Strong Convergence Theorem for the Lexicographic Ky Fan Inequality

Ánh, Phạm Ngọc; Thuy, Le Quang; Anh, Tran T. H.

doi:10.1007/s10013-017-0253-z

Cited by 11 publications

(7 citation statements)

References 24 publications

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“…Using the equilibrium convex function

g

, in Yen et al, for each

x = false(x_{1}, …, x_{5} false)

is formulated as

g (x) = x^{T} G x + \sum_{k = 1}^{5} c_{k} (x_{k}), c_{k} (x_{k}) = \frac{1}{2} p_{k} x_{k}^{2} + q_{k} x_{k}, p_{k} = \sqrt{k}, q_{k} = \frac{1}{k} \forall k = 1, …, 5,

G

is a diagonal matrix, where

G = {false(g_{i j} false)}_{5 \times 5}

and

g_{i i} = \sqrt[3]{2 i + 1}

for all

i = 1, …, 5

. The nonexpansive mappings

S_{i} false(i = 1, 2 false)

, in Anh et al, are given as

S_{1} (x) = (\frac{1}{3} x_{1} - \frac{2}{3}, x_{2}, …, x_{5}), S_{2} (x) = (x_{1}

…”

Section: Computational Experimentsmentioning

confidence: 87%

“…This section provides some several numerical experiments to illustrate strong convergence of the proposed algorithm and compare them with others such as the subgradient projection algorithm (SPA) of Anh et al 3 ans A is a matrix of order n, B is a n × n skew-symmetric matrix, C is a n × n diagonal matrix, and q is a vector in  n . Thus, M is positive semidefinite.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…This section provides some several numerical experiments to illustrate strong convergence of the proposed algorithm and compare them with others such as the subgradient projection algorithm (SPA) of Anh et al and the projected viscosity subgradient algorithm (PVSA) of Maingé . All the programs are written in Matlab R2014a and performed on a Intel(R) Core(TM) i5 CPU @ 2.30GHz, RAM 8.00 GB.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…In the case

S_{i} false(i \in I false)

is the identity, Problem is formulated in the form of Problem . These problems have closely related to many other, for example, as the variational inequality problem

V I s

in the case

F = 0

, the multivalued variational inequality problem when

f = 0

, bilevel variational inequality problems, some classes of mathematical programs with equilibrium constraints, bilevel minimization problems, minimum‐norm problems of the solution set of variational inequalities, and bilevel convex programming models . An important case of Problem is the lexicographic variational inequalities with single‐valued mapping

F

.…”

Section: Introductionmentioning

confidence: 99%

“…Using the equilibrium convex function g, in Yen et al, for each x = (x 1 , … , x 5 ) is formulated asg(x) = x T Gx + k=1 c k (x k ), c k (x k ) = 1 2 p k x 2 k + q k x k , p k = √ k, q k = 1 k ∀k = 1, … , 5,G is a diagonal matrix, where G = (g i ) 5×5 and g ii = 2i + 1 for all i = 1, … , 5. The nonexpansive mappings S i (i = 1, 2), in Anh et al,3 are given as S 1 (x) = ( (x) = (x 1 , − cos(x 2 + 1), x 3 , … , x 5 ).…”

mentioning

confidence: 99%

See 4 more Smart Citations

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

Ánh

Thi

2019

Math Methods in App Sciences

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In this paper, building upon projection methods and parallel splitting‐up techniques with using proximal operators, we propose new algorithms for solving the multivalued lexicographic variational inequalities in a real Hilbert space. First, the strong convergence theorem is shown with Lipschitz continuity of the cost mapping, but it must satisfy a strongly monotone condition. Second, the convergent results are also established to the multivalued lexicographic variational inequalities involving a finite system of demicontractive mappings under mild assumptions imposed on parameters. Finally, some numerical examples are developed to illustrate the behavior of our algorithms with respect to existing algorithms.

show abstract

“…Using the equilibrium convex function

g

, in Yen et al, for each

x = false(x_{1}, …, x_{5} false)

is formulated as

g (x) = x^{T} G x + \sum_{k = 1}^{5} c_{k} (x_{k}), c_{k} (x_{k}) = \frac{1}{2} p_{k} x_{k}^{2} + q_{k} x_{k}, p_{k} = \sqrt{k}, q_{k} = \frac{1}{k} \forall k = 1, …, 5,

G

is a diagonal matrix, where

G = {false(g_{i j} false)}_{5 \times 5}

and

g_{i i} = \sqrt[3]{2 i + 1}

for all

i = 1, …, 5

. The nonexpansive mappings

S_{i} false(i = 1, 2 false)

, in Anh et al, are given as

S_{1} (x) = (\frac{1}{3} x_{1} - \frac{2}{3}, x_{2}, …, x_{5}), S_{2} (x) = (x_{1}

…”

Section: Computational Experimentsmentioning

confidence: 87%

Section: Computational Experimentsmentioning

confidence: 99%

“…This section provides some several numerical experiments to illustrate strong convergence of the proposed algorithm and compare them with others such as the subgradient projection algorithm (SPA) of Anh et al and the projected viscosity subgradient algorithm (PVSA) of Maingé . All the programs are written in Matlab R2014a and performed on a Intel(R) Core(TM) i5 CPU @ 2.30GHz, RAM 8.00 GB.…”

Section: Computational Experimentsmentioning

confidence: 99%

“…In the case

S_{i} false(i \in I false)

is the identity, Problem is formulated in the form of Problem . These problems have closely related to many other, for example, as the variational inequality problem

V I s

in the case

F = 0

, the multivalued variational inequality problem when

f = 0

F

.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

Ánh

Thi

2019

Math Methods in App Sciences

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show abstract

Convergence of the projection and contraction methods for solving bilevel variational inequality problems

Thang

Ánh²,

Truong

2023

Math Methods in App Sciences

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In this work, we analyze some convergent properties of a projection and contraction algorithm for solving a variational inequality problem, where the feasible domain is the solution set of an affine variational inequality problem. We prove that, for solving the problem where the second cost mapping is affine and not necessary for monotone properties, any iterative sequence generated by the algorithm converges to a unique solution provided that the first cost mapping is strongly monotone and Lipschitz continuous. Computational errors of the algorithm are showed. Finally, some preliminary numerical experiences and comparisons are also reported.

show abstract

DC auxiliary principle methods for solving lexicographic equilibrium problems

Ánh

Ansari

2022

J Glob Optim

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Strong Convergence Theorem for the Lexicographic Ky Fan Inequality

Cited by 11 publications

References 24 publications

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

Modified parallel projection methods for the multivalued lexicographic variational inequalities using proximal operator in Hilbert spaces

Convergence of the projection and contraction methods for solving bilevel variational inequality problems

DC auxiliary principle methods for solving lexicographic equilibrium problems

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