Continuous Method For Solving Fixed Point Problem On Square Grid By Cubic Spline Boundary

Fixed point problem can be used as a model of solving the optimization problem, such as facility location or gravity center problem on Euclidean space. In the actual application, the plane area under study is simplified by square grid. Thus, the constraint set consisting of feasible points is a connected grid of such squares. The outer edges of all squares of an edge connected square grid are enclosed into a polygon most of which are nonempty compact but non-convex set. Weaker conditions that constraint set can be extended to the non-convex bounded set for the fixed point problem are given. The upper and lower bounds function of the inequality are treated by the cubic spline interpolation function. The numerical method combined cubic spline boundary interpolation and tracing the homotopy pathway can be used to solve the fixed point problem on square grid. The numerical example shows that this method is much effective.