A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping

Iiduka, Hideaki; Yamada, Isao

doi:10.1137/070702497

Cited by 97 publications

(55 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This set is represented as the fixed point set of the mapping composed of the metric projections onto the C i s [ [2], Proposition 4.2]. Iterative algorithms have been presented in [2][3][4] for the convex optimization problem with a fixed point constraint along with proof that these algorithms converge strongly to the unique solution of problems with a strongly monotone operator. The strong monotonicity condition guarantees the uniqueness of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, a number of mathematical programs and iterative algorithms have been developed to resolve complex real world problems. In particular, monotone variational inequalities with a fixed point constraint [2][3][4] include such practical problems as signal recovery [3], beamforming [5], and power control [6], and many iterative algorithms for solving them have been presented.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Zeng

Wong

Yao

2012

Fixed Point Theory Appl

View full text Add to dashboard Cite

Up to now, a large number of practical problems such as signal processing and network resource allocation have been formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms for solving these problems have been proposed. The purpose of this article is to investigate a monotone variational inequality with variational inequality constraint over the fixed point set of one or finitely many nonexpansive mappings, which is called the triple-hierarchical constrained optimization. Two relaxed hybrid steepest-descent algorithms for solving the triple-hierarchical constrained optimization are proposed. Strong convergence for them is proven. Applications of these results to constrained generalized pseudoinverse are included.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Zeng

Wong

Yao

2012

Fixed Point Theory Appl

View full text Add to dashboard Cite

show abstract

“…To exploit the existing methods of solving Problem 1, as presented in [5,13,14,17,20,22,41,42], we shall consider the following convex optimization problem: minimize f (x) := 1 2 x, Q x + b, x subject to x ∈Ĉ := m i=1 C i , where Q ∈ R K ×K is positive definite, b ∈ R K , and C i ⊂ R K (i = 1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

Fixed point optimization algorithm and its application to power control in CDMA data networks

Iiduka

2010

Math. Program.

Self Cite

View full text Add to dashboard Cite

We discuss the variational inequality problem for a continuous operator over the fixed point set of a nonexpansive mapping. One application of this problem is a power control for a direct-sequence code-division multiple-access data network. For such a power control, each user terminal has to be able to quickly transmit at an ideal power level such that it can get a sufficient signal-to-interference-plus-noise ratio and achieve the required quality of service. Iterative algorithms to solve this problem should not involve auxiliary optimization problems and complicated computations. To ensure this, we devise a fixed point optimization algorithm for the variational inequality problem and perform a convergence analysis on it. We give numerical examples of the algorithm as a power control.

show abstract

“…In the last years, many authors studied the problems of finding a common element of the set of fixed points of nonlinear operator and the set of solutions of an equilibrium problem (and the set of solutions of variational inequality problem) in the framework of Hilbert spaces, see, for instance, [2,11,18,20,24,31] and the references therein. The motivation for studying such a problem is in its possible application to mathematical models whose constraints can be expressed as fixed-point problems and/or equilibrium problem.…”

Section: Introductionmentioning

confidence: 99%

“…The motivation for studying such a problem is in its possible application to mathematical models whose constraints can be expressed as fixed-point problems and/or equilibrium problem. This happens, in particular, in the practical problems as signal processing, network resource allocation, and image recovery; see, for instance, [24,31].…”

Section: Introductionmentioning

confidence: 99%

Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups

Latif¹,

Eslamian²

2017

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

In this paper, we present a new algorithm for the split equality problem for finding a common element of solution of equilibrium problem, solution of variational inequality problem for monotone and Lipschitz continuous operators, and common fixed point of nonexpansive semigroups. We establish strong convergence of the algorithm in an infinite dimensional Hilbert spaces. Our results improve and generalize some recent results in the literature.

show abstract

A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping

Cited by 97 publications

References 25 publications

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Fixed point optimization algorithm and its application to power control in CDMA data networks

Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups

Contact Info

Product

Resources

About