An improved definition of proper efficiency for vector maximization with respect to cones

Benson, Harold P.

doi:10.1016/0022-247x(79)90226-9

Cited by 243 publications

(73 citation statements)

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“…The concept of proper efficiency was introduced for the first time by KUHN and TUCKER in [42], but since then other well-known definitions have been given by GEOFFRION [20], BORWEIN [6], BENSON [3] and HENIG [25]. By the results presented by SAWARAGI, NAKAYAMA and TANINO in [57], for the optimization problem presented in this work, all these four concepts turn out to be equivalent.…”

Section: S That Can Be Subject Of a Number Of Constraints Defined Bymentioning

confidence: 84%

Bibliographical Description

1992

Rede Me and Be Nott Wrothe

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Duality for convex composed programming problemsDissertation, 102 pages, Chemnitz University of Technology, Faculty of Mathematics, 2004 Report The theory of duality represents an important research area in optimization. The goal of this work is to present a conjugate duality treatment of composed programming as well as to give an overview of some recent developments in both scalar and multiobjective optimization.In order to do this, first we study a single-objective optimization problem, in which the objective function as well as the constraints are given by composed functions. By means of the conjugacy approach based on the perturbation theory, we provide different kinds of dual problems to it and examine the relations between the optimal objective values of the duals. Given some additional assumptions, we verify the equality between the optimal objective values of the duals and strong duality between the primal and the dual problems, respectively. Having proved the strong duality, we derive the optimality conditions for each of these duals. As special cases of the original problem, we study the duality for the classical optimization problem with inequality constraints and the optimization problem without constraints.The second part of this work is devoted to location analysis. Considering first the location model with monotonic gauges, it turns out that the same conjugate duality principle can be used also for solving this kind of problems. Taking in the objective function instead of the monotonic gauges several norms, investigations concerning duality for different location problems are made.We finish our investigations with the study of composed multiobjective optimization problems. In doing like this, first we scalarize this problem and study the scalarized one by using the conjugacy approach developed before. The optimality conditions which we obtain in this case allow us to construct a multiobjective dual problem to the primal one. Additionally the weak and strong duality are proved. In conclusion, some special cases of the composed multiobjective optimization problem are considered. Once the general problem has been treated, particularizing the results, we construct a multiobjective dual for each of them and verify the weak and strong dualities.

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Section: S That Can Be Subject Of a Number Of Constraints Defined Bymentioning

confidence: 84%

Bibliographical Description

1992

Rede Me and Be Nott Wrothe

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“…Later Geoffrion defined proper efficiency by eliminating unbounded trade-offs between objectives and studied their relation to Kuhn and Tucker's proper efficiency and linear scalarization [6]. This concept was generalized by Borwein [2,3] and Benson [1] to problems that the objective space is ordered by closed convex cones. There exist so many definitions and interpretations of proper efficiency, so we refer to the papers [7][8][9]16].…”

Section: Introductionmentioning

confidence: 99%

Linear and conic scalarizations for obtaining properly efficient solutions in multiobjective optimization

Karimi

2017

Math Sci

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The existence of equivalent scalar problems for properly efficient point of a given multiobjective optimization problem over arbitrary cones is studied by so many authors. This paper emphasizes two scalarizations, i.e., linear scalarization and conic scalarization, and studies geometrical viewpoint on the relationship between proper efficiency and these scalarizations. We also show that conic scalarization is a generalization of linear scalarization based on augmented dual cone which provides a new type of trade-off for properly efficient solutions.

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“…As observed by Kuhn, Tucker and later by Geoffrion, some efficient points exhibit certain abnormal properties. To eliminate such abnormal efficient points, there are many papers to introduce various concepts of proper efficiency; see [1][2][3][4][5][6][7][8] . Particularly, Zaffaroni 9 introduced the concept of tightly proper efficiency and used a special scalar function to characterize the tightly proper efficiency, and obtained some properties of tightly proper efficiency.…”

Section: Introductionmentioning

confidence: 99%

Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

2011

J Inequal Appl

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A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly cone-subconvexlike set-valued maps. A dual is proposed, and some duality results are obtained in terms of tightly properly efficient solutions. A new type of saddle point, which is called tightly proper saddle point of an appropriate set-valued Lagrange map, is introduced and is used to characterize tightly proper efficiency.

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An improved definition of proper efficiency for vector maximization with respect to cones

Abstract: Journal of Mathematical Analysis and Applications 71 (1979) 232-241. doi:10.1016/0022-247X(79)90226-9Received by publisher: 0000-01-01Harvest Date: 2016-01-04 12:20:16DOI: 10.1016/0022-247X(79)90226-9Page Range: 232-24

Cited by 243 publications

References 8 publications

Bibliographical Description

Bibliographical Description

Linear and conic scalarizations for obtaining properly efficient solutions in multiobjective optimization

Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

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