An inexact multiple proximal bundle algorithm for nonsmooth nonconvex multiobjective optimization problems
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An inexact multiple proximal bundle algorithm for nonsmooth nonconvex multiobjective optimization problems N. Hoseini Monjezi1
· S. Nobakhtian1,2
Accepted: 21 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a class of nonsmooth nonconvex multiobjective problems, we develop an inexact multiple proximal bundle method. In our approach instead of scalarization, we find descent direction for every objective function separately by utilizing the inexact proximal bundle method. Then we attempt to find a common descent direction for all objective functions. We study the effect of the inexactness of the objective and subgradient values on the new proposed method and obtain the reasonable convergence properties. We further consider a class of difficult nonsmooth nonconvex problems, made even more difficult by inserting the inexactness in the available information. At the end, to demonstrate the efficiency of the proposed algorithm, some encouraging numerical experiments are provided. Keywords Multiobjective optimization · Proximal bundle method · Inexact information Mathematics Subject Classification 90C29 · 90C26 · 49J52
1 Introduction Multiobjective optimization arises in many real life applications for instance, in the fields of economics, engineering, mechanics, statistics, internet routing, location problem and etc.; for instance see Doolittle et al. (2018) and Liao et al. (2011). Along with the multiobjective nature, many practical applications have nonsmooth structure, for instance, in the fields of the optimal shape design (Mäkelä and Neittaanmäki 1992), economics (Outrata et al. 1998), mechanics (Moreau et al. 1988) and machine learning (Jin 2006), to name but a few. Beside the nonsmoothness, some of the practical multiobjective problems have only access
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S. Nobakhtian [email protected] N. Hoseini Monjezi [email protected]
1
Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran
2
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
123
Annals of Operations Research
to inexact information. This is common in the large-scale Lagrangian relaxation (Emiel and Sagastizábal 2010; Lemaréchal 2001; Sagastizábal 2012), the Lagrangian relaxation with on-demand accuracy (de Oliveira and Sagastizábal 2014) and in the stochastic programming (Ruszczy´nski 2003). In addition, we refer to Sagastizábal (2012) for some examples in the energy sector with inexact information. We consider a nonsmooth multiobjective optimization problem of the form min { f 1 (x), f 2 (x), . . . , f m (x)} x ∈ Rn ,
(1)
where the objective functions f i : Rn → R, i ∈ I := {1, 2, . . . , m} are locally Lipschitz but potentially nonsmooth and nonconvex. One of the basic approaches to solve multiobjective optimization problems is scalarization, which transforms a multiobjective problem into a single objective one. To this end different procedures are usually applied for i
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