Abstract
In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graña Drummond and Iusem, since it admits relative errors on the search directions. At each iteration, a decrease of the objective value is obtained by means of an Armijo-like rule. The convergence results of this new method extend those obtained by Fukuda and Graña Drummond for the exact version. For partial orders induced by both pointed and nonpointed cones, under some reasonable hypotheses, global convergence to weakly efficient points of all sequences generated by the inexact projected gradient method is established for convex (respect to the ordering cone) objective functions. In the convergence analysis we also establish a connection between the so-called weighting method and the one we propose.
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Acknowledgements
We are grateful to the referees for their helpful comments. This research was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (grant 2010/20572-0), Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (grant Cientistas do Nosso Estado 2008) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (Universal grant 480101/2008-6).
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Fukuda, E.H., Graña Drummond, L.M. Inexact projected gradient method for vector optimization. Comput Optim Appl 54, 473–493 (2013). https://doi.org/10.1007/s10589-012-9501-z
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DOI: https://doi.org/10.1007/s10589-012-9501-z