Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

Gutiérrez, César; Huerga, Lidia; Novo, Vicente

doi:10.1111/itor.12398

Cited by 10 publications

(12 citation statements)

References 29 publications

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“…They also designed a Newton-type algorithm for solving it. Very recently, Gutiérrez et al [15,16] characterized the several kind of exact and approximate efficient solutions of a class of multiobjective optimization problems with partial order provided by a polyhedral cone. From a computational point of view, they showed that the results in [15,16] based on the ordering cone generated by some matrix are attractive.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, Gutiérrez et al [15,16] characterized the several kind of exact and approximate efficient solutions of a class of multiobjective optimization problems with partial order provided by a polyhedral cone. From a computational point of view, they showed that the results in [15,16] based on the ordering cone generated by some matrix are attractive. Using the partial order considered in [16], Hai et al [17] continued to investigating vector equilibrium problems with a polyhedral ordering cone.…”

Section: Introductionmentioning

confidence: 99%

“…From a computational point of view, they showed that the results in [15,16] based on the ordering cone generated by some matrix are attractive. Using the partial order considered in [16], Hai et al [17] continued to investigating vector equilibrium problems with a polyhedral ordering cone. In [17], variants of the Ekeland variational principle for a type of approximate proper solutions of those vector equilibrium problems were also provided.…”

Section: Introductionmentioning

confidence: 99%

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Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

Hùng¹,

Novo

Tâm

2021

J Glob Optim

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The aim of this paper is to establish new results on the error bounds for a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix. We first propose some regularized gap functions of this problem using the concept of $$\mathcal {G}_{A}$$ G A -convexity of a vector-valued function. Then, we derive error bounds for vector equilibrium problems with partial order given by a polyhedral cone in terms of regularized gap functions under some suitable conditions. Finally, a real-world application to a vector network equilibrium problem is given to illustrate the derived theoretical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

Hùng¹,

Novo

Tâm

2021

J Glob Optim

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“…Sawaragi et al [1985] reduce the problem to finding Pareto efficient points of a set in the objective function's image space. More recently, for example, Gutierrez, et al [19] define a scalarization by perturbing the same matrix.…”

Section: Introductionmentioning

confidence: 99%

Scalarizations for Maximization with Respect to Polyhedral Cones

Corley¹

2017

JAAM

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Abstract. Efficient points are obtained for cone-ordered maximizations in n R using the method of scalarization. Various scalarizations are presented for ordering cones in general and then for the important special case of polyhedral cones. For polyhedral cones, it is shown how to find vectors in the positive dual cone that are needed for a scalarized objective function. Instructive examples are presented. Keywords:Multiobjective optimization, polyhedral cones, cone maximization, Pareto maximization, scalarization. IntroductionMultiobjective optimization is applied in various fields of science, engineering, and economics when decisions involve two or more conflicting objectives. For example, a pharmaceutical company may wish to determine a dose for a new drug that would maximize its therapeutic benefits while minimizing its deleterious side effects. The result would be a multiobjective optimization problem. In many other situations, however, a decision may not be adequately characterized by such standard criteria as Pareto optimality, goal programming, or lexicographic maximization. For this reason, cone-ordered maximization was developed, of which the previously mentioned criteria are special cases. . Such results are difficult to implement, however. For this reason, scalarization is the method of choice for actually obtaining a numerical solution to a cone-ordered maximization. A scalarization is an optimization problem with a single scalar objective function for which an optimal solution solves the cone-ordered problem. Examples are the scalarizations of Borwein [6] and Jahn [10] in terms of the dual cone and that of Soland [11] for Pareto maximization involving both the dual cone and aspiration levels for the individual objective functions. References [12][13][14][15][16][17][18] consolidate the theory and solution techniques for cone-ordered maximizations. However, specific results for polyhedral cones are scarce. For such cones, Yu [5] proposes a specialized scalarization based on finding cone extreme points. Sawaragi et al. [1985] reduce the problem to finding Pareto efficient points of a set in the objective function's image space. More recently, for example, Gutierrez, et al. [19] define a scalarization by perturbing the same matrix.We consider a decision problem with an n-dimensional objective function, where the optimization criterion is maximization with respect to a partial order induced by a pointed convex cone in .n R The purpose here is to present scalarizations for such problems in general, with the emphasis on polyhedral cones. Methods are also developed for finding dual vectors of a polyhedral cone as required for scalarized objective functions. One such approach involves linear programming. The paper is organized as follows. In Section 2 the requisite preliminaries concerning cones are established, while the notion of coneordered maximization is defined in Section 3. In Section 4 various scalarizations are presented. Section 5 then addresses the problem of obtaining the required dual vector...

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“…There are three main factors involved with the profit maximization of the company: waste minimization, surplus minimization, and cost minimization. Therefore, a multiobjective modelling and optimization (Boros, Fehér, Lakner, Niroomand, & Vizvári, 2016;Cardillo, Cascini, Frillici, & Rotini, 2013;Franco, Jablonsky, Leopold-Wildburger, & Montibeller, 2009;Ghasemi & Varaee, 2017;Gholizadeh & Baghchevan, 2017;Gutiérrez, Huerga, & Novo, 2018;Jablonsky, 2014;Jiménez, Bilbao-Terol, & Arenas-Parra, 2018;Kovács & Marian, 2002;Özmen, Karakaya, & Köksalan, 2018;Sanei, Mahmoodirad, Niroomand, Jamalian, & Gelareh, 2017;Taassori et al, 2015) is a suitable approach for incorporating the above-mentioned objectives. Furthermore, when some parameters of the problem have uncertain nature, it must be reflected in the model as well (Fullér, Canós-Darós, & Canós-Darós, 2012;Fullér & Majlender, 2004;Moloudzadeh, Allahviranloo, & Darabi, 2013;Niroomand, Mahmoodirad, Heydari, Kardani, & Hadi-Vencheh, 2017;Salahshour & Allahviranloo, 2013;Salmasnia, Khatami, Baradaran Kazemzadeh, & Zegordi, 2015;Taassori, Niroomand, Uysal, Hadi-Vencheh, & Vizvari, 2016;Wang, Zhou, Li, Zhang, & Chen, 2014).…”

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confidence: 99%

A hybrid solution approach for fuzzy multiobjective dual supplier and material selection problem of carton box production systems

Niroomand¹,

Mahmoodirad

Mosallaeipour

2018

Expert Systems

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A novel optimization problem of carton box manufacturing industries is introduced in this paper. A mixed integer linear formulation with multiple objective functions is developed in order to determine the value of some criteria of carton raw sheets such as size, amount, and supplier under simultaneous minimization of multiple goals such as purchasing cost of raw sheets under discount policy, wastage remained from raw sheets, and quantity of surplus of carton boxes. In order to cope with the unstable market of this sector, some parameters of the proposed formulation such as demand value of the products and price given for raw sheets are assumed to be fuzzy numbers. To tackle such fuzzy multiobjective problem, first, the fuzzy problem is converted to a crisp form using the concepts of necessity‐based chance‐constrained modelling approach. Then a new hybrid form of the fuzzy programming approach is proposed to solve the obtained crisp multiobjective problem effectively. Computational experiments on a real case given by a carton box factory show the superior result of the proposed solution approach compared with the well‐known multiobjective solution methods taken from the literature.

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Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

Cited by 10 publications

References 29 publications

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

Scalarizations for Maximization with Respect to Polyhedral Cones

A hybrid solution approach for fuzzy multiobjective dual supplier and material selection problem of carton box production systems

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