Global Optimization for Polynomial Programming Problems Usingm-homogeneous Polynomial Homotopies

A polynomial programming problem is a nonlinear programming problem where the objective function, inequality constraints, and equality constraints all consist of polynomial functions. The necessary optimality conditions for such a problem can be formulated as a polynomial system of equations, among whose zeros the global optimum must lie. This note applies the theory of m-homogeneous polynomials in Cartesian product projective spaces and recent homotopy algorithms to significantly reduce the work of a naive homotopy approach to the polynomial system formation of the mecessary optimality conditions. The m-homogeneous approach, providing the global optimum, is practical for small problems. For example, the geometric modeling problem of finding the distance between two polynomial surfaces is a polynomial programming problem. Also discussed is a prototype structural design problem.