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Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties

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Abstract

The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first- or second-order point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, M-stationarity, or C-stationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic-mode formulation is solved inexactly.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

    Mihai Anitescu

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Paul Tseng

  3. Computer Sciences Department, University of Wisconsin, 1210 West Dayton Street, Madison, WI, 53706, USA

    Stephen J. Wright

Authors
  1. Mihai Anitescu
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  2. Paul Tseng
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  3. Stephen J. Wright
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Correspondence to Mihai Anitescu.

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The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory (“Argonne”) under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

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Anitescu, M., Tseng, P. & Wright, S.J. Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties. Math. Program. 110, 337–371 (2007). https://doi.org/10.1007/s10107-006-0005-4

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  • DOI: https://doi.org/10.1007/s10107-006-0005-4

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