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Multiplier convergence in trust-region methods with application to convergence of decomposition methods for MPECs

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Abstract

We study piecewise decomposition methods for mathematical programs with equilibrium constraints (MPECs) for which all constraint functions are linear. At each iteration of a decomposition method, one step of a nonlinear programming scheme is applied to one piece of the MPEC to obtain the next iterate. Our goal is to understand global convergence to B-stationary points of these methods when the embedded nonlinear programming solver is a trust-region scheme, and the selection of pieces is determined using multipliers generated by solving the trust-region subproblem. To this end we study global convergence of a linear trust-region scheme for linearly-constrained NLPs that we call a trust-search method. The trust-search has two features that are critical to global convergence of decomposition methods for MPECs: a robustness property with respect to switching pieces, and a multiplier convergence result that appears to be quite new for trust-region methods. These combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trust-region constraint, or more complex methods that require a separate subproblem for multiplier estimation.

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Authors and Affiliations

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

    Giovanni Giallombardo

  2. Judge Business School, University of Cambridge, Cambridge, CB2 1AG, UK

    Daniel Ralph

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  1. Giovanni Giallombardo
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  2. Daniel Ralph
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Correspondence to Giovanni Giallombardo.

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Giallombardo, G., Ralph, D. Multiplier convergence in trust-region methods with application to convergence of decomposition methods for MPECs. Math. Program. 112, 335–369 (2008). https://doi.org/10.1007/s10107-006-0020-5

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  • DOI: https://doi.org/10.1007/s10107-006-0020-5

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