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Non-Linear Programming Via Penalty Functions


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  • Willard I. Zangwill

    (University of California, Berkeley)

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    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.

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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 13 (1967)
    Issue (Month): 5 (January)
    Pages: 344-358

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    Handle: RePEc:inm:ormnsc:v:13:y:1967:i:5:p:344-358

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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. 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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