A saddle point characterization of efficient solutions for interval optimization problems

2017 ◽  
Vol 58 (1-2) ◽  
pp. 193-217 ◽  
Author(s):  
Debdulal Ghosh ◽  
Debdas Ghosh ◽  
Sushil Kumar Bhuiya ◽  
Lakshmi Kanta Patra
Keyword(s):  
Saddle Point ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Interval Optimization

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

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.

∊-STRICTLY EFFICIENT SOLUTIONS OF VECTOR OPTIMIZATION PROBLEMS WITH SET-VALUED MAPS

2007 ◽  
Vol 24 (06) ◽  
pp. 841-854 ◽  
Author(s):  
TAIYONG LI ◽  
YIHONG XU ◽  
CHUANXI ZHU
Keyword(s):  
Saddle Point ◽  
Vector Optimization ◽  
Efficient Solution ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Lagrangian Multiplier ◽  
Multiplier Theorem ◽  
Lagrangian Multiplier Theorem

In this paper, the notion of ∊-strictly efficient solution for vector optimization with set-valued maps is introduced. Under the assumption of the ic-cone-convexlikeness for set-valued maps, the scalarization theorem, ∊-Lagrangian multiplier theorem, ∊-saddle point theorems and ∊-duality assertions are established for ∊-strictly efficient solution.

scholarly journals Spectral analysis of saddle-point matrices from optimization problems with elliptic PDE constraints

2020 ◽  
Vol 36 (36) ◽  
pp. 773-798
Author(s):  
Fabio Durastante ◽  
Isabella Furci
Keyword(s):  
Spectral Analysis ◽  
Saddle Point ◽  
Spectral Properties ◽  
Optimization Problems ◽  
Elliptic Partial Differential Equations ◽  
Elliptic Pde ◽  
The One ◽  
Saddle Point Matrices ◽  
Saddle Point Matrix

The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. They uncover the existence of an hidden structure in these matrix sequences, namely, they show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. They show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that they deal with saddle--point matrices. Finally, they exploit it to propose an optimal preconditioner strategy for the GMRES, and Flexible--GMRES methods.

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

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