A boundary perturbation interior point homotopy method for solving fixed point problems

Abstract

In this paper, a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general Brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets. Compared with the previous results, by using the newly proposed method, initial points can be chosen in the whole space of R n , which may improve greatly the computational efficiency of reduced predictor-corrector algorithms resulted from that method. Some numerical examples are given to illustrate the results of this paper.

Solving Fixed-Point Problems with Inequality and Equality Constraints via a Non-Interior Point Homotopy Path-Following Method

“…In [20], we applied appropriate perturbations to the constraint functions and developed a new homotopy method to expand the scope of initial point selection, but involving the inequality constraint cases only. In [21], using similar perturbations to the inequality constraints in [20], we mainly extended the results in [18] to unbounded cases by providing a set of unbounded conditions.…”

“…In [20], we applied appropriate perturbations to the constraint functions and developed a new homotopy method to expand the scope of initial point selection, but involving the inequality constraint cases only. In [21], using similar perturbations to the inequality constraints in [20], we mainly extended the results in [18] to unbounded cases by providing a set of unbounded conditions. It should be pointed out that the results in [18,19] excluded the initial point selection; in addition, the researchers excluded the equality constraint cases, although the results in [20,21] expanded the scope of initial point selection.…”

“…In [21], using similar perturbations to the inequality constraints in [20], we mainly extended the results in [18] to unbounded cases by providing a set of unbounded conditions. It should be pointed out that the results in [18,19] excluded the initial point selection; in addition, the researchers excluded the equality constraint cases, although the results in [20,21] expanded the scope of initial point selection. It is difficult to 2 Mathematical Problems in Engineering construct appropriate perturbations and to guarantee the regularity of the homotopy matrix for the existence of the equality constraints.…”

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

“…Since the interior point combined homotopy method requires that the initial point must be chosen in the original feasible set, to enlarge the chosen scope of initial points, in 2011, Su et al [11] presented a boundary perturbation homotopy method on non-convex bounded sets with only inequality constraints. 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.…”

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

“…In 2013, to relax the bounded condition and weaken the normal cone condition, Zhu et al [21] constructed a modified combined homotopy for computing the fixed point of a self-mapping on the general unbounded non-convex sets with both equality constraints and inequality constraints and the existence and global convergence of the smooth homotopy pathway was proven under much weaker pseudo cone condition. Since the interior point combined homotopy requires that the initial point must be in the original feasible set, to enlarge the chosen scope of initial points, in 2011, Su et al [11] presented a boundary perturbation interior point homotopy method for solving the fixed point problems on the non-convex bounded sets with only inequality constraints. 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.…”