pith. sign in

arxiv: 2406.12436 · v3 · submitted 2024-06-18 · 🧮 math.OC

Affordable mixed-integer Lagrangian methods: optimality conditions and convergence analysis

Alberto De Marchi This is my paper

Pith reviewed 2026-05-23 23:39 UTC · model grok-4.3

classification 🧮 math.OC
keywords mixed-integer nonlinear optimizationLagrangian methodsoptimality conditionssequential minimizationconvergence analysisaugmented Lagrangiannonconvex optimization
0
0 comments X

The pith

Necessary optimality conditions in Lagrangian form are extended to mixed-integer nonlinear optimization without convexity assumptions.

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

The paper extends necessary optimality conditions in Lagrangian form and the sequential minimization framework to mixed-integer nonlinear optimization without any convexity assumptions. This builds on a recent notion of local optimality for polyhedral and integrality constraints to characterize local minimizers and critical points even with nonlinear constraints. Such an extension matters because it provides the basis for developing practical sequential minimization algorithms that guarantee convergence to critical points starting from arbitrary points. The work also advances a primal-dual perspective, local saddle point properties, and connections to the proximal point algorithm for problems with integer variables.

Core claim

By building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, the paper provides a characterization of local minimizers and critical points for mixed-integer nonlinear optimization problems that include nonlinear constraints. This characterization enables the extension of Lagrangian necessary optimality conditions and the sequential minimization framework without requiring convexity, laying foundations for affordable algorithms with convergence guarantees.

What carries the argument

A recently developed notion of local optimality for problems with polyhedral and integrality constraints, which supports the derivation of Lagrangian conditions and sequential minimization for mixed-integer problems.

If this is right

  • Sequential minimization algorithms achieve convergence guarantees to critical points from arbitrary initializations.
  • A primal-dual perspective and local saddle point property apply in the presence of integer variables.
  • Dual relationships with the proximal point algorithm are established for integer variables.
  • Preliminary numerical results demonstrate the approach for augmented Lagrangian and interior point methods.

Where Pith is reading between the lines

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

  • This framework could support the development of hybrid optimization solvers combining Lagrangian methods with integer programming techniques.
  • Applications in engineering design or logistics might benefit from guaranteed convergence in nonconvex mixed-integer settings.
  • Further research could test the method on larger-scale problems to assess computational affordability.

Load-bearing premise

The approach depends on the validity of the recently developed notion of local optimality for polyhedral and integrality constraints.

What would settle it

Finding a mixed-integer nonlinear problem where a local minimizer violates the proposed Lagrangian necessary conditions would falsify the extension.

Figures

Figures reproduced from arXiv: 2406.12436 by Alberto De Marchi.

Figure 1
Figure 1. Figure 1: Binary optimal control problem (5): solutions obtained with relaxed integrality (NLP), combinatorial integral approximation (CIA), and mixed-integer linearization scheme (MILA). KKT-criticality implicitly requires feasibility, since the normal cone NC(c(¯x)) must be nonempty. Moreover, by (4) the first condition can be rewritten as min x∈X ∩BPL(¯x,∆) h∇f(¯x) + Jc(¯x) ⊤y, x − x¯i = 0, meaning that the Lagra… view at source ↗
read the original abstract

Necessary optimality conditions in Lagrangian form and the sequential minimization framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, a characterization of local minimizers and critical points is given for problems including also nonlinear constraints. This approach lays the foundations for developing affordable sequential minimization algorithms with convergence guarantees to critical points from arbitrary initializations. A primal-dual perspective, a local saddle point property, and the dual relationships with the proximal point algorithm are also advanced in the presence of integer variables. Preliminary numerical results are presented for an augmented Lagrangian and an interior point method.

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

0 major / 2 minor

Summary. The paper extends necessary optimality conditions in Lagrangian form and the sequential minimization framework to mixed-integer nonlinear optimization without any convexity assumptions. Building on a recently developed notion of local optimality for polyhedral and integrality constraints, it characterizes local minimizers and critical points for problems that also include nonlinear constraints. The work advances a primal-dual perspective including a local saddle-point property and dual relationships with the proximal point algorithm in the presence of integer variables, and presents preliminary numerical results for an augmented Lagrangian method and an interior-point method.

Significance. If the derivations are correct, the result is significant because it supplies theoretical foundations for affordable sequential-minimization algorithms that converge to critical points from arbitrary initializations on nonconvex MINLPs. The absence of convexity assumptions and the explicit primal-dual analysis distinguish the contribution from most existing Lagrangian approaches; the preliminary numerics provide initial evidence of practicality.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'problems including also nonlinear constraints' is slightly ambiguous; a brief parenthetical clarifying whether the nonlinear constraints are assumed smooth or merely continuous would improve precision.
  2. [Introduction] The manuscript relies on a recently developed local-optimality notion; the introduction should contain an explicit forward reference to the precise statement of that notion (including its theorem number) so that readers can locate the foundational assumption without searching the bibliography.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, the recognition of the paper's contributions to nonconvex MINLP theory, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Minor self-citation of foundational local optimality notion; central extension independent

full rationale

The paper extends Lagrangian optimality conditions and sequential minimization to nonconvex MINLPs by building on a cited prior notion of local optimality for polyhedral/integrality constraints. This is a self-citation to recently developed work by the same author, but the extension itself (characterizations of minimizers/critical points, primal-dual properties, convergence analysis) consists of new derivations that do not reduce to the input by construction, fitting, or renaming. No self-definitional loops, fitted predictions called results, or ansatz smuggling appear in the described framework. The derivation chain is self-contained as a theoretical extension whose validity rests on the external correctness of the referenced notion rather than internal reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on an external notion of local optimality for polyhedral and integrality constraints; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption A recently developed notion of local optimality for problems with polyhedral and integrality constraints is valid and applicable.
    The paper explicitly builds its characterization upon this notion as stated in the abstract.

pith-pipeline@v0.9.0 · 5624 in / 1101 out tokens · 18618 ms · 2026-05-23T23:39:04.540720+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Birgin, Jos´ e M

    Roberto Andreani, Ernesto G. Birgin, Jos´ e M. Mart ´ ınez , and Maria Laura Schuverdt. On aug- mented Lagrangian methods with general lower–level constr aints. SIAM Journal on Optimization , 18(4):1286–1309, 2008

    work page 2008

  2. [2]

    Mito, Alb erto Ramos, and Leonardo D

    Roberto Andreani, Gabriel Haeser, Leonardo M. Mito, Alb erto Ramos, and Leonardo D. Secchin. On the best achievable quality of limit points of augmented Lag rangian schemes. Numerical Algorithms, 90(2):851–877, 2022

    work page 2022

  3. [3]

    Bauschke and Patrick L

    Heinz H. Bauschke and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Cham, CH, 2017

    work page 2017

  4. [4]

    Mixed-integer nonlinear optimization

    Pietro Belotti, Christian Kirches, Sven Leyffer, Jeff Lin deroth, James Luedtke, and Ashutosh Ma- hajan. Mixed-integer nonlinear optimization. Acta Numerica, 22:1–131, 2013

    work page 2013

  5. [5]

    Birgin and Jos´ e M

    Ernesto G. Birgin and Jos´ e M. Mart ´ ınez.Practical Augmented Lagrangian Methods for Constrained Optimization. Society for Industrial and Applied Mathematics, Philadel phia, PA, US, 2014

    work page 2014

  6. [6]

    A Gauss-Newton-based decomposition algorithm for nonlinea r mixed-integer optimal control problems

    Adrian B¨ urger, Clemens Zeile, Angelika Altmann-Diese s, Sebastian Sager, and Moritz Diehl. A Gauss-Newton-based decomposition algorithm for nonlinea r mixed-integer optimal control problems. Automatica, 152:110967, 2023

    work page 2023

  7. [7]

    pycombina: An open-source tool for solving combinat orial approximation problems arising in mixed-integer optimal control

    Adrian B¨ urger, Clemens Zeile, Mirko Hahn, Angelika Alt mann-Dieses, Sebastian Sager, and Moritz Diehl. pycombina: An open-source tool for solving combinat orial approximation problems arising in mixed-integer optimal control. IF AC-PapersOnLine, 53(2):6502–6508, 2020. 21st IF AC World Congress

    work page 2020

  8. [8]

    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

    work page 2005

  9. [9]

    A storm of feasibility pumps for nonconvex MINLP

    Claudia D’Ambrosio, Antonio Frangioni, Leo Liberti, an d Andrea Lodi. A storm of feasibility pumps for nonconvex MINLP. Mathematical Programming, 136(2):375–402, 2012. 17

    work page 2012

  10. [10]

    Implicit augmented Lagrangian and g eneralized optimization

    Alberto De Marchi. Implicit augmented Lagrangian and g eneralized optimization. Journal of Applied and Numerical Optimization , 6(2):291–320, 2024

    work page 2024

  11. [11]

    Mixed-integer linearity in nonline ar optimization: a trust region approach

    Alberto De Marchi. Mixed-integer linearity in nonline ar optimization: a trust region approach. arXiv:2310.17285, 2024. Under review after minor revisions

    work page arXiv 2024

  12. [12]

    Constrained composite optimization and augmented Lagrangian methods

    Alberto De Marchi, Xiaoxi Jia, Christian Kanzow, and Pa trick Mehlitz. Constrained composite optimization and augmented Lagrangian methods. Mathematical Programming, 201(1):863–896, 2023

    work page 2023

  13. [13]

    Local propertie s and augmented Lagrangians in fully nonconvex composite optimization

    Alberto De Marchi and Patrick Mehlitz. Local propertie s and augmented Lagrangians in fully nonconvex composite optimization. Journal of Nonsmooth Analysis and Optimization , 5, 2024

    work page 2024

  14. [14]

    An interior pro ximal gradient method for nonconvex optimization

    Alberto De Marchi and Andreas Themelis. An interior pro ximal gradient method for nonconvex optimization. Open Journal of Mathematical Optimization , 5:1–22, 2024

    work page 2024

  15. [15]

    A comparative study of SQP-type algo- rithms for nonlinear and nonconvex mixed-integer optimiza tion

    Oliver Exler, Thomas Lehmann, and Klaus Schittkowski. A comparative study of SQP-type algo- rithms for nonlinear and nonconvex mixed-integer optimiza tion. Mathematical Programming Com- putation, 4(4):383–412, 2012

    work page 2012

  16. [16]

    Fiacco and Garth P

    Anthony V. Fiacco and Garth P. McCormick. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York, NY, US, 1968

    work page 1968

  17. [17]

    A Voronoi-based mixed-integer Gauss-Newton algorithm for MINLP arising in optimal control

    Andrea Ghezzi, L´ eo Simpson, Adrian B¨ urger, Clemens Z eile, Sebastian Sager, and Moritz Diehl. A Voronoi-based mixed-integer Gauss-Newton algorithm for MINLP arising in optimal control. In 2023 European Control Conference (ECC) , pages 1–7, 2023

    work page 2023

  18. [18]

    On the compl exity of an augmented Lagrangian method for nonconvex optimization

    Geovani Nunes Grapiglia and Ya-xiang Yuan. On the compl exity of an augmented Lagrangian method for nonconvex optimization. IMA Journal of Numerical Analysis , 41(2):1546–1568, 2020

    work page 2020

  19. [19]

    Lagrange theory of discrete-continuous nonlinear optimiza- tion

    Johannes Jahn and Martin Knossalla. Lagrange theory of discrete-continuous nonlinear optimiza- tion. Journal of Nonlinear and Variational Analysis , 2(3):317–342, 2018

    work page 2018

  20. [20]

    Hybrid optimal control with mixed- integer Lagrangian methods

    Viktoriya Nikitina, Alberto De Marchi, and Matthias Ge rdts. Hybrid optimal control with mixed- integer Lagrangian methods. arXiv:2403.06842, 2024

    work page arXiv 2024

  21. [21]

    Sequential quadr atic programming algorithm for real-time mixed-integer nonlinear MPC

    Rien Quirynen and Stefano Di Cairano. Sequential quadr atic programming algorithm for real-time mixed-integer nonlinear MPC. In 2021 60th IEEE Conference on Decision and Control (CDC) , pages 993–999, Austin, TX, US, 2021

    work page 2021

  22. [22]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Convergence of augmented lagr angian methods in extensions beyond non- linear programming. Mathematical Programming, 199(1):375–420, 2023

    work page 2023

  23. [23]

    Tyrrell Rockafellar and Roger J

    R. Tyrrell Rockafellar and Roger J. B. Wets. Variational Analysis, volume 317 of Grundlehren der mathematischen Wissenschaften . Springer, Heidelberg, DE, 2009. 3rd printing

    work page 2009

  24. [24]

    Combinatorial integral approximation

    Sebastian Sager, Michael Jung, and Christian Kirches. Combinatorial integral approximation. Math- ematical Methods of Operations Research , 73(3):363–380, 2011

    work page 2011

  25. [25]

    OpEn: Code generation for embedded nonconvex optimization

    Pantelis Sopasakis, Emil Fresk, and Panagiotis Patrin os. OpEn: Code generation for embedded nonconvex optimization. IF AC-PapersOnLine, 53(2):6548–6554, 2020

    work page 2020

  26. [26]

    Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces

    Daniel Steck. Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces . PhD thesis, Universit¨ at W¨ urzburg, W¨ urzburg, DE, 2018

    work page 2018

  27. [27]

    Andreas W¨ achter and Lorenz T. Biegler. On the implemen tation of an interior-point filter line- search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25–57, 3 2006

    work page 2006

  28. [28]

    Mixed-integer optimal control under minimum dwell time constraints

    Clemens Zeile, Nicol´ o Robuschi, and Sebastian Sager. Mixed-integer optimal control under minimum dwell time constraints. Mathematical Programming, 188(2):653–694, 2021. 18

    work page 2021