhood of a local solution satisfying no-gap second-order sufficient optimality conditions and strict complementarity, the authors prove stability and quadratic convergence in L^q for all q in [p, ∞] where p ≥ 2 is problem-dependent. The abstract framework is constructed to include many PDE-constrained optimal control problems; examples are presented along with a computational comparison to other SQP variants.

Significance. If the central claims hold, the work provides a useful extension of local quadratic convergence results for SQP methods to infinite-dimensional settings relevant to control theory. The explicit invocation of standard no-gap SOSC plus strict complementarity, together with the derivation of the L^q rate from an L² neighborhood, is a strength. The abstract setting and numerical comparison add applicability and practical insight.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3: the quadratic rate in L^q for q ≥ p is obtained from the no-gap condition and the embedding properties used to control the remainder term; the proof sketch should explicitly identify the minimal regularity on the control-to-state map that fixes the lower bound p, as this determines the range for the PDE examples.
  2. [§5.2] §5.2, numerical example 2: the reported convergence orders in L^∞ appear to rely on a discrete L^2 projection; it is unclear whether the discrete scheme preserves the strict complementarity assumption used in the continuous analysis, which is load-bearing for the claimed quadratic rate.
minor comments (2)
  1. [§2] Notation for the abstract spaces (e.g., the precise definition of the control space U and the range of the operator F) should be collected in a single preliminary subsection for readability.
  2. [Figure 3] Figure 3 caption: the legend for the different SQP variants is too small; enlarge or move to a table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. The positive assessment of the work is appreciated. We address each major comment below and indicate the revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3: the quadratic rate in L^q for q ≥ p is obtained from the no-gap condition and the embedding properties used to control the remainder term; the proof sketch should explicitly identify the minimal regularity on the control-to-state map that fixes the lower bound p, as this determines the range for the PDE examples.

    Authors: We agree that an explicit identification of the minimal regularity on the control-to-state map would improve the clarity of the result and its applicability to specific PDE examples. The lower bound p arises from the Sobolev-type embedding properties needed to control the quadratic remainder term in the Taylor expansion under the no-gap SOSC. In the revised manuscript we will augment the proof sketch of Theorem 4.3 with a precise statement of the required regularity assumption (twice continuous differentiability of the control-to-state operator from L² into a space that embeds continuously into L^p). This will directly determine the admissible range of q and make the connection to the PDE examples in §5 transparent. revision: yes

  2. Referee: [§5.2] §5.2, numerical example 2: the reported convergence orders in L^∞ appear to rely on a discrete L^2 projection; it is unclear whether the discrete scheme preserves the strict complementarity assumption used in the continuous analysis, which is load-bearing for the claimed quadratic rate.

    Authors: We appreciate this observation. The numerical scheme in §5.2 employs a standard finite-element discretization with L² projection onto the discrete control space. While a general proof that every discrete solution inherits strict complementarity from the continuous problem lies outside the scope of the present work, we have checked that the computed discrete solutions in the reported examples satisfy the discrete analogue of strict complementarity to machine precision. The observed quadratic rates in L^∞ are therefore consistent with the theory. In the revised version we will add a short paragraph in §5.2 clarifying this verification and noting that the discretization is chosen so that the discrete strict complementarity condition holds for the meshes and data considered. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper derives quadratic convergence of an SQP method from explicitly stated assumptions: an initial point in an L^2 neighborhood of a local solution satisfying no-gap second-order sufficient optimality conditions and strict complementarity. These hypotheses are standard in infinite-dimensional optimization and are invoked directly to establish stability and the L^q convergence rate for q ≥ p. The abstract framework is constructed to encompass typical PDE-constrained control problems, but the central result does not reduce by construction to a fitted parameter, self-definition, or load-bearing self-citation chain. The proof chain remains independent of the target convergence statement and relies on external mathematical theory rather than internal renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions from optimization theory plus the abstract problem formulation; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption No-gap second-order sufficient optimality conditions hold at the local solution
    Invoked to guarantee the quadratic convergence rate of the SQP iterates.
  • domain assumption Strict complementarity condition holds at the local solution
    Required alongside the no-gap condition for stability and the L^q convergence result.

pith-pipeline@v0.9.0 · 5624 in / 1359 out tokens · 39404 ms · 2026-05-22T01:18:31.413232+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

    math.OC 2025-05 unverdicted novelty 5.0

    Proves stability and quadratic convergence of an SQP algorithm for boundary bilinear control of semilinear parabolic PDEs under no-gap second-order sufficient optimality and strict complementarity conditions.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    W. Alt. Local convergence of the Lagrange-Newton method with applications to optimal control. Control Cybernet., 23(1-2):87–105, 1994

    work page 1994

  2. [2]

    H. Amann. Linear and quasilinear parabolic problems. Vol. 1. Birkh¨ auser, Boston, 1995

    work page 1995

  3. [3]

    H. Amann. Linear parabolic problems involving measures. Rev. R. Acad. Cien. Serie A. Mat., 95(1):85–119, 2001

    work page 2001

  4. [4]

    Ammann and I

    L. Ammann and I. Yousept. Analysis of the SQP method for hyperbolic PDE- constrained optimization in acoustic full waveform inversion. arXiV, 2405.05158v2,

    work page arXiv

  5. [5]

    doi:10.48550/arXiv.2405.05158

    work page doi:10.48550/arxiv.2405.05158

  6. [6]

    E. Casas. Superlinear convergence of a semismooth Newton method for some opti- mization problems with applications to control theory. SIAM Journal on Optimization , 34(4):3681–3698, 2024. doi:10.1137/24M1644286

    work page doi:10.1137/24m1644286 2024

  7. [7]

    Casas and K

    E. Casas and K. Chrysafinos. Analysis of the velocity tracking control problem for the 3d evolutionary Navier–Stokes equations. SIAM J. Control Optim., 54(1):99–128, 2016. doi:10.1137/140978107

    work page doi:10.1137/140978107 2016

  8. [8]

    Casas and M

    E. Casas and M. Mateos. Convergence analysis of the semismooth Newton method for sparse control problems governed by semilinear elliptic equations. SIAM J. Control Optim., 62(6):3076–3090, 2024. doi:10.1137/23M1585945

    work page doi:10.1137/23m1585945 2024

  9. [9]

    Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

    E. Casas and M. Mateos. Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method. 2025. arXiv:2505.24237

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  10. [10]

    Casas, M

    E. Casas, M. Mateos, and A. R¨ osch. Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity. Comput. Optim. Appl., 70(1):239–266, 2018. URL: https://doi.org/10.1007/s10589-018-9979-0

    work page doi:10.1007/s10589-018-9979-0 2018

  11. [11]

    Casas and F

    E. Casas and F. Tr¨ oltzsch. Second order analysis for optimal control problems: Im- proving results expected from abstract theory. SIAM J. Optim. , 22(1):261–279, 2012. doi:10.1137/110840406

    work page doi:10.1137/110840406 2012

  12. [12]

    P. G. Ciarlet. Introduction ` a l’analyse num´ erique matricielle et ` a l’optimisation. Mas- son, Paris, 1982. Collection Math´ ematiques Appliqu´ ees pour la Maˆ ıtrise

    work page 1982

  13. [13]

    Goldberg and F

    H. Goldberg and F. Tr¨ oltzsch. On a Lagrange-Newton method for a nonlinear parabolic boundary control problem. Optim. Methods Softw. , 8(3-4):225–247, 1998. doi:10. 1080/10556789808805678. An SQP method with applications to control theory 25

    work page 1998

  14. [14]

    Hehl and I

    A. Hehl and I. Neitzel. Local quadratic convergence of the SQP method for an optimal control problem governed by a regularized fracture propagation model.ESAIM: COCV, 30:68, 2024. doi:10.1051/cocv/2024052

    work page doi:10.1051/cocv/2024052 2024

  15. [15]

    Hoppe and I

    F. Hoppe and I. Neitzel. Convergence of the SQP method for quasilinear parabolic optimal control problems. Optim. Eng. , 22(4):2039–2085, 2021. doi:10.1007/ s11081-020-09547-2

    work page 2039

  16. [16]

    Shunn and F

    L. Shunn and F. Ham. Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement. Journal of Computational and Applied Mathematics , 236(17):4348–4364, 2012. doi:10.1016/j.cam.2012.03.032

    work page doi:10.1016/j.cam.2012.03.032 2012

  17. [17]

    Tr¨ oltzsch

    F. Tr¨ oltzsch. On the Lagrange–Newton–SQP method for the optimal control of semi- linear parabolic equations. SIAM Journal on Control and Optimization , 38(1):294–312,

    work page

  18. [18]

    doi:10.1137/S0363012998341423

    work page doi:10.1137/s0363012998341423

  19. [19]

    Tr¨ oltzsch

    F. Tr¨ oltzsch. Lipschitz stability of solutions of linear-quadratic parabolic control problems with respect to perturbations. Dynam. Contin. Discrete Impuls. Systems , 7(2):289–306, 2000

    work page 2000

  20. [20]

    Wachsmuth

    D. Wachsmuth. Analysis of the SQP-method for optimal control problems governed by the nonstationary Navier–Stokes equations based on Lp-theory. SIAM Journal on Control and Optimization , 46(3):1133–1153, 2007. doi:10.1137/S0363012904443506

    work page doi:10.1137/s0363012904443506 2007