Bernstein Least Squares Modeling of Convergent Affine Bounds for Optimizing Control Functions

In this paper, we investigate global optimization problems that involve lower bounds for high dimensional polynomial and rational polynomial functions over a simplex. Affine lower bounds (ALB) in a rigorous style should be ensured to stay tightly under the original monomial function. We use the method of least squares approximation of Bernstein control points for constructing a convergent ALB for both polynomials and rational control functions. The method based upon the convex optimization property of Bernstein control points and the number of Bernstein coefficients. Subsequently, we show that these bounds obtain linear and quadratic rate of convergence to the original functions. Finally, we compare the methods and approximate all error bounds by finite values of maximum degrees.