Connectedness and Stability of the Solution Sets in Linear Fractional Vector Optimization Problems

Yen, Nguyen Dong; Phuong, Ta Duy

doi:10.1007/978-1-4613-0299-5_29

Cited by 21 publications

(10 citation statements)

References 10 publications

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“…Remark 5.5 In Section 3, the method of [23] for proving the result in Corollary 5.1 has been developed furthermore to obtain Theorem 3.1.…”

Section: Connectedness Of the Solution Setsmentioning

confidence: 98%

“…Next, let us recall some basic information about the linear fractional vector optimization problem (or LFVOP). More details can be found in [20,23] and [12,Chap. 8].…”

Section: Definition 23mentioning

confidence: 99%

“…Our results imply several facts on solution stability and connectedness of the solution sets of convex quadratic vector optimization problems and of linear fractional vector optimization problems. Note that the latter form an interesting class of nonconvex problems in vector optimization which have been studied by several authors (see [2,3,[6][7][8][9]16,20,23] and the *Corresponding author. Email: ndyen@math.ac.vn yDedicated to Professor Franco Giannessi in celebration of his 75th birthday.…”

Section: Introductionmentioning

confidence: 99%

“…This valuable observation of [23], which has been used also in [7], will allow us to derive several results on (VP 2 ) from the corresponding results on monotone AVVIs.…”

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confidence: 97%

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Monotone affine vector variational inequalities

Yen

Yao

2011

Optimization

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By using a stability theorem of S.M. Robinson and a scalarization method, we establish sufficient conditions for the upper semicontinuity of the solution maps of parametric monotone affine vector variational inequalities. As a by-product, some new topological properties of the solution sets of these problems are obtained. Our results imply several facts for the solution stability and connectedness of the solution sets of convex quadratic vector optimization problems and of linear fractional vector optimization problems.

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“…Remark 5.5 In Section 3, the method of [23] for proving the result in Corollary 5.1 has been developed furthermore to obtain Theorem 3.1.…”

Section: Connectedness Of the Solution Setsmentioning

confidence: 98%

“…Next, let us recall some basic information about the linear fractional vector optimization problem (or LFVOP). More details can be found in [20,23] and [12,Chap. 8].…”

Section: Definition 23mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…This valuable observation of [23], which has been used also in [7], will allow us to derive several results on (VP 2 ) from the corresponding results on monotone AVVIs.…”

mentioning

confidence: 97%

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Monotone affine vector variational inequalities

Yen

Yao

2011

Optimization

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“…⊓ ⊔ Remark 3.2 In [11], there is a conjecture that the maximal number of connected components of the solution sets of a monotone AVVI equals to m -the number of the criteria in the problem. Theorem 3.2 shows that the conjecture is wrong when m = 2 and n ≥ 4 because, in that case, n 2 + 1 ≥ 3 > m. [14], the necessary and sufficient condition for a point to be a Pareto solution of a linear fractional vector optimization problem (LFVOP) can be regarded as the condition for that point to be a Pareto solution of a skew-symmetric AVVI. In [14], the authors also observed that the necessary and sufficient condition for a point to be a weak Pareto solution of a linear fractional vector optimization problem (LFVOP) can be regarded as the condition for that point to be a weak Pareto solution of a skew-symmetric AVVI.…”

Section: Unconstrained Problemsmentioning

confidence: 99%

Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

Hieu¹

2018

J Glob Optim

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This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question 2 in [N.D. Yen and J.-C. Yao, Monotone affine vector variational inequalities, Optimization 60 (2011), pp. 53-68] and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation.Keywords Monotone affine vector variational inequality · Solution set · Number of connected components · Scalarization formula · Skew-symmetric matrix Mathematics Subject Classification (2000) 49J40 · 47H05 · 90C29 · 90C33 Recently, thanks to several results from Real Algebraic Geometry, the investigation of the connectedness structure of the solution sets of polynomial

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