An infeasible homotopy method for solving fixed point problems on a general unbounded set

Fan, Xiaona; Gao, Furong; Xu, Tingting

doi:10.1007/s12190-014-0775-4

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…By using this new homotopy continuation method, we can not only remove the assumption that the matrix ∇ (∑ =1 ( , )) is of full row rank in [18] but also can make the initial point (0) only satisfy ( ) < 0, without satisfying the equation ℎ( ) = 0. Moreover, we mainly extend the results in [19,20] to more general nonconvex sets and provide a practical perturbation for the inequality constraints compared with the results in [20]. Under commonly used conditions in the literature, we obtain the global convergence of the parameter perturbation continuation method.…”

Section: Introductionmentioning

confidence: 91%

A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

Shang

Zhu

2017

Mathematical Problems in Engineering

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In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.

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Section: Introductionmentioning

confidence: 91%

A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

Shang

Zhu

2017

Mathematical Problems in Engineering

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“…In 2015, Su and Qian [10] constructed a modified combined homotopy method by a perturbation on the inequality constraints of the general non-convex unbounded sets. In 2015, Fan et al [5] proposed an infeasible interior point homotopy method for enlarging the choice scope of initial points by a perturbation on equality constraints on the general unbounded sets, but some conditions were lost in the proof of the global convergence. In 2017, Zhu et al [22] proposed a constraint set swelling homotopy method for computing the fixed point of self-mapping on the general non-convex set.…”

Section: Introductionmentioning

confidence: 99%

A modified infeasible homotopy algorithm for computing fixed point in general non-convex set

Zhu¹,

Wu²,

Xing³

2017

J. Nonlinear Sci. Appl.

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In this paper, to find a fixed point of self-mapping in the general non-convex set with both equality constraints and inequality constraints, a modified infeasible homotopy for perturbing only inequality constraints is constructed and the global convergence of the smooth homotopy pathways is proved under some much weaker conditions. The advantage of the modified homotopy is that the initial point needs to be only in the shifted set with only inequality constraints, not necessarily, a feasible point in the original set, and hence it is more convenient to be implemented than the existing methods. The feasibility and effectiveness of the modified homotopy method is shown by some numerical tests.

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“…In 2015, Su and Qian [10] proposed a modified combined homotopy method for computing the fixed point of a self-mapping by a perturbation on the inequality constraints on the general non-convex unbounded sets. In 2015, for solving the fixed point problems of self-mapping, Fan et al [4] proposed an infeasible interior point homotopy method for enlarging the choice scope of initial points by a perturbation on equality constraints on the general unbounded sets. In 2016, Zhu and Yang [19] proposed a more conveniently constraint shifting homotopy method for computing the fixed point of self-mapping on the non-convex set with only inequality constraints.…”

Section: Introductionmentioning

confidence: 99%

A homotopy algorithm for computing the fixed point of self-mapping with inequality and equality constraints

Zhu¹,

Yang²,

Xing³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, to compute the fixed point of self-mapping on general non-convex sets, a modified constraint shifting homotopy algorithm for perturbing simultaneously both equality constraints and inequality constraints is proposed and the global convergence of the smooth homotopy pathways is proven under some mild conditions. The advantage of the newly constructed homotopy is that the initial point needs to be only in the shifted feasible set, not necessarily, an interior point in the original feasible set, and hence it is more convenient to be implemented than the existing results. Some numerical examples are also given to show its feasibility and effectiveness.

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An infeasible homotopy method for solving fixed point problems on a general unbounded set

Cited by 3 publications

References 8 publications

A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

A modified infeasible homotopy algorithm for computing fixed point in general non-convex set

A homotopy algorithm for computing the fixed point of self-mapping with inequality and equality constraints

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