ε-Properly Efficiency of Multiobjective Semidefinite Programming with Set-Valued Functions

Mathematical Problems in Engineering ◽

10.1155/2017/5978130 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7

Author(s):

Chun Hong Yuan ◽

Wei Dong Rong

Keyword(s):

Saddle Point ◽

Semidefinite Programming ◽

Lagrange Multiplier ◽

Alternative Theorem ◽

Generalized Cone Subconvexlikeness ◽

Multiplier Theorems

In this paper, we study the ε-properly efficiency of multiobjective semidefinite programming with set-valued functions. Firstly, we obtain the scalarization theorems under the condition of the generalized cone-subconvexlikeness. Then, we establish the alternative theorem which contains matrixes and vectors, the ε-Lagrange multiplier theorems, and the ε-proper saddle point theorems of the primal programming under some suitable conditions.

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Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation

Nonlinear Analysis ◽

10.1016/0362-546x(79)90079-8 ◽

1979 ◽

Vol 3 (2) ◽

pp. 239-248 ◽

Cited By ~ 14

Author(s):

Ivan Singer

Keyword(s):

Convex Optimization ◽

Lagrange Multiplier ◽

Best Approximation ◽

Duality Theorem ◽

Multiplier Theorems ◽

New Applications

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ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces

The Scientific World JOURNAL ◽

10.1155/2013/403642 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Zhi-Ang Zhou

Keyword(s):

Saddle Point ◽

Optimization Problem ◽

Optimization Problems ◽

Saddle Points ◽

Linear Spaces ◽

Equivalent Characterization ◽

Generalized Cone Subconvexlikeness ◽

Real Linear ◽

The Relationship

We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization ofϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between theϵ-Henig saddle point of the Lagrangian set-valued map and theϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.

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Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

Abstract and Applied Analysis ◽

10.1155/2013/570918 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Xiaohong Hu ◽

Zhimiao Fang ◽

Yunxuan Xiong

Keyword(s):

Saddle Point ◽

Vector Optimization ◽

Optimization Problem ◽

Efficient Solutions ◽

Vector Optimization Problem ◽

Well Posedness ◽

Scalar Problem ◽

New Type ◽

Strict Efficiency ◽

Multiplier Theorems

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

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