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On solving the Lagrangian dual of integer programs via an incremental approach

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Abstract

The Lagrangian dual of an integer program can be formulated as a min-max problem where the objective function is convex, piecewise affine and, hence, nonsmooth. It is usually tackled by means of subgradient algorithms, or multiplier adjustment techniques, or even more sophisticated nonsmooth optimization methods such as bundle-type algorithms.

Recently a new approach to solving unconstrained convex finite min-max problems has been proposed, which has the nice property of working almost independently of the exact evaluation of the objective function at every iterate-point.

In the paper we adapt the method, which is of the descent type, to the solution of the Lagrangian dual. Since the Lagrangian relaxation need not be solved exactly, the approach appears suitable whenever the Lagrangian dual must be solved many times (e.g., to improve the bound at each node of a branch-and-bound tree), and effective heuristic algorithms at low computational cost are available for solving the Lagrangian relaxation.

We present an application to the Generalized Assignment Problem (GAP) and discuss the results of our numerical experimentation on a set of standard test problems.

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Authors and Affiliations

  1. Dipartimento di Elettronica Informatica e Sistemistica, Università della Calabria, 87036, Rende, (CS), Italy

    Manlio Gaudioso, Giovanni Giallombardo & Giovanna Miglionico

Authors
  1. Manlio Gaudioso
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  2. Giovanni Giallombardo
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  3. Giovanna Miglionico
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Corresponding author

Correspondence to Giovanni Giallombardo.

Additional information

This research has been partially supported by the Italian “Ministero dell’Università e della Ricerca Scientifica,” under PRIN2005 research project “Numerical methods for global optimization and for some classes of nondifferentiable optimization” (2005017083_002).

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Gaudioso, M., Giallombardo, G. & Miglionico, G. On solving the Lagrangian dual of integer programs via an incremental approach. Comput Optim Appl 44, 117–138 (2009). https://doi.org/10.1007/s10589-007-9149-2

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  • DOI: https://doi.org/10.1007/s10589-007-9149-2

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