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Non-Linear Programming Via Penalty Functions


  • Willard I. Zangwill

    (University of California, Berkeley)


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

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    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. 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".
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    5. 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.
    6. 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.
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    12. 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.
    13. repec:bpj:jossai:v:4:y:2016:i:1:p:87-96:n:6 is not listed on IDEAS
    14. 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.
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    17. 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.

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