New relaxations for composite functions

2019-08-13T19:53:37Z (GMT) by Taotao He
Mixed-integer nonlinear programs are typically solved using branch-and-bound algorithms. A key determinant of the success of such methods is their ability to construct tight and tractable relaxations. The predominant relaxation strategy used by most state-of-the-art solvers is the factorable programming technique. This technique recursively traverses the expression tree for each nonlinear function and relaxes each operator over a bounding box that covers the ranges for all the operands. While it is versatile, and allows finer control over the number of introduced variables, the factorable programming technique often leads to weak relaxations because it ignores operand structure while constructing the relaxation for the operator.
In this thesis, we introduce new relaxations, called composite relaxations, for composite functions by convexifying the outer-function over a polytope, which models an ordering structure of outer-approximators of inner functions. We devise a fast combinatorial algorithm to separate the hypograph of concave-extendable supermodular outer-functions over the polytope, although the separation problem is NP-Hard in general. As a consequence, we obtain large classes of inequalities that tighten prevalent factorable programming relaxations. The limiting composite relaxation obtained with infinitely many outer-approximators for each inner-function is shown to be related to the solution of an optimal transport problem. Moreover, composite relaxations can be seamlessly embedded into a discretization scheme to relax nonlinear programs with mixed-integer linear programs. Combined with linearization, composite relaxations provide a framework for deriving cutting planes used in relaxation hierarchies and more.