Improvement sets and vector optimization

Gutiérrez, César; Jiménez, Bienvenido; Novo, Vicente

doi:10.1016/j.ejor.2012.05.050

Cited by 58 publications

(38 citation statements)

References 15 publications

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“…A very interesting topic in vector optimization is to unify several classical concepts of optimality (usually defined by means of an ordering cone) via a general notion of optimality (see, e.g., Flores-Bazán and Hernández [13], Gutiérrez, Jiménez and Novo [17], and Gutiérrez et al [18]). Let X and Y be two real topological linear spaces, Y = {0 Y }.…”

Section: K-extremal Pointsmentioning

confidence: 99%

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Unifying local–global type properties in vector optimization

Bagdasar

Popovici

2018

J Glob Optim

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It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function's domain. The aim of this paper is to show that these "local min -global min" and "local maxglobal min" type properties can be extended and unified by a single general localglobal extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local-global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.

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Section: K-extremal Pointsmentioning

confidence: 99%

“…Remark 3 a) Besides the concepts of C-minimality and C-maximality presented in Example 2, some other optimality notions can be also defined for suitable choices of K, as shown by Flores-Bazán and Hernández [13], Gutiérrez, Jiménez and Novo [17], and Gutiérrez et al [18].…”

Section: K-extremal Pointsmentioning

confidence: 99%

Unifying local–global type properties in vector optimization

Bagdasar

Popovici

2018

J Glob Optim

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“…In this way, we can deal with efficient, weakly efficient, or approximate ε-efficient points depending on the choice of E (see [3], Remark 2.2 or [2], Remark 4.2).…”

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confidence: 99%

“…This concept has been generalized in [2] to a linear topological space ordered via a convex cone K , i.e., E is an improvement set with respect to K iff the origin does not belong to E, and it is invariant with respect to the sum with K .…”

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confidence: 99%

Improvement Sets and Convergence of Optimal Points

Oppezzi

Rossi

2014

J Optim Theory Appl

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The aim of this paper is to give sufficient conditions for the existence of optimal points with respect to an improvement set, in the framework of Banach spaces and by using a recent definition of such sets, given by M.Chicco et al. and by C.Gutiérrez et al. The study of an economic model is provided as example of application of our achievements.\ud The lower and the upper convergence of optimal points of a convergent sequence of sets, in finite and infinite dimensional settings, are also considered, improving previous results.\ud Finally, some sufficient conditions for the stability of optimal points are developed, discussing their importance via several examples

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“…Scalarization for cone-ordered optimization, including the Pareto case, has been studied extensively. For example, see [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], which present scalarizations of varying degrees of abstraction. Few involve polyhedral cones.…”

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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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Improvement sets and vector optimization

Cited by 58 publications

References 15 publications

Unifying local–global type properties in vector optimization

Unifying local–global type properties in vector optimization

Improvement Sets and Convergence of Optimal Points

Scalarizations for Maximization with Respect to Polyhedral Cones

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