On inexact projected gradient methods for solving variable vector optimization problems
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On inexact projected gradient methods for solving variable vector optimization problems J. Y. Bello-Cruz1 · G. Bouza Allende2 Received: 7 April 2020 / Revised: 31 August 2020 / Accepted: 7 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, inexact projected gradient methods for solving smooth constrained vector optimization problems on variable ordered spaces are presented. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. The convergence results are also derived in the particular case in which the problem is unconstrained and even if inexact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map coincides with the image of the objective function. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed. Keywords Gradient method · K -convexity · Variable order · Vector optimization · Weakly efficient points Mathematics Subject Classification 90C29 · 90C52 · 65K05 · 35E10
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J. Y. Bello-Cruz [email protected] G. Bouza Allende [email protected]
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Department of Mathematical Sciences, Northern Illinois University, Watson Hall 366, DeKalb, IL 60115, USA
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Facultad de Matemática y Computación, Universidad de La Habana, 10400 La Habana, Cuba
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J. Y. Bello-Cruz, G. B. Allende
1 Introduction Variable order structures are a natural extension of the well-known fixed (partial) order given by a closed, pointed and convex cone; see Eichfelder (2014). This kind of orderings model situations in which the comparison between two points depends on a set-valued map. These problems have recently received much attention from the optimization community due to their broad application to several different areas. Variable order structures (VOS), given by a point-to-cone valued map, were well studied in Eichfelder (2014), Eichfelder (2011), Engau (2008), motivated by important applications. VOS appear in medical diagnosis (Eichfelder 2014), portfolio optimization (Wiecek 2007), capability theory of well-being (Bao et al. 2015b), psychological modeling (Bao et al. 2015a), consumer preferences (John 2001, 2006) and location theory, etc; see, for instance, Baatar and Wiecek (2006), Engau (2008). The main goal is to model elements of a certain set such that their objective function evaluation cannot be improved by the image of any other feasible point with respect to the variable order. So, their mathematical descriptio
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