Quadratic Posylognomials: An Extension of Posynomial Geometric Programming

Abstract

The theory of geometric programming is extended to include a new function with logarithmic exponents. The function, defined as a Quadratic Posylognomial (QPL), is a series of nonlinear product terms with positive coefficients and positive variables. A QPL may be created by adding a linear function of the logarithms of the variables to the constant exponents of a posynomial. The logarithm of each nonlinear term is a quadratic form in the logarithms of the primal variables, whereas the logarithm of a general term of a posynomial is linear in the logarithms of the variables. The primal-dual relationships and necessary conditions are developed, and special cases where sufficient conditions exist are derived. The theory is the basis for solution of a machining economics problem using a more accurate QPL tool life equation and has the potential for solution of other engineering problems where a posynomial formulation is inadequate. The extended theory may also prove useful for condensing posynomial geometric programming problems by reducing the “degree of difficulty,” a measure of the number of decision variables in the dual problem.