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Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization

Author

Listed:
  • D.P. Bertsekas

    (Massachusetts Institute of Technology)

  • A.E. Ozdaglar

    (Massachusetts Institute of Technology)

Abstract

We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the Fritz–John theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and the Farkas lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.

Suggested Citation

  • D.P. Bertsekas & A.E. Ozdaglar, 2002. "Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 287-343, August.
    • Handle: RePEc:spr:joptap:v:114:y:2002:i:2:d:10.1023_a:1016083601322
    DOI: 10.1023/A:1016083601322
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    References listed on IDEAS

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    1. Frank H. Clarke, 1976. "A New Approach to Lagrange Multipliers," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 165-174, May.
    2. Willard I. Zangwill, 1967. "Non-Linear Programming Via Penalty Functions," Management Science, INFORMS, vol. 13(5), pages 344-358, January.
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    Cited by:

    1. X. Q. Yang & Z. Q. Meng, 2007. "Lagrange Multipliers and Calmness Conditions of Order p," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 95-101, February.
    2. 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.
    3. Max Bucher & Alexandra Schwartz, 2018. "Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 383-410, August.
    4. 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.
    5. A. Pascoletti & P. Serafini, 2007. "Differential Conditions for Constrained Nonlinear Programming via Pareto Optimization," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 399-411, September.
    6. Christian Kanzow & Andreas B. Raharja & Alexandra Schwartz, 2021. "An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 793-813, June.
    7. Sjur D. Flåm & Jan-J. Rückmann, 2022. "The Lagrangian, constraint qualifications and economics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 215-232, October.
    8. X. Q. Yang & Y. Y. Zhou, 2010. "Second-Order Analysis of Penalty Function," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 445-461, August.
    9. Kuang Bai & Yixia Song & Jin Zhang, 2023. "Second-Order Enhanced Optimality Conditions and Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1264-1284, September.
    10. Giorgio Giorgi, 2017. "Minimum Principle-Type Necessary Optimality Conditions in Scalar and Vector Optimization. An Account," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 9(4), pages 168-184, August.
    11. R. Andreani & J. M. Martinez & M. L. Schuverdt, 2005. "On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 473-483, May.
    12. Giorgio Giorgi, 2018. "A Guided Tour in Constraint Qualifications for Nonlinear Programming under Differentiability Assumptions," DEM Working Papers Series 160, University of Pavia, Department of Economics and Management.
    13. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.

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