Global Optimization of Bilinear Programs by Elementary Functions Substitutions

In this paper, we present an approach to global optimization of bilinear programs based on a transformation of bilinear terms to elementary functions of single variables. The substitutions relocate bilinear terms to additional equality constraints. The complicating variables in these constraints are separated by logarithm operations. Finally, the log-linear constraints are relaxed by piecewise linear approximations and/or linearizations. The transformation produces a lower-bounding mixed-integer linear program, which is used in conjunction with the original bilinear program in a global optimization algorithm. The approach was tested by pooling problems from the literature. The global optimum was obtained in all the test problems; however, more elaborate algorithmic schemes would be needed to make the proposed approach computationally efficient.