Solving global optimization problems using reformulations and signomial transformations

In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C2-functions the α reformulation (αR) technique as well as more generally the method of difference of convex functions can be applied. In the αR, the αBB convex underestimator can be used. The framework is utilized in the α signomial global optimization (αSGO) algorithm to find the ϵ-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy.