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Optimality conditions and global convergence for nonlinear semidefinite programming

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Abstract

Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush–Kuhn–Tucker (AKKT), well known in nonlinear optimization. The second one, called Trace-AKKT, is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided. We propose an augmented Lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed, weaker than previously considered ones, which are sufficient for the global convergence of the algorithm to a stationary point.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas, Campinas, SP, Brazil

    Roberto Andreani

  2. Department of Applied Mathematics, University of São Paulo, São Paulo, SP, Brazil

    Gabriel Haeser & Daiana S. Viana

  3. Center of Exact and Technological Sciences, Federal University of Acre, Rio Branco, AC, Brazil

    Daiana S. Viana

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  1. Roberto Andreani
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  2. Gabriel Haeser
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  3. Daiana S. Viana
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Correspondence to Gabriel Haeser.

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This work was supported by FAPESP (Grants 2013/05475-7 and 2017/18308-2), CNPq and CAPES.

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Andreani, R., Haeser, G. & Viana, D.S. Optimality conditions and global convergence for nonlinear semidefinite programming. Math. Program. 180, 203–235 (2020). https://doi.org/10.1007/s10107-018-1354-5

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