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On Optimality Conditions for Nonlinear Conic Programming

Author

Listed:
  • Roberto Andreani

    (Department of Applied Mathematics, University of Campinas, Campinas 13083-970, Brazil)

  • Walter Gómez

    (Department of Mathematical Engineering, Universidad de la Frontera, Temuco 4811230, Chile)

  • Gabriel Haeser

    (Department of Applied Mathematics, University of São Paulo, São Paulo 05508-090, Brazil)

  • Leonardo M. Mito

    (Department of Applied Mathematics, University of São Paulo, São Paulo 05508-090, Brazil)

  • Alberto Ramos

    (Department of Mathematics, Federal University of Paraná, Curitiba 81530-015, Brazil)

Abstract

Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush–Kuhn–Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.

Suggested Citation

  • Roberto Andreani & Walter Gómez & Gabriel Haeser & Leonardo M. Mito & Alberto Ramos, 2022. "On Optimality Conditions for Nonlinear Conic Programming," Mathematics of Operations Research, INFORMS, vol. 47(3), pages 2160-2185, August.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:3:p:2160-2185
    DOI: 10.1287/moor.2021.1203
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Alberto Ramos, 2019. "Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 566-591, November.
    3. Gábor Pataki, 2007. "On the Closedness of the Linear Image of a Closed Convex Cone," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 395-412, May.
    4. L. F. Bueno & G. Haeser & F. Lara & F. N. Rojas, 2020. "An Augmented Lagrangian method for quasi-equilibrium problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 737-766, July.
    5. Roberto Andreani & José Mario Martínez & Alberto Ramos & Paulo J. S. Silva, 2018. "Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 693-717, August.
    6. J.M. Martínez & B.F. Svaiter, 2003. "A Practical Optimality Condition Without Constraint Qualifications for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 117-133, July.
    7. Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2016. "Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 70-89, October.
    8. William Hager & Delphine Mico-Umutesi, 2014. "Error estimation in nonlinear optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 327-341, July.
    9. Gabriel Haeser, 2018. "A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms," Computational Optimization and Applications, Springer, vol. 70(2), pages 615-639, June.
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