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Non-Linear Programming Via Penalty Functions

Author

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  • Willard I. Zangwill

    (University of California, Berkeley)

Abstract

The non-linear programming problem seeks to maximize a function f(x) where the n component vector x must satisfy certain constraints g i (x) - 0, i - 1, ..., m 1 and g i (z) \geqq 0, i - m 1 + 1, ..., m. The algorithm presented in this paper solves the non-linear programming problem by transforming it into a sequence of unconstrained maximization problems. Essentially, a penalty is imposed whenever x does not satisfy the constraints. Although the algorithm appears most useful in the concave case, the convergence proof holds for non-concave functions as well. The algorithm is especially interesting in the concave case because the programming problem reduces to a single unconstrained maximization problem or, at most, to a finite sequence of unconstrained maximization problems. In addition, the paper presents a new class of dual problems, and the algorithm is shown to be a dual feasible method. Another property of the algorithm is that it appears particularly well suited for large-scale problems with a sizable number of constraints.

Suggested Citation

  • Willard I. Zangwill, 1967. "Non-Linear Programming Via Penalty Functions," Management Science, INFORMS, vol. 13(5), pages 344-358, January.
    • Handle: RePEc:inm:ormnsc:v:13:y:1967:i:5:p:344-358
    DOI: 10.1287/mnsc.13.5.344
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    Cited by:

    1. Marco Corazza & Stefania Funari & Riccardo Gusso, 2012. "An evolutionary approach to preference disaggregation in a MURAME-based credit scoring problem," Working Papers 5, Department of Management, Università Ca' Foscari Venezia.
    2. M. V. Dolgopolik, 2018. "A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 728-744, March.
    3. Tiago Andrade & Nikita Belyak & Andrew Eberhard & Silvio Hamacher & Fabricio Oliveira, 2022. "The p-Lagrangian relaxation for separable nonconvex MIQCQP problems," Journal of Global Optimization, Springer, vol. 84(1), pages 43-76, September.
    4. Pan, Yan & Duan, Fabing & Xu, Liyan & Chapeau-Blondeau, François, 2019. "Benefits of noise in M-estimators: Optimal noise level and probability density," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).
    5. T. Antczak, 2013. "A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 437-453, November.
    6. Marco Corazza & Giovanni Fasano & Riccardo Gusso, 2011. "Particle Swarm Optimization with non-smooth penalty reformulation for a complex portfolio selection problem," Working Papers 2011_10, Department of Economics, University of Venice "Ca' Foscari".
    7. 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.
    8. Roberto Andreani & Ellen H. Fukuda & Paulo J. S. Silva, 2013. "A Gauss–Newton Approach for Solving Constrained Optimization Problems Using Differentiable Exact Penalties," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 417-449, February.
    9. Giorgio Giorgi & Bienvenido Jiménez & Vicente Novo, 2014. "Some Notes on Approximate Optimality Conditions in Scalar and Vector Optimization Problems," DEM Working Papers Series 095, University of Pavia, Department of Economics and Management.
    10. Ellen H. Fukuda & L. M. Graña Drummond & Fernanda M. P. Raupp, 2016. "An external penalty-type method for multicriteria," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 493-513, July.
    11. Hua Zhou & Kenneth Lange, 2015. "Path following in the exact penalty method of convex programming," Computational Optimization and Applications, Springer, vol. 61(3), pages 609-634, July.
    12. Xinhua Mao & Jianwei Wang & Changwei Yuan & Wei Yu & Jiahua Gan, 2018. "A Dynamic Traffic Assignment Model for the Sustainability of Pavement Performance," Sustainability, MDPI, vol. 11(1), pages 1-19, December.
    13. Kaiwen Meng & Xiaoqi Yang, 2015. "First- and Second-Order Necessary Conditions Via Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 720-752, June.
    14. Duan Yaqiong & Lian Shujun, 2016. "Smoothing Approximation to the Square-Root Exact Penalty Function," Journal of Systems Science and Information, De Gruyter, vol. 4(1), pages 87-96, February.
    15. A. J. Zaslavski, 2014. "An Approximate Exact Penalty in Constrained Vector Optimization on Metric Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 649-664, August.
    16. Rao, K.S. Rama & Sunderan, T. & Adiris, M. Ref'at, 2017. "Performance and design optimization of two model based wave energy permanent magnet linear generators," Renewable Energy, Elsevier, vol. 101(C), pages 196-203.
    17. Tadeusz Antczak & Najeeb Abdulaleem, 2023. "On the exactness and the convergence of the $$l_{1}$$ l 1 exact penalty E-function method for E-differentiable optimization problems," OPSEARCH, Springer;Operational Research Society of India, vol. 60(3), pages 1331-1359, September.
    18. T. Antczak, 2018. "Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 205-224, January.
    19. Amaioua, Nadir & Audet, Charles & Conn, Andrew R. & Le Digabel, Sébastien, 2018. "Efficient solution of quadratically constrained quadratic subproblems within the mesh adaptive direct search algorithm," European Journal of Operational Research, Elsevier, vol. 268(1), pages 13-24.
    20. Antczak, Tadeusz, 2009. "Exact penalty functions method for mathematical programming problems involving invex functions," European Journal of Operational Research, Elsevier, vol. 198(1), pages 29-36, October.
    21. 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.

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