Finite convergence into a convex polytope via facet reflections

summary:The problem of utilizing facet reflections to bring a point outside of a convex polytope to inside has not been studied explicitly in the literature. Here we introduce two algorithms that complete the task in finite iterations. The first algorithm generates multiple solutions on the plane, and can be readily utilized in creating games on a plane or as a level generation method for video games. The second algorithm is a new efficient way to bring infeasible starting points of an optimization problem to inside a feasible region defined by constraints. Using simulations, we demonstrate many desirable properties of the algorithm. Specifically, more edges do not lead to more iterations in $\mathbb {R}^2$, the algorithm is extremely efficient in high dimensions, and it can be employed to discretize the feasibility region using a grid of points outside the region