Abstract
Vector optimization problems are a significant extension of multiobjective optimization, which has a large number of real life applications. In vector optimization the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative orthant. We consider extensions of the projected gradient gradient method to vector optimization, which work directly with vector-valued functions, without using scalar-valued objectives. We provide a direction which adequately substitutes for the projected gradient, and establish results which mirror those available for the scalar-valued case, namely stationarity of the cluster points (if any) without convexity assumptions, and convergence of the full sequence generated by the algorithm to a weakly efficient optimum in the convex case, under mild assumptions. We also prove that our results still hold when the search direction is only approximately computed.
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Drummond, L.G., Iusem, A. A Projected Gradient Method for Vector Optimization Problems. Computational Optimization and Applications 28, 5–29 (2004). https://doi.org/10.1023/B:COAP.0000018877.86161.8b
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DOI: https://doi.org/10.1023/B:COAP.0000018877.86161.8b