Many problems in science and engineering can be reduced to the problem of
finding optimum bounds for the range of a multivariable polynomial on a
specified domain. Local optimization is an important tool for solving polynomial
problems, but there is no guarantee of global optimality.
For polynomial optimization problems, an alternate approach is based on the
Bernstein form of the polynomial. If a polynomial is written in the Bernstein
basis over a box, then the range of the polynomial is bounded by the values of
the minimum and maximum Bernstein coefficients. Global optimization based on the
Bernstein form does not require the iterative evaluation of the objective
function. Moreover, the coefficients of the Bernstein form are needed to be
computed only once, i.e., only on the initial domain box. The Bernstein
coefficients for the subdivided domain boxes can then be obtained from the
initial box itself. Capturing these beautiful properties of the Bernstein
polynomials, global optimum for the polynomial on the given domain can be
obtained.
The toolbox is developed based on the above ideas.
