In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise li...In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.展开更多
文摘In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.
